# Bar-Ilan Algebra seminar

Time: Wednesday 10:30 (unless noted otherwise)
Place: Math building, Top floor seminar room (327)
(How to get there).

## Academic year   2014-15

### Organizers:   L.H. Rowen, M. Schein and U. Vishne

 DATE: June 24, 2015 Speaker: Prof. Marianne Akian (INRIA Saclay--Ile-de-France and CMAP, Ecole Polytechnique) Title: Majorization inequalities for valuations of eigenvalues using tropical algebra Abstract. We consider a matrix with entries over the field of Puiseux series, equipped with its non-archimedean valuation (the leading exponent). We establish majorization inequalities relating the sequence of the valuations of the eigenvalues of a matrix with the tropical eigenvalues of its valuation matrix (the latter is obtained by taking the valuation entrywise). We also show that, generically in the leading coefficients of the Puiseux series, the precise asymptotics of eigenvalues, eigenvectors and condition numbers can be determined. For this, we apply diagonal scalings constructed from the dual variables of a parametric optimal assignment constructed from the valuation matrix. Next, we establish an archimedean analogue of the above inequalities, which applies to matrix polynomials with coefficients in the field of complex numbers, equipped with the modulus as its valuation. In particular, we obtain log-majorization inequalities for the eigenvalues which involve combinatorial constants depending on the pattern of the matrices. (joint work with Ravindra Bapat, St phane Gaubert, Andrea Marchesini, and Meisam Sharify.) DATE: June 17, 2015 Speaker: Dr. Adi Niv (INRIA Saclay Ile-de-France and Ecole Polytechnique) Title: Tropical totally positive matrices Abstract. We start by presenting Gaubert's symmetrized tropical semiring, which defines a tropical additive-inverse and uses it to resolve tropical singularity. Then, we recall properties of totally positive matrices over rings, define tropical total positivity and total non-negativity of matrices using the symmetrized structure, and state combinatorial and algebraic properties of these matrices. By studying the tropical semiring via valuation on the field of Puiseux series, we relate the tropical properties to the classical ones. (joint work with Stephane Gaubert.) DATE: June 3, 2015 Speaker: Prof. Hau-Wen Huang (Hebrew University of Jerusalem) Title: The Askey-Wilson algebra Abstract. Motivated by the Racah coefficients, the Askey-Wilson algebra was introduced by the theoretical physicist Zhedanov. The algebra is named after Richard Askey and James Wilson because this algebra also presents the hidden symmetry between the three-term recurrence relation and $q$-difference equation of the Askey-Wilson polynomials. In this talk, I will present the progression on the finite-dimensional irreducible modules for Askey-Wilson algebra. DATE: May 27, 2015 Speaker: Prof. Sefi Ladkani (Ben-Gurion University) Title: From groups to clusters Abstract. I will present a new combinatorial construction of finite-dimensional algebras with some interesting representation-theoretic properties: they are of tame representation type, symmetric and have periodic modules. The quivers we consider are dual to ribbon graphs and they naturally arise from triangulations of oriented surfaces with marked points. The class of algebras that we get contains in particular the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other hand, it contains also the Jacobian algebras of the quivers with potentials associated by Fomin-Shapiro-Thurston and Labardini to triangulations of closed surfaces with punctures. Hence our construction may serve as a bridge between modular representation theory of finite groups and cluster algebras. All notions will be explained during the talk. DATE: May 20, 2015 Speaker: Prof. Nicola Sambonet (Technion) Title: On the exponent of the Schur multiplier Abstract. The Schur multiplier is a very interesting invariant, being the archetype of group cohomology. An explicit description of the multiplier is often too difficult a task. Therefore it is of interest to obtain information about its arithmetical features, such as the order, the rank, and the exponent. I will present the problem of bounding the exponent of the multiplier of a finite group, introducing the new concept of unitary cover. DATE: May 13, 2015 Speaker: Prof. David Corwin (MIT) Title: Elliptic curves with maximal Galois action on torsion points Abstract. Let E be an elliptic curve over a number field K with algebraic closure K'. For an integer n, the set of n-torsion points in E(K') forms a group isomorphic to Z/n X Z/n, which carries an action of G_K=Gal(K'/K). A result of Serre shows that if K=Q, then the associated homomorphism from G_K to GL_2(Z/n) cannot be surjective for all n. The result is false, however, over other number fields. The 2010 PhD thesis of Greicius found the first counterexample, over a very special non-Galois cubic extension of Q. In this talk I will describe the above background and then describe more recent results of the speaker and others allowing one to find such elliptic curves over a more general class of number fields. As time allows, I may describe other results from our paper about finding elliptic curves with maximal (but not surjective) Galois action given certain constraints, or a forthcoming paper doing similar work for abelian surfaces. DATE: May 6, 2015 Speaker: Beeri Greenfeld (Bar-Ilan University) Title: Non-commutative graded algebras with restricted growth Abstract. Graded algebras play a major role in many topics, including algebraic geometry, topology, and homological algebra, besides classical ring theory. These are algebras which admit a decomposition into a sum of homogeneous components which 'behave well' with respect to multiplication. In this talk we present several structure-theoretic results concerning affine (that is, finitely generated) Z-graded algebras which grow 'not too fast'. In particular, we bound the classical Krull dimension both for algebras with quadratic growth and for domains with cubic growth, which live in the heart of Artin's proposed classification of non-commutative projective surfaces. We also prove a dichotomy result between primitive and PI-algebras, relating a graded version of a question of Small. From a radical-theoretic point of view, we prove that unless a graded affine algebra has infinitely many zero homogeneous components, its Jacobson radical vanishes. Under a suitable growth restriction, we prove a stability result for graded Brown-McCoy radicals of Koethe conjecture type: they remain Brown-McCoy even after being tensored with some arbitrary algebra.Finally, we pose several open questions which could be seen as graded versions of the Kurosh and Koethe conjectures. (joint work with A. Leroy, A. Smoktunowicz and M. Ziembowski.) DATE: May 6, 2015 Speaker: Prof. Mark Shusterman (Tel Aviv University) Title: Free profinite subgroups and Galois representations Abstract. The talk is going to be about the work carried out as part of my MSc thesis. Motivated by recent arithmetic results, we will consider new and improved results on the freeness of subgroups of free profinite groups:1.The Intermediate Subgroup Theorem - A subgroup (of infinite index) in a nonabelian finitely generated free profinite group, is contained in a free profinite group of infinite rank. 2. The Verbal Subgroup Theorem - A subgroup containing the normal closure of a (finite) word in the elements of a basis for a free profinite group, is free profinite. These results shed light on several theorems in Field Arithmetic and may be combined with the twisted wreath product approach of Haran, an observation on the action of compact groups, and a rank counting argument to prove a generalization of a result of Bary-Soroker, Fehm, and Wiese on the profinite freeness of subgroups arising from Galois representations. DATE: Apr 29, 2015 Speaker: Dr. Rony Bitan (Bar-Ilan University) Title: The Hasse principle for bilinear symmetric forms over the ring of integers of a global function field Abstract. Let C be a smooth projective curve defined over a finite field F_q (q is odd), and let K=F_q(C) be its (global) function field. This field is considered as the geometric analogue of a number field.Removing one closed point from C results in an affine curve C^af. The ring of regular functions over C^af is an integral domain, over which we consider a non-degenerate bilinear and symmetric form f of any rank n.We express the number c(f) of isomorphism classes in the genus of f in cohomological terms and use it to present a sufficient and necessary condition depending only on C^af, under which f admits the Hasse local-global principle. We say that f admits the Hasse local-global principle if c(f)=1, namely, |Pic(C^\af)| is odd for any n other than 2 and equal to 1 for n=2. This result emphasizes the difference between Galois cohomology and etale cohomology. Examples are provided. DATE: Apr 15, 2015 Speaker: Prof. Uriya First (University of British Columbia) Title: Rationally isomorphic quadratic objects Abstract. Let R be a discrete valuation ring with fraction field F. Two algebraic objects (say, quadratic forms) defined over R are said to be rationally isomorphic if they become isomorphic after extending scalars to F. In the case of unimodular quadratic forms, it is a classical result that rational isomorphism is equivalent to isomorphism. This has been recently extended to "almost umimodular" forms by Auel, Parimala and Suresh. I will present further generalizations to hermitian forms over (certain) involutary R-algebras and quadratic spaces equipped with a group action ("G-forms"). The results can be regarded as versions of the Grothendieck-Serre conjecture for certain non-reductive groups. (joint work with Eva Bayer-Fluckiger) DATE: Mar 25, 2015 Speaker: Prof. Gabriella D'Este (University of Milan) Title: Some remarks on tilting theory Abstract. In the first part of my talk I will describe with few words and many pictures some more or less combinatorial results on tilting modules, bimodules and complexes, almost always obtained by means of elementary tools of two types:- Linear Algebra arguments (that is, comparison of the dimensions of the underlying vector spaces of certain Hom and Ext groups);- Representation Theory arguments (that is, analysis of the Auslander - Reiten quivers of suitable finite dimensional algebras, almost always admitting only finitely many indecomposable modules up to isomorphism). In the second part of my talk I will describe other results (suggested by quivers) concerning reflexive modules (not necessarity belonging to the tilting and cotilting worlds) and multiplicities of simple modules in the socle of certain injective cogenerators. Almost all the results and examples are illustrated in two preprints available at http://arxiv.org/abs/1401.2085 and http://arxiv.org/abs/1411.4418 . DATE: Mar 25, 2015 Speaker: Prof. Darrell Haile (Indiana University) Title: On the Teichmuller map and a class of nonassociative algebras Abstract. Let K/F be a finite Galois extension with Galois group G. The Teichmuller map is a function that associates to every central simple K-algebra B normal over F an element of H^3(G, K*). The value of the function is trivial precisely when the class of B is restricted from F. The classical definition of this map involves the use of a crossed-product algebra over B. The associativity of this algebra is also equivalent to the class of B being restricted from F. The aim of this lecture is to elucidate the nature of the nonassociative algebras that arise when B is normal but not restricted. It turns out that the resulting theory is remarkably similar to the theory of associative algebras arising from the noninvertible cohomology of a Galois extension L/F such that L contains K, and I want to explain that relationship. (joint work with Yuval Ginosar) DATE: Mar 18, 2015 Speaker: Moshe Newman Title: Some facts about the Gieseking group Abstract. The Gieseking group is a one-relator group defined by the equation aab=bba. It is also the fundamental group of a certain 3-dimensional manifold. As a non-topologist trying to make use of the latter fact, I learned some things the hard way, which I will share with the audience. DATE: Mar 3, 2015 Speaker: Prof. Mikhail Borovoi (Tel Aviv University) Title: Real Galois cohomology of simply connected groups Abstract. By the celebrated Hasse principle of Kneser, Harder and Chernousov, calculating the Galois cohomology H^1(K,G) of a simply connected simple K-group over a number field K reduces to calculating H^1(R,G) over the field of real numbers R. For some cases, in particular, for the split simply connected R-group G of type E_7, the first calculations of H^1(R,G) appeared only in 2013 and 2014 in preprints of Jeffry Adams, of Brian Conrad, and of the speaker and Zachi Evenor. All these calculations used the speaker's note of 1988.In the talk I will explain the method of Kac diagrams of calculating H^1(R,G) for a simply connected simple R-group G by the examples of groups of type E_7. The talk is based on a work in progress with Dmitry A. Timashev. No preliminary knowledge of Galois cohomology or of groups of type E_7 is assumed. DATE: Mar 3, 2015 Speaker: Prof. Viktor Batyrev (Universitat Tubingen) Title: Stringy Chern classes of toric varieties and their applications Abstract. Stringy Chern classes of singular projective algebraic varieties can be defined by some explicit formulas using a resolution of singularities. It is important that the output of these formulas does not depend on the choice of a resolution. The proof of this independence is based on nonarchimedean motivic integration. The purpose of the talk is to explain a combinatorial computation of stringy Chern classes for singular toric varieties. As an application one obtains combinatorial formulas for the intersection numbers of stringy Chern classes with toric Cartier divisors and some interesting combinatorial identities for convex lattice polytopes. DATE: Jan 28, 2015 Speaker: Prof. Alessandra Cherubini (Politecnico di Milano) Title: Decidability vs. undecidability for the word problem in amalgams of inverse semigroups Abstract. In 2012 J. Meakin posed the following question: under what conditions is the word problem for amalgamated free products of inverse semigroups decidable? Some positive results were interrupted by a result of Radaro and Silva showing that the problem is undecidable even under some nice conditions. Revisiting the proofs of decidability, we discuss whether positive results can be achieved for wider classes of inverse semigroups and show how small the distance is between decidability and undecidability. DATE: Jan 28, 2015 Speaker: Dr. Mark Berman (ORT Braude College) Title: Pro-isomorphic zeta functions of groups and solutions to congruence equations Abstract. Zeta functions of groups were introduced by Grunewald, Segal and Smith in 1988. They have proved to be a powerful tool for studying the subgroup structure and growth of certain groups, especially finitely generated nilpotent groups. Three types of zeta function have received special attention: those enumerating all subgroups, normal subgroups or "pro-isomorphic" subgroups: subgroups isomorphic to the original group after taking profinite completions. Of particular interest is a striking symmetry observed in many explicit computations, of a functional equation for local factors of the zeta functions. Inspired by wide-reaching results, due to Voll, for the first two types of zeta function, I will talk about recent progress on the functional equation for local pro-isomorphic zeta functions. Thanks to work of Igusa and of du Sautoy and Lubotzky, these local zeta functions can be analysed by translating them into integrals over certain points of an automorphism group of a Lie algebra associated to the nilpotent group and then applying a p-adic Bruhat decomposition due to Iwahori and Matsumoto. While this technique proves a functional equation for certain classes of such integrals, it is difficult to relate these results back to the nilpotent groups they arise from. In particular, it is not known whether the local pro-isomorphic zeta functions of all finitely generated groups of nilpotency class 2 enjoy local functional equations. I will discuss recent explicit calculations of pro-isomorphic zeta functions for specific nilpotent groups. Interesting new features include an example of a group whose local zeta functions do not satisfy functional equations, a family of groups whose global zeta functions have non-integer abscissae of convergence of arbitrary denominator, and an example whose calculation requires solving congruence equations modulo p^n for a prime p. The latter sheds new light on the types of automorphism groups that can be expected to arise. This is joint work with Benjamin Klopsch and Uri Onn. DATE: Jan 21, 2015 Speaker: Efrat Bank (Tel Aviv University) Title: Prime polynomial values of linear functions in short intervals Abstract. In this talk I will present a function field analogue of a conjecture in number theory. This conjecture is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. I prove an asymptotic formula for the number of simultaneous prime values of $n$ linear functions, in the limit of a large finite field. A key role is played by the computation of some Galois groups. DATE: Jan 14, 2015 Speaker: Prof. Arkady Tsurkov (Federal University of Rio Grande do Norte) Title: Automorphic equivalence in varieties of representations of Lie algebras DATE: Dec 31, 2014 Speaker: Dr. Klim Efremenko (University of California, Berkeley) Title: Arithmetic circuits and algebraic geometry Abstract. The goal of this talk is to show that natural questions in complexity theory raise very natural questions in algebraic geometry. More precisely, we will show how to adapt an approach introduced by Landsberg and Ottaviani, called Young Flattening, to questions about arithmetic circuits. We will show that this approach generalizes the method of shifted partial derivatives introduced by Kayal to show lower bounds for shallow circuits. We will also show how one can calculate shifted partial derivatives of the permanent using methods from homological algebra, namely by calculating a minimal free resolution of an ideal generated by partial derivatives. I will not assume any previous knowledge about arithmetic circuits. (joint work with J.M. Landsberg, H Schenck and J Weyman) DATE: Dec 24, 2014 Speaker: Dr. Adam Chapman (Michigan State University) Title: Subfields of quaternion algebras in characteristic 2 Abstract. We discuss the situation where two quaternion algebras over a field of characteristic 2 share the same genus, i.e. have the same set of isomorphism classes of quadratic field extension of the center. We provide examples of pairs of nonisomorphic quaternion algebras that satisfy this property. We also show that over global fields and the fields of Laurent series over perfect fields the quaternion algebras are uniquely determined by their subfields. DATE: Dec 24, 2014 Speaker: Dr. Shaul Zemel (Technische Universitt Darmstadt) Title: Lattices over valuation rings of arbitrary rank Abstract. We show how the simple property of 2-Henselianity suffices to reduce the classification of lattices over a general valuation ring in which 2 is invertible (with no restriction on the value group) to classifying quadratic spaces over the residue field. The case where 2 is not invertible is much more difficult. In this case we present the generalized Arf invariant of a unimodular rank 2 lattice, and show how in case the lattice contains a primitive vector with norm divisible by 2, a refinement of this invariant and a certain class suffice for classifying these lattices. DATE: Dec 10, 2014 Speaker: Prof. Adina Cohen (Hebrew University of Jerusalem) Title: Morphisms of Berkovich analytic curves and the different function Abstract. In this talk we will study the topological ramification locus of a generically etale morphism f : Y --> X between quasi-smooth Berkovich curves. We define a different function \delta f : Y --> [0,1] which measures the wildness of the morphism. It turns out to be a piecewise monomial function on the curve, satisfying a balancing condition at type 2 points analogous to the classical Riemann-Hurwitz formula. We also explain how \delta can be used to explicitly construct the simultaneous skeletons of X and Y. The talk will begin with a quick background on Berkovich curves. All terms will be defined. (joint work with M. Temkin and Dr. D. Trushin) DATE: Dec 3, 2014 Speaker: Prof. Grisha Soifer (Bar-Ilan University) Title: Free subgroups of linear groups: geometry, algebra, and dynamics Abstract. In a celebrated paper, J. Tits proved the following fundamental dichotomy for a finitely generated linear group: Let G be a finitely generated linear group over an arbitrary field. Then either G is virtually solvable, or G contains a free non-abelian subgroup. Let G be a non-virtually solvable subgroup of a linear group. We will discuss the following problem(s): is it possible to find a free subgroup of G that fulfills additional (topological, algebraic, and dynamical) conditions? DATE: Nov 26, 2014 Speaker: Prof. Ido Efrat (Ben Gurion University) Title: Massey products in Galois theory Abstract. We will report on several recent works on Massey products in Galois cohomology and explain how they reveal new information on the structure of absolute Galois groups of fields. DATE: Nov 19, 2014 Speaker: Dr. Eli Matzri (Ben Gurion University) Title: Diophantine and cohomological dimensions Abstract. We give explicit linear bounds on the p-cohomological dimension of a field in terms of its Diophantine dimension. In particular, we show that for a field of Diophantine dimension at most 4, the 3-cohomological dimension is less than or equal to the Diophantine dimension. DATE: Nov 12, 2014 Speaker: Prof. Arie Levit (Weizmann Institute of Science) Title: Counting commensurability classes of hyperbolic manifolds Abstract. Subgroup growth usually means the asymptotic behavior of the number of subgroups of index n of a given finitely generated group as a function of n. We generalize this to discrete (torsion-free) subgroups of the Lie group G=SO+(n,1) for which the quotient admits finite volume, as a function of the co-volume. Conjugacy classes of such discrete subgroups correspond geometrically to n-dimensional hyperbolic manifolds of finite volume.By a classical result of Wang, for n >=4 there are only finitely many such conjugacy classes up to any given finite volume V. More recently, Burger, Gelander, Lubotzky and Mozes showed that this number grows like V^V.In this talk we focus on counting commensurability classes. Two subgroups are commensurable if they admit a common finite index subgroup (in our context, up to taking conjugates). We show that surprisingly, for n >= 4 this number grows like V^V as well. Since the number of arithmetic commensurability classes grows ~polynomially (Belolipetsky), our result implies that non-arithmetic subgroups account for most" commensurability classes.Our proof uses a mixture of arithmetic, hyperbolic geometry and some combinatorics. In particular, recall that a quadratic form of signature (n,1) over a totally real number field, whose conjugates are positive definite, defines an arithmetic discrete subgroup of finite covolume in G. As in the classical construction of Gromov--Piatetski-Shapiro, several non-similar quadratic forms can be combined to construct amalgamated non-arithmetic subgroups. (joint work with Tsachik Gelander) DATE: Nov 5, 2014 Speaker: Dr. Shai Shechter (Ben Gurion University) Title: Representation zeta functions of norm one subgroups of a local division algebra Abstract. Let D be a central division algebra of dimension $\ell^2$ over a non-archemedian local field K. Let O denote the maximal compact subring and P the maximal ideal of D. In his 1981 paper, H. Koch investigated the notion of conjugation by invertible elements of O modulo powers of P. We focus our attention on the group SL1(O) of elements of reduced-norm 1 in D and its first congruence subgroup SL^1_1(O) = SL1(O)(1+P). Applying some of the methods described in Koch, we compute the orbits of the action of SL_1^1(O) on quotients of the Lie-lattice of elements of reduced trace 0 in P modulo powers of P, in the case where  is a prime number distinct from the residual characteristic of K.Under some additional assumption on the field K, we will present a connection between the character theory of SL_1^1(O) and the aforementioned orbits, via the Kirrilov orbit method for a specific class p-adic groups. In particular, by applying a p-adic formalism developed by Avni, Klopsch, Onn and Voll for such groups, we will obtain the representation zeta function of SL_1^1(O). Time permitting, we will discuss the connection to the representation zeta function of SL_1(O)

## Academic year   2013-14

### Organizers:   L.H. Rowen, M. Schein and U. Vishne

 DATE: Oct 29, 2014 Speaker: Dr. Oren Ben-Bassat (Oxford University and Haifa University) Title: Banach Algebraic Geometry Abstract. I will present a 'categorical' way of doing analytic geometry in which analytic geometry is seen as a precise analogue of algebraic geometry. Our approach works for both complex analytic geometry and p-adic analytic geometry in a uniform way. I will focus on the idea of an 'open set' as used in these various areas of math and how it is characterised categorically. In order to do this, we need to study algebras and their modules in the category of Banach spaces. The categorical characterization that we need uses homological algebra in these 'quasi-abelian' categories which is work of Schneiders and Prosmans. In fact, we work with the larger category of Ind-Banach spaces for reasons I will explain. This gives us a way to establish foundations of analytic geometry and to compare with the standard notions such as the theory of affinoid algebras, Grosse-Klonne's theory of dagger algebras (over-convergent functions) and others. If time remains I will explain how this extends to a formulation of derived analytic geometry following the relative algebraic geometry approach of Toen, Vaquie and Vezzosi. (joint work with Federico Bambozzi (Regensburg) and Kobi Kremnizer (Oxford)) DATE: May 14, 2014 Speaker: Prof. Christopher Voll (Bielefeld University) Title: Representation zeta functions of finitely generated nilpotent groups and generating functions for hyperoctahedral groups Abstract. The representation zeta function of a finitely generated nilpotent group is the Dirichlet generating series enumerating the group's irreducible finite-dimensional complex characters up to twists by one-dimensional characters. A simple example is the Heisenberg group over the integers: here the relevant arithmetic function is just Euler's totient function. In general, these zeta functions have natural Euler product decompositions, indexed by the places of a number field. The Euler factors are rational functions with interesting arithmetic properties, such as palindromic symmetries.In my talk -- which reports on joint work with Alexander Stasinski -- I will (A) explain some general facts about representation zeta functions of finitely generated nilpotent groups, and (B) discuss in detail some specific classes of examples, including groups generalizing the free class-2-nilpotent groups. One reason for interest in these classes of groups is the fact that their representation growth exhibits intriguing connections with some statistics on the hyperoctahedral groups (Weyl groups of type B).My talk will assume no specialist knowledge on zeta functions, representation theory, algebraic combinatorics, or in fact anything advanced DATE: April 30, 2014 Speaker: Prof. Ido Efrat (Ben-Gurion University) Title: Filtrations of absolute Galois groups Abstract. A profinite group is equipped with various standard filtrations by closed normal subgroups, such as the lower central series, the lower p-central series, and the p-Zassenhaus filtration. In the case of an absolute Galois group of a field, these filtrations are related to the arithmetic structure of the field, as well as to its Galois cohomology. We will describe some recent results on these connections, in particular with the Massey product in Galois cohomology. DATE: April 2, 2014 Speaker: Dr. Lior Bary-Soroker (Tel Aviv University) Title: Morse polynomials and Galois theory Abstract. Many problems in algebra and number theory reduce to the problem of calculating Galois groups. In this talk, I will focus on the proof of the following theorem:Thm: Let x |--> f(x) be a polynomial map from the Riemann sphere to itself of degree n=deg f.Assume that f(x) is Morse (in the sense that the critical points are non-degenerate and the critical values are distinct). Then the Galois group is the full symmetric group. The proof involves some geometry and some finite group theory. DATE: Mar 26, 2014 Speaker: Gili Golan Title: Tarski numbers of groups. Abstract. The Tarski number of a group G is the minimal number of pieces in a paradoxical decomposition of it. We investigate how Tarski numbers may change under various group-theoretic operations. Using these estimates and known properties of Golod-Shafarevich groups, we show that the there are 2-generated groups with property (T) and arbitrarily large Tarski numbers. We also prove that there exist groups with Tarski number 6. These provide the rst examples of non-amenable groups without free subgroups whose Tarski number has been computed precisely. (joint work with Mikhail Ershov and Mark Sapir, http://arxiv.org/abs/1401.2202) DATE: Mar 19, 2014 Speaker: Prof. Darrell Haile (Indiana University) Title: Noninvertible cohomology and the Teichmuller cocycle Abstract. Noninvertible cohomology refers to Galois cohomology in which the values of the cocycles are allowed to be noninvertible. In this talk I will describe an application of this theory to the following problem: Given L/F, a finite separable extension of fields, and an L-central simple algebra B, classify those F-algebras A containing B that are "tightly connected to B" in a sense I will make precise. The answer uses the Teichmuller cocycle. This is a three-cocycle that is the obstruction, when L/F is Galois, to a normal L/F central simple algebra (i.e. a central simple L-algebra B with the property that every element of Gal(L/F) extends to an automorphism of B) having the property that its Brauer class in Br(L) is restricted from B(F). This is mostly work of two of my students, Holly Attenborough and Kevin Foster. DATE: Mar 12, 2014 Speaker: Prof. Alexandre Zalesski (National Academy of Sciences of Belarus) Title: Subgroups of algebraic groups with a rational element Abstract. I shall report on some results obtained in a joint work with Donna Testerman.Let H be a simple algebraic group defined over the algebraically closed field F of characteristic p >= 0. The subgroups of H containing a maximal torus are called subgroups of maximal rank. They play a substantial role in the structure theory of algebraic groups. A well known and important classical result (going back to Dynkin and Borel-De Siebenthal) states that every subgroup of maximal rank is either contained in a parabolic subgroup of H or belongs to the normalizer of a subsystem subgroup.A subsystem subgroup G is a semisimple subgroup of H whose root system is, in a natural way, a subset of the root system of H. Our aim was to generalize this result by replacing a maximal torus with a regular torus, that is, a torus whose centralizer in H is a maximal torus of H. In other words, we would like to determine the algebraic subgroups G of H whose maximal torus T is regular in H. The case where T itself is a maximal torus of H is therefore the subject of the Dynkin and Borel-De Siebenthal classification, and hence need not be considered. In such generality it is impossible to arrive at a tractable result, and we specialize the problem by assuming that G is a maximal simple subgroup of H. I will discuss the results obtained and explain the method used. Additionally, we determined all irreducible representations of a simple algebraic group all but one of whose weights are of multiplicity one. DATE: Feb 26, 2014 Speaker: Prof. Boris Kunyavskii (Bar-Ilan University) Title: Geometry and arithmetic of word equations in simple matrix groups and algebras Abstract. We will discuss various geometric and arithmetic properties of matrix equations of the formP(X_1,...,X_d)=A,where the left-hand side is either an associative noncommutative monomial in X_i's and their inverses, or an associative or Lie polynomial in X_i's, and the right-hand side is a fixed matrix. Solutions are sought in some subgroup G of GL(n,R), or some associative or Lie subalgebra of M(n,R), respectively. We will focus on the case where the group (algebra) under consideration is simple, or close to such.We will give a survey of classical and recent results and open problems concerning this equation, concentrating around the following questions (posed for geometrically and/or arithmetically interesting rings and fields R):- is it solvable for any A?- is it solvable for a typical'' A?- does it have many'' solutions?- does the set of solutions possess good'' local-global properties?- to what extent does the set of solutions depend on A? DATE: Jan 29, 2014 Speaker: Dr. Ivan Mitrofanov (Moscow State University) Title: Substitutional systems and algorithmic problems Abstract. Let A=$\{a_1,\dots,a_n\}$ be a finite alphabet. Consider a substitution $S: a_i\to v_i; i=1,\dots, n$, where $v_i$ are some words.A DOL-system is an infinite word (superword) $W$ obtained by iteration of $S$. An HDOL-system is $V$ an image of $W$ under some other substitution $a_i\to u_i; i=1,\dots, n$.The general problem is: suppose we have 2 HDOL-systems. Do they have the same set of finite subwords? This problem is open so far, but the author proved a positive solution of the periodicity problem (is $U$ periodic?) and uniformly recurrence problem http://arxiv.org/abs/1110.4780. This result was obtained independently by Fabien Durand http://arxiv.org/abs/1111.3268 using different method. see also http://arxiv.org/abs/1107.0185. DATE: Jan 14, 2014 Speaker: Dr. Anton Khoroshkin (Higher School of Economics, Moscow) Title: Macdonald polynomials and highest weight categories Abstract. The goal of the talk is to explain an approach to the problem of categorification of Macdonald polynomials based on derived categories of modules over the Lie algebra of currents. First, I recall the definition of Macdonald polynomials as the orthogonalisation of the linear monomial basis in the ring of symmetric functions with respect to a certain given pairing depending on two parameters.I will give the relationship of the latter pairing with the Grothendieck ring of the category of modules over the Lie algebra of currents. Second, I will explain the orthogonalisation procedure in derived categories and give a hint on the categorification problem. Third, I will discuss when it is possible to avoid the derived setting and get different applications for the category of modules itself, in the literature this property is known under the name "Bernstein-Gelfand-Gelfand reciprocity".The category of modules over the Lie algebra $g\otimes \mathbb{C}[x]$ with $g$-semisimple will be the main example with this property. The talk is based on math.arXiv:1312.7053 DATE: Jan 7, 2014 Speaker: Dr. Rishi Vyas (Ben-Gurion University) Title: Minimal injective resolutions of modules over noetherian rings Abstract. In many cases, minimal injective resolutions can be viewed as realizations of the geometry' of a noetherian ring. In this talk, we aim to discuss certain aspects of the shape of the minimal injective resolution of a finitely generated module over some noetherian rings. We begin by giving an overview of what is known for commutative Gorenstein rings; this material is classical. We then describe the obstructions to generalizing this theory to noncommutative rings, and finish by giving some positive results for certain families of noncommutative noetherian rings. DATE: Dec 25, 2013, 10am Speaker: Dr. Adam Chapman (Universite Catholique de Louvain) Title: Division algebras with involution over fields of cohomological dimension 2 Abstract. We prove several results on division algebras of degree a power of 2 and exponent 2 over fields of cohomological 2-dimension 2. Some of them were known for characteristic not 2 and we provide proofs for characteristic 2 as well, such as the fact that every such algebra decomposes as the tensor product of some quaternion algebras, and the fact that for each subfield that is a quadratic or biquadratic extension of the center, the algebra decomposes in such a way that each of the Artin-Schreier generators of the subfield is contained in a different quaternion algebra. We also say a few words on the chain lemma for these algebras. DATE: Dec 25, 2013, 11am Speaker: Dr. Danny Neftin (University of Michigan) Title: The absolute Galois group of Q and its Sylow subgroups Abstract. Understanding the rich structure of the absolute Galois groups of the field Q of rational numbers and of the fields Q_p of p-adic numbers is a central goal in number theory with application to many other areas. Following Serre's question, the Sylow subgroups of the absolute Galois group of Q_p were studied and completely understood by Labute. However, the structure of the p-Sylow subgroups of the absolute Galois group of Q is much more subtle and mysterious. We shall discuss the first steps towards its determination via a surprisingly simple decomposition. DATE: Dec 18, 2013 Speaker: Prof. Mikhail Kharitonov (Moscow State University) Title: Subexponential estimates in the height theorem and estimates on numbers of periodic parts of small periods Abstract. We present subexponential estimations in Shirshov's Height theorem. We show that the set of non n-divisible words over \ell letters has bounded height < $2^{96} \ell n^{12 \log_{3}n + 36 \log_3\log_3 n + 91}$. Every word over \ell letters is either n-divisible or contains d-th power as a subword, once the length is > $2^{27} \ell (nd)^{3 \log_3(nd) + 9 \log_3\log_3(nd) + 36}$. We use Latyshev's idea of Dilworth's theorem on anti-chains. DATE: Dec 12, 2013 Speaker: Dr. Michael Schein (Bar-Ilan University) Title: The structure of universal supersingular representations of p-adic general linear groups Abstract. The mod p representation theory of the groups GL(n,F), where F is a p-adic field, is essential to the emerging mod p and p-adic local Langlands correspondences. The irreducible supersingular representations are the building blocks of the theory, analogous to the supercuspidal representations in the classical theory over C. For the group GL(2, Q_p), the supersingular representations have been classified completely. In all other cases, we do not know a single explicit example of such a representation. However, they all arise as quotients of certain universal modules. We will present some results about the rich internal structure of these modules. All necessary notions will be defined in the talk. DATE: Dec 4, 2013 Speaker: Dr. Zhihua Chang (Bar-Ilan University) Title: Automorphisms and twisted forms of differential conformal superalgebras Abstract. The non-abelian cohomology theory of affine group schemes over the Laurent polynomial ring has been successfully applied to the study of twisted affine Kac-Moody algebras by A. Pianzola and his collaborators in the recent decade. V. Kac, M. Lau, and A. Pianzola further developed the theory of differential conformal superalgebras which used the same strategy to investigate conformal superalgebras. In my talk, I will briefly introduce their methods and present my joint work with A. Pianzola about the concrete classifications of twisted loop conformal superalgebras based on each of the N=1,2,3 and large N=4 conformal superalgebras. DATE: Nov 27, 2013 Speaker: Dr. Shaul Zemel (Technische Universitt Darmstadt) Title: A Gross-Kohnen-Zagier type theorem for higher-codimensional Heegner cycles Abstract. The multiplicative Borcherds singular theta lift is a well-known tool for obtaining automorphic forms with known zeros and poles on quotients of orthogonal symmetric spaces. This has been used by Borcherds in order to prove a generalization of the Gross-Kohnen-Zagier Theorem, stating that certain combinations of Heegner points behave, in an appropriate quotient of the Jacobian variety of the modular curve, like the coeffcients of a modular form of weight 3/2. The same holds for certain CM (or Heegner) divisors on Shimura curves.The moduli interpretation of Shimura and modular curves yields universal families (Kuga-Sato varieties) over them, as well as variations of Hodge structures coming from these universal families. In these universal families one defines the CM cycles, which are vertical cycles of codimension larger than 1 in the Kuga-Sato variety. We will show how a variant of the additive lift, which was used by Borcherds in order to extend the Shimura correspondence, can be used in order to prove that the (fundamental cohomology classes of) higher codimensional Heegner cycles become, in certain quotient groups, coefficients of modular forms as well. Explicitly, by taking the $m$th symmetric power of the universal family, we obtain a modular form of the desired weight $3/2+m$. Along the way we obtain a new singular Shimura-type lift, from weakly holomorphic modular forms of weight 1/2-m to meromorphic modular forms of weight 2m+2. DATE: Nov 20, 2013 Speaker: Dr. Oren Ben-Bassat (University of Oxford and University of Haifa) Title: Analytic geometry as relative algebraic geometry Abstract. I will review symmetric monoidal categories and explain how one can work with "algebras and modules" in such a category. Toen, Vaquie, and Vezzosi promoted the study of algebraic geometry relative to a closed symmetric monoidal category. By considering the closed symmetric monoidal category of Banach spaces, we recover various aspects of Berkovich analytic geometry. The opposite category to commutative algebra objects in a closed symmetric monoidal category has a few different notions of a Zariski toplogy. We show that one of these notions agrees with the G-topology of Berkovich theory and embed Berkovich analytic geometry into these abstract versions of algebraic geometry. We will describe the basic open sets in this topology and what algebras they correspond to. These algebras play the same role as the basic localizations which you get from a ring by inverting a single element. In our context, the quasi-abelian categories of Banach spaces or modules as developed by Schneiders and Prosmans are very helpful. This is joint work with Kobi Kremnizer (Oxford). DATE: Nov 13, 2013 Speaker: Sergey Malev (Bar-Ilan University) Title: The images of non-commutative polynomials evaluated on 2 x 2 matrices over the real numbers Abstract. Let p be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set M_n(K) of n-by-n matrices is either zero, or the set of scalar matrices, or the set sl_n(K) of matrices of trace 0, or all of M_n(K). I prove the conjecture when K is the field of real numbers and n=2, and give a partial solution for an arbitrary field K. DATE: Nov 6, 2013 Speaker: Tal Perri (Bar-Ilan University) Title: Algebraic aspects in tropical mathematics Abstract. Tropical geometry is concerned with transforming classical algebraic varieties into polyhedral objects called tropical varieties, which in turn retain some information about the original varieties. Using mainly combinatorial tools, tropical geometry has been shown to be a useful approach for studying algebraic varieties.In our study, much like in the theory of algebraic geometry, we have established a connection between tropical geometry and algebra. Namely, we establish a correspondence between tropical varieties (more accurately a generalized version called corner loci') and algebraic structures named "kernels" in the semifield of fractions (viewed as a subset of the semifield of functions over the max-plus semifield). The kernels serve as the analogue for ideals while the semifield of fractions takes the place of the polynomial ring.We have found a way to represent the corner loci defined by a tropical polynomial by an equation in the semifield of fractions. The resulting equation gives rise to a congruence which is encapsulated in the structure of a kernel (like zero sets and ideals in algebraic geometry). Being an idempotent semifield, the semifield of fractions is also a "lattice ordered group" (a group whose elements are endowed with a lattice structure compatible with the group operation) and the kernels are normal convex l-subgroups there. The theory of lattice ordered groups has been studied extensively by Birkhoff, Medvedev, Kopytov, Steinberg, Anderson, Feil and many other mathematicians. It offers a whole new set of algebraic tools to study tropical varieties (which are currently mainly studied using combinatorial tools as mentioned above). DATE: October 31, 2013 Speaker: Dr. Eli Matzri (Ben Gurion University) Title: Symbol length in the Brauer group of a C_m field Abstract. We bound the symbol length of elements in the Brauer group of a $C_m$ field $F$ and solve the local exponent-index problem for these fields. In particular we show that every $F$ central simple algebra of exponent $p^t$ is similar to the tensor product of at most $t(p^{m-1}-1)$ symbol algebras of degree $p^t$. We then use this bound on the symbol length to show that the index of such algebras is bounded by $(p^t)^{t(p^{m-1}-1)}$, which in turn gives a bound for any algebra of exponent $n$ via the primary decomposition. DATE: October 23, 2013 Speaker: Dr. Uriya First (Hebrew University of Jerusalem) Title: Azumaya algebras with involution Abstract. A well-known theorem of Albert states that a central simple algebra A of period 2 in the Brauer group admits an involution of the first kind. This was generalized by Saltman to Azumaya algebras with the difference that the conclusion is that there is some algebra in the Brauer class of A admitting an involution of the first kind. Albert and Saltman also gave necessary and sufficient conditions for existence of involutions of the second kind. Another proof of Saltman's Theorem was later given by Knus, Primala and Srinivas.<\br>I will discuss a new approach to Saltman's Theorem based on bilinear forms. This will give a new perspective on the proof of Knus-Primala-Srinivas and also give rise to further results and generalizations. In particular, we shall apply this approach to construct non-trivial Azumaya algebras of period 2 without involution and to show that, under certain assumptions, rings which are Morita equivalent to their opposite are Morita equivalent to a ring with an anti-automorphism.<\br>Several open questions and their implications will be posed at the end. DATE: 16/10/2013 Speaker: Prof. Yoav Segev Title: On infinite sharply 2-transitive groups Abstract. Let G be a sharply 2-transitive permutation group on a set X. Then, it is easy to see that G contains many'' involutions (i.e. elements of order 2). Let Inv(G) be the set of all involutions in G. <\br>In all known examples of G as above the SET Inv(G)^2 of all products of two involutions in G form an abelian normal regular (i.e. sharply 1-transitive) subgroup of G.However the efforts to prove, or disprove that this is true for every G as above have failed consistently, for a long time now, in spite of efforts made by some well known mathematicians.I will discuss this conjecture and present some known facts and some partial new results. DATE: 2/10/2013 Speaker: Prof. Letterio Gatto (Dipartimento di Scienze Matematiche, Politecnico di Torino) Title: Linear ODEs and the Boson-Fermion Correspondence Abstract. The purpose of the talk is to show an application of the algebraic treatment of the elementary theory of linear ODEs with constant coefficients based on previous joint work with I. Scherbak. Let $B_r:={\Bbb Q}[x_1,\ldots,x_r]$ be the polynomial ring in the $r$ indeterminates $x_1,\ldots, x_r$ with rational coefficents. Let $z$ be an indeterminate over $B_r$. Certain vertex operators $\Gamma(z), \Gamma^\vee(z):B_r\rightarrow B_r[[z]]$ associated to a linear ODE of finite order $r$ will be defined and computed. It will be shown that their limits as $r\rightarrow \infty$ are the well known vertex operators associated to the Boson-Fermion correspondence arising in the representation theory of the Heisenberg algebra. The needing vocabulary will be defined along the talk.

## Academic year   2012-13

### Organizers:   L.H. Rowen, M. Schein and U. Vishne

 DATE: August 28, 2013 Speaker: Dr. David Hume (University of Oxford) Title: Metric geometry of finitely generated groups and related spaces Abstract. We give a rough overview of the study of finitely generated groups as metric spaces, with an emphasis towards hyperbolicity and related properties. DATE: June 5, 2013 Speaker: Prof. Nikolai Gordeev (Herzen State Pedagogical University, St. Petersburg) Title: Linearly Kleiman groups and related topics Abstract. Let V denote a finite-dimensional vector space over some field. We say that a linear group G < GL(V) is a linearly Kleiman group if, for every pair of linear subspaces v and u of V, there is an element g of G such that the subspaces g(v) and u are in general position. The main result is the classification of connected linear algebraic groups over a field of characteristic zero which are linearly Kleiman. We also consider some properties of linearly Kleiman groups. DATE: May 29 Speaker: Prof. Amnon Yekutieli (BGU) Title: Introduction to Derived Categories Abstract. We outline the construction of the derived category D(M) of an abelian category M. We then define left and right derived functors. We introduce K-projective, K-injective and K-flat resolutions, and prove existence of some derived functors when M is either the category of modules over a ring, or the category of sheaves of modules over a sheaf of rings. DG algebras and their derived categories will also be mentioned.Next we discuss some more specialized topics: dualizing complexes (commutative and noncommutative), two-sided tilting complexes, derived Morita theory, and rigid dualizing complexes. DATE: May 22 Speaker: Prof. Soli Vishkautsan (Bar-Ilan University) Title: Residual periodicity on algebraic varieties Abstract. We present "residual periodicity", a relatively new concept in arithmetic dynamics, as defined by Bandman, Grunewald and Kunyavskii. A rational self-map of a quasiprojective variety defined over a number field is strongly residually periodic if its minimal periods are bounded modulo almost every prime. We discuss some interesting examples, and present results about residual periodicity on cubic surfaces. DATE: May 1 Speaker: Prof. Lior Bary-Soroker (Tel Aviv University) Title: Prime polynomials in short intervals and long arithmetic progressions Abstract. In this talk I will present a joint work with Efrat Bank and Lior Rosenzweig in which we prove function field analogs of the following classical conjectures in the theory of prime numbers. The first conjecture predicts that the number of primes xAll relevant notions will be explained during the talk. DATE: April 24 Speaker: Dr. Arkady Tsurkov (Bar-Ilan) Title: Automorphic equivalence of the many-sorted algebras Abstract. Universal algebras $H_{1}$, $H_{2}$ of the variety $\Theta$ are geometrically equivalent if they have same structure of the algebraic closed sets. Automorphic equivalence of algebras is a generalization of this notion. We can say that universal algebras $H_{1}$, $H_{2}$ of the variety $% \Theta$ are geometrically equivalent if the structures of the structures of the algebraic closed sets of these algebras coincides up to changing of coordinates defined by some automorphism of the category $\Theta ^{0}$. $% \Theta ^{0}$ is a category of the free finitely generated algebras of the variety $\Theta$. The quotient group $\mathfrak{A/Y}$ determines the difference between geometric and automorphic equivalence of algebras of the variety $\Theta$, where $\mathfrak{A}$ is a group of the all automorphisms of the category $\Theta ^{0}$, $\mathfrak{Y}$ is a group of the all inner automorphisms of this category. The method of the verbal operations was worked out in: B. Plotkin, G. Zhitomirski, \textit{On automorphisms of categories of free algebras of some varieties}, 2006 - for the calculation of the group $\mathfrak{A/Y}$. By this method the automorphic equivalence was reduced to the geometric equivalence in: A. Tsurkov, \textit{Automorphic equivalence of algebras}, 2007. All these results were true for the one-sorted algebras: groups, semigroups, linear algebras... Now we reprove these results for the many-sorted algebras: representations of groups, actions of semigroups over sets and so on. DATE: April 17 Speaker: Prof. James B. Wilson (Colorado State University, visiting Hebrew University of Jerusalem) Title: A genetic approach to p-groups Abstract. One hopeful prospect to study groups is to understand their basic building blocks. For p-groups, Z/pZ is not a very helpful building block. So instead we introduce new decompositions of p-groups into larger components such as bilinear maps which are no longer p-groups but inform us about features such as characteristic subgroups and automorphisms. The properties of these components are passed along to quotients and central extensions so they are in this sense genetic'' properties of p-groups. One of the most compelling aspects of this approach is its dependence on other areas. This already includes ring theory, semisimple Lie and Jordan algebras, and nonassociative division rings. DATE: April 10 Speaker: Dr. Arno Fehm (Universitt Konstanz) Title: Hilbert's irreducibility theorem and generalizations Abstract. The classical Hilbert irreducibility theorem on irreducible specializations of polynomials has immediate applications in number theory, arithmetic geometry, and Galois theory. I will give an introduction to this topic and discuss generalizations, namely Hilbertian fields and thin subsets of varieties in the sense of Serre. I will then present recent results concerning algebraic groups (joint with Bary-Soroker and Petersen) and related work of Borovoi on homogeneous spaces. DATE: April 3 Speaker: Prof. Amiram Braun (Haifa University) Title: Factoriality for the Zassenhaus variety, or how lucky can one be? Abstract. The center of the enveloping algebra of a reductive Lie algebra, in prime characteristic, is a factorial domain. This 2010 result is due to R.Tange. We shall explain the non-commutative ring-theoretic origin of this result and outline a new proof for it. This also enabled us to prove the same for its quantum analog (at the root of unity case). The unlikely chain of events which led to this proof, involving modular invariants, Grothendieck-Serre correspondence etc, will be also described. DATE: March 13 Speaker: Dr. Uriya First (Hebrew University of Jerusalem) Title: Non-reflexive hermitian categories Abstract. A hermitian category is a triplet (H,*,w) such that H is an additive category, * is a contravariant functor from H to itself, and w: id --> ** is a natural isomorphism satisfying a certain identity (e.g. take H to be the category of finite-dimensional vector spaces and * to be the functor sending V to V*). Hermitian categories are categorical frameworks for quadratic and bilinear forms that allow one to prove results about them in great generality. Let (H,*,w) be a hermitian category. I call H non-reflexive if w : id --> ** is only assumed to be a natural transformation, rather than a natural isomorphism. Most results about hermitian categories only apply to the reflexive case (i.e. when w is an isomorphism). In this talk I will show that given a non-reflexive category (H,*,w), there exists a reflexive category (H',*',w') such that the category of arbitrary bilinear forms over (H,*,w) (even non-symmetric forms) is equivalent to the category of symmetric regular (=unimodular) bilinear forms over (H',*',w').Next, I will show how systems of bilinear forms can be understood as a single bilinear form in an appropriate non-reflexive hermitian category.Combining both observations leads to numerous applications for systems of bilinear forms and also to hermitian forms over rings which are defined over non-reflexive modules. Among these applications are Witt's Cancellation Theorem, a version of Springer's Theorem, various finiteness results, etc. (joint work with E. Bayer-Fluckiger and D. Moldovan) DATE: March 6, 2013 Speaker: Dr. Dmitry Kerner (Ben-Gurion University) Title: Finite determinacy of maps and matrices Abstract. If a (smooth/analytic/formal) function of one variable has order p at the origin, then, after a change of coordinates and scaling, it becomes a monomial, f(x)=x^p. More generally, the functions are often determined (locally, up to equivalence) by just a few first terms of their Taylor expansion. This phenomenon is called 'finite determinacy'. Jointly with G.Belitski we study the finite determinacy of matrices over local rings (aka modules over analytic/formal/smooth germs). In this case finite determinacy implies algebraizability (when does a given module arise as the stalk of an algebraic sheaf?)The most common equivalences are: the two-sided action on a matrix (the group GL(m,R)\times GL(n,R)) and the change of variables (i.e. the automorphisms of the ring, Aut(R)). We give simple necessary and sufficient criteria for a matrix to be finitely determined. These criteria imply results of Mather's type: for some scenarios (the size of matrices, the ring, the group) the non-finitely determined matrices are very rare (their set is of infinite codimension), while for the others, there are no finitely determined matrices. DATE: 27/2/2013 Speaker: Dr. Shai Sarussi Title: Maximal covers of chains of prime ideals Abstract. Suppose $f:S \rightarrow R$ is a ring homomorphism such that $f[S]$ is contained in the center of $R$. We study the connections between chains in $\text{Spec} (S)$ and chains in $\text{Spec} (R)$. We focus on the properties LO (lying over), INC (incomparability), GD (going down), GU (going up) and SGB (strong going between). We provide a sufficient condition for every maximal chain in $\text{Spec} (R)$ to cover a maximal chain in $\text{Spec} (S)$. We prove some necessary and sufficient conditions for $f$ to satisfy each of the properties GD, GU and SGB, in terms of maximal $\mathcal D$-chains, where $\mathcal D \subseteq \text{Spec} (S)$ is a chain. We show that if $f$ satisfies all of the properties above, then every maximal $\mathcal D$-chain is a perfect cover of $\mathcal D$. Finally, we provide equivalent conditions for the following property: for every chain $\mathcal D \subseteq {\text Spec} (S)$ and for every maximal $\mathcal D$-chain $\mathcal C \subseteq {\text Spec} (R)$, $\mathcal C$ and $\mathcal D$ are of the same cardinality. DATE: 6/2/2013 Speaker: Prof. David Savitt (University of Arizona) Title: Lattices in the cohomology of Shimura curves Abstract. I will discuss joint work with Matthew Emerton and Toby Gee, in which we relate the geometry of tamely potentially Barsotti-Tate deformation rings for two-dimensional Galois representations to the integral structure of the cohomology of Shimura curves. As a consequence, we establish some conjectures of Breuil regarding this integral structure. DATE: 16/1/2013 Speaker: Prof. Pasha Zusmanovich (Tallinn University of Technology) Title: Lie algebras coming from dual operads and their invariants Abstract. I will discuss Lie algebras arising as tensor products of two algebras over binary quadratic operads Koszul dual to each other. Many Lie algebras appearing in physics - including "Poisson brackets of hydrodynamic type" of Balinsky-Novikov, Schr\"odinger-Virasoro algebra, etc. - admit representation in such a form for a suitable pair of dual operads. I will also speculate on a possible role of this tensor product construction in the structure theory of Lie algebras in small characteristics. A simple linear-algebraic method, somewhat resembling separation of variables of differential equations, allows, in some situations, to compute invariants of such Lie algebras which are important for physics and structure theory - such as low-degree cohomology, invariant bilinear forms, etc. DATE: January 9, 2013 Speaker: Dr. Anton Khoroshkin (Stony Brook University) Title: Shuffle operads and pattern avoidance Abstract. An algebra of some type is a set with some operations on it. Let us remove the underlying set, what remains is the collection of all operations one can define. This collection with the prescribed rules of composition is what one calls an operad.In this talk I will explain the notion of operads and the theory of monomials for operads. The combinatorics of monomials in operads is governed by avoidance problems. In particular, the Hilbert series of dimensions of certain class of operads coincides with the generating series of permutations avoiding a given set of patterns. I will state several general results and conjectures about the class of generating series for monomial operads providing some computational algorithms for these series. (joint work with V.Dotsenko, B.Shapiro and D.Piontkovsky; see http://arxiv.org/abs/1109.2690, http://arxiv.org/abs/1202.5170, http://arxiv.org/abs/1009.5308) DATE: 26/12/2012 Speaker: Gili Schul (Hebrew University of Jerusalem) Title: Fourier expansion of word maps Abstract. Frobenius has observed that the irreducible characters of a group determine the number of times an element in the group is obtained as a commutator. More generally, for any word w the number of times an element is obtained by substitution in w is a class function. Thus, it has a presentation as a combination of irreducible characters, called the Fourier expansion of w. In this talk I will present formulas regarding the Fourier expansion of words in which some letters appear twice. These formulas give simple proofs for classical results, as well as new ones. (joint work with Ori Parzanchevski.) DATE: 26/12/2012, 4:00pm Speaker: Prof. Eliyahu Rips (Hebrew University of Jerusalem) Title: Canonical words for small cancellation groups DATE: 19/12/2012 Speaker: Prof. Lenny Makar-Limanov Title: A new proof of the AMS (Abhyankar-Moh-Suzuki) theorem and related open problems Abstract. One of the most famous theorems on polynomials in one variable is the following statement: if two polynomials $f$ and $g$ from $K[z]$, where $K$ is a field, generate all $K[z]$ then either the degree of $f$ divides the degree of $g$ or the degree of $g$ divides the degree of $f$ (if the characteristic of $K$ is finite additional restrictions are necessary). There are about a dozen published proofs of this fact, some of them wrong, and I'll discuss a new proof, and some related open problems. DATE: 12/12/2012 Speaker: Ori Parzanchevski (Hebrew University of Jerusalem) Title: Simplicial Hodge theory and isoperimetric inequalities Abstract. Simplicial Hodge theory, initiated by B. Eckmann, relates the simplicial homology of complexes to corresponding spaces of harmonic functions. We will explain this theory, and recent applications of it: isoperimetric inequalities generalizing the Cheeger inequalities in graph theory, and high-dimensional generalizations of random walks. No previous knowledge will be assumed. (joint work with Ron Rosenthal and Ran Tessler.) DATE: 5/12/2012 Speaker: Prof. Stuart Margolis (Bar-Ilan University) Title: A topological approach to the global dimension of left regular band algebras Abstract. In the last 15 years it has been realized that a number of combinatorial/geometric structures admit a multiplication that is a left regular band. This is a semigroup that satisfies the identities x^2 = x and xyx = xy. Such structures include real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids.Markov chains on such objects have been successfully analyzed by the representation theory of left regular bands. This in turn has stimulated intensive development of the representation theory and cohomology of these and other finite monoids.The purpose of this talk is to show that if M is a finite left regular band then Ext spaces between simple M-modules are reduced cohomology spaces of associated subcomplexes of the order complex of the poset of principal right ideals of M. In particular, we easily compute the Ext-quiver and the global dimension of left regular band algebras. (joint work with Franco Saliola and Benjamin Steinberg.) DATE: 28/11/2012 Speaker: Dr. Dima Trushin (Hebrew University of Jerusalem) Title: Algebraization of a Cartier divisor Abstract. I will talk about my recent results extending to pairs classical theorems of R. Elkik on lifting of homomorphisms and algebraization. This solves affirmatively a problem raised by M. Temkin and has applications to desingularization theory. DATE: 21/11/2012 Speaker: Uriya First (Hebrew University) Title: Bilinear forms and anti-endomorphisms Abstract. A fundamental theorem in the theory of simple algebras with involution asserts that there is a one-to-one correspondence between (non-symmetric) regular bilinear forms over a finite-dimensional F-vector space V (considered up to scalar multiplication) and F-anti-automorphisms of End(V). Furthermore, under that correspondence, symmetric and alternating forms correspond to orthogonal and symplectic involutions, respectively. The theorem admits generalizations to sesquilinear forms over simple algebras with involution.In this talk I will introduce a new notion of bilinear forms over arbitrary non-commutative rings (no involution on the base ring is needed) and then discuss in how to generalize the previous correspondence into this new context.We will see that there is indeed a canonical way to obtain a correspondence between regular bilinear forms on an R-module M and anti-endomorphisms of End(M) and under that correspondence, involutions correspond to "symmetric" forms. However, in contrast to the case when R is a field, this correspondence might fail, unless some assumptions are made on M and R. For example, it is guaranteed to hold when M is finite projective or a generator the correspondence hold.The lecture will include many examples and several open problems will be presented.If time permits, we will also see an application of the previous correspondence to the following problem: Let R be a ring which is Morita equivalent to its opposite. Does there exists a ring with an anti-automorphism which is Morita equivalent to R? (Partial answer: yes if R is semiperfect).The lecturer will also give a related (and independent) talk at the Amitsur algebra seminar on 29/11. It will concern the isomorphism problem of bilinear forms over rings and it is titled "Solution to The Isomorphism Problem of Systems of Bilinear Forms Over Certain Rings". DATE: 14/11/2012 Speaker: Dr. Oz Ben-Shimol (Bar-Ilan University) Title: On Dixmier-Duflo isomorphism in positive characteristic - the classical nilpotent case Abstract. Let $\mathfrak{g}$ be the nil radical of the Borel subalgebra of one of the classical simple Lie algebras over a field $F$ of characteristic $p\geq 0$. For $p > 0$ we find an explicit realization of the center $Z(\mathfrak{g})$ of the enveloping algebra $U(\mathfrak{g})$ by generators and relations. This constructive approach yields an explicit isomorphism between $Z(\mathfrak{g})$ and the polynomial invariants algebra $S(\mathfrak{g})^\mathfrak{g}$. While realizing $Z(\mathfrak{g})$, we also prove that $Z(\mathfrak{g})$ is a complete intersection ring. Moreover, it leads to an explicit realization of $Z(\mathfrak{g})$ and $S(\mathfrak{g})^\mathfrak{g}$ for $p = 0$ as well. This extends a result of Dixmier in type $A_n$. DATE: 7/11/2012, 10:00 Speaker: Prof. Alexander Guterman (Moscow State University) Title: On the conversion between permanent and determinant Abstract. Two important functions in matrix theory, determinant and permanent, look very similar: $$\det A= \sum_{\sigma\in { S}_n} sgn({\sigma}) a_{1\sigma(1)}\cdots a_{n\sigma(n)} \quad \mbox{ and } \quad \per A= \sum_{\sigma\in { S}_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)}$$ here $A=(a_{ij})\in M_n(F)$ is an $n\times n$ matrix and ${ S}_n$ denotes the set of all permutations of the set $\{1,\ldots, n\}$. The value $sgn(\sigma)\in \{-1,1\}$ is the signum of the permutation $\sigma$. While the computation of the determinant can be done in a polynomial time, it is still an open question, if there are such algorithms to compute the permanent. Due to this reason, starting from the work by P\'olya, 1913, different approaches to convert the permanent into the determinant were under the intensive investigation. In this regard the following two definitions naturally appear: A transformation $T$ on a certain matrix set $S$ is called a {\em conversion on $S$\/} if $\per A=\det T(A)$ for all $A\in S$. A single matrix $A$ is called {\em sign-convertible\/} if there exists a $(+1,-1)$ matrix $X$ such that $\per A = \det (X \circ A)$, where $X\circ A$ is the entrywise product of matrices. Among our results we prove that there are no bijective maps converting permanent to the determinant converters for the matrices over finite fields. Also we investigate Gibson barriers (the maximal and minimal numbers of non-zero elements) for convertible $(0,1)$-matrices. (joint work with Mikhail Budrevich, Gregor Dolinar, Bojan Kuzma, and Marko Orel) DATE: 7/11/2012, 11:00 Speaker: Prof. Elena Kreines (Moscow State University) Title: Embedded graphs on Riemann surfaces and beyond Abstract. Belyi pair is a smooth connected algebraic curve together with a non-constant meromorphic function on it with no more than 3 critical values. Belyi pairs are closely related with tamely embedded graphs on Riemann surfaces, so-called Grothendieck dessins d'enfants. Introduction to the theory will be given including the modern applications. In particular, we will discuss the generalized hebyshev polynomials and their geometry, visualization of the Galois group action and its invariants, Grothendieck dessins d'enfants on reducible curves. (joint work with Natalia Amburg and George Shabat.) DATE: 31/10/2012 Speaker: Dr. Michael Schein (Bar-Ilan University) Title: Zeta functions of Heisenberg groups over number rings Abstract. This is a report on work in progress with Mark Berman (Bar-Ilan) and Christopher Voll (Bielefeld). Let G be a finitely generated group, and let a_n be the number of subgroups of G of index n, which is always finite. The zeta function Z(s) = \sum a_n n^{-s} counts the finite index subgroups of G and has been an object of active study for the past 25 years. The zeta function splits into an Euler product of local factors, and in some cases these factors possess a striking symmetry (a functional equation). It is an interesting and deep problem to explain this symmetry in terms of the algebraic properties of G.We consider the special case of a Heisenberg group over a number ring. Let K be a number field with ring of integers O. The Heisenberg group H(O) consists of upper triangular matrices with entries in O and ones on the diagonal. We have studied the local zeta factors of the group H(O). In some cases we have explicit formulae for these factors. In the remaining cases, we study an algorithm for computing them that provides an interesting connection to the combinatorics of Dyck words. The local zeta factor at any prime p appears to satisfy a functional equation that depends on the ramification of p in K.The talk will give an introduction to the theory of zeta functions and an exposition of our results about H(O). No prior knowledge of the subject is assumed. DATE: 24/10/2012 Speaker: Dr. Adam Gamzon (Hebrew University of Jerusalem) Title: Local torsion on abelian surfaces Abstract. Fix an integer d >= 1. In 2008, David and Weston showed that, on average, an elliptic curve over Q picks up a nontrivial p-torsion point defined over a finite extension K of the p-adics of degree at most d for only finitely many primes p. This talk will discuss a generalization of their work, analyzing the frequency with which a principally polarized abelian surface A over Q with real multiplication by Q(\sqrt{5}) has a nontrivial p-torsion point defined over K. Averaging by height, the main result shows that A picks up a nontrivial p-torsion point over K for only finitely many p. Furthermore, we will indicate how this result connects with the deformation theory of modular Galois representations.

## Academic year   2011-12

### Organizers:   L.H. Rowen, M. Schein and U. Vishne

 DATE: 2/8/2012 Speaker: Prof. Boris M. Schein (University of Arkansas) Title: Infinitesimal bases and transitive representations of inverse semigroups Abstract. Every group is isomorphic to a group of permutations of a suitable set. Groups form a subclass of the class of inverse semigroups, the most important class of semigroups besides groups. Not every inverse semigroup is isomorphic to a transitive inverse semigroup of partial one-to-one transformations of a set. The problem of describing those inverse semigroups that admit such an isomorphism was raised in 1952. The speaker presents a new solution to this problem. DATE: 25/7/2012, at 12:00. Speaker: Prof. George Glauberman (University of Chicago) Title: Taking limits in finite p-groups Abstract. One of the beautiful aspects of mathematics is the way that different subjects shed light on each other. Borel's Fixed Point Theorem in algebraic group theory asserts that a solvable algebraic group acting on a set in a particular way must have a fixed point. We plan to discuss how special cases of this theorem have been adapted to yield new results and open problems for finite p-groups. (Knowledge of algebraic group theory is not necessary.) DATE: 20/6/2012 Speaker: Prof. Ido Efrat (Ben-Gurion University) Title: Filtrations of absolute Galois groups Abstract. We will discuss the following question: Given a field $F$ with absolute Galois group $G_F$, how much Galois theory is needed to determine the mod-$m$ cohomology ring of $G_F$? This will allow us to get new restrictions on the structure of absolute Galois groups. (joint work with Jan Minac.) DATE: 6/6/2012 Speaker: Dr. Rony Bitan (Tel Aviv University) Title: A building-theoretic approach to relative Tamagawa numbers of quasi-split semisimple groups over global function fields Abstract. Let G be a semisimple quasi-split group (e.g. SL_n) defined over a global function field K. The adelic group G(A) being locally compact admits a Haar measure which is unique up to a scalar multiplication. One such normalized measure is the Tamagawa measure \tau. The volume of G(A)/G(K) with respect to \tau is called the Tamagawa number of G. We express the (relative) Tamagawa number of G in terms of local data including the number of types of special vertices in one orbit of the Bruhat--Tits building of G at some place of infinity of K and the class number of G with respect to this point. DATE: 30/5/2012 Speaker: Prof. Gabor Wiese (University of Luxembourg) Title: On modular Galois representations modulo prime powers Abstract. In the talk we motivate the study of modular Galois representations modulo prime powers by a natural number theoretic question. We go on to state and explain a result, obtained together with Imin Chen and Ian Kiming, about removing powers of p from the level of a Hecke eigenform modulo p^m, at the expense of working with so-called dc-weak eigenforms (dc = divided congruences). DATE: 23/5/2012 Speaker: Dr. Santosha Pattanayak (Weizmann Institute of Science) Title: Projective normality of G. I. T. quotient varieties modulo finite solvable groups and Weyl groups Abstract. We prove that for any finite dimensional vector space $V$ over an algebraically closed field $K$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the $|G|$ is a unit in $K$, the projective variety $P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$, where $\mathcal O(1)$ denotes the ample generator of the Picard group of $\mathbb P(V)$. We also prove that for the standard representation $V$ of the Weyl group $W$ of a semi-simple algebraic group of type $A_n , B_n , C_n , D_n , F_4$ and $G_2$ over $\mathbb C$, the projective variety $\mathbb P(V^m)/W$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |W|}$, where $V^m$ denote the direct sum of $m$ copies of $V$. DATE: 16/5/2012 Speaker: Dr. Eli Matzri (Bar-Ilan University) Title: Z_3 x Z_3 crossed products Abstract. Let $A$ be the generic abelian crossed product with respect to $Z_3 \times Z_3$. We show that $A$ is similar to the tensor product of 4 symbols. We use this to show that the essential 3-dimension of the class of $A$ is at most 6. DATE: 9/5/2012 Speaker: Menny Aka (Hebrew University of Jerusalem) Title: Arithmetic groups with isomorphic finite quotients Abstract. Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the fibers are of unbounded size.In the talk I will concentrate on giving interesting families of non-isomorphic arithmetic groups which are profinitely isomorphic and explain how they fit in the scheme of my proof. DATE: 28/3/2012 Speaker: Prof. Bangteng Xu Title: Some Structure Theory of Table Algebras and Association Schemes Abstract. In this talk we start with some basic concepts about table algebras, closed subsets, quotient table algebras, and table algebra homomorphisms. Then we present the isomorphism theorems for table algebras. We will also talk about the direct products of closed subsets in a table algebra, as well as the Krull-Schmidt type theorem. Finally, we discuss the combinatorial isomorphisms, isomorphism theorems, and Krull-Schmidt type theorems for association schemes. DATE: March 21, 2012 Speaker: Prof. Eugene Plotkin (Bar-Ilan University) Title: Equations over algebras: how logical geometry appears Abstract. The aim of the talk is to describe the ideas of logical geometry in more or less plain words. We will trace how some algebraic/model-theoretic problems arise in a geometrical way. We also compare the methods of solution of equations over simple algebras with those over free algebras. (joint work with E. Aladova and B. Plotkin.) DATE: March 14, 2012 Speaker: Dr. Rizos Sklinos (Hebrew University of Jerusalem) Title: Geometric thoughts on stable groups Abstract. After Sela and Kharlampovich-Myasnikov independently proved that non abelian free groups share the same common theory, model theoretic interest for the subject arose. Moreover, Sela's techniques established a connection between geometric group theory and logic. In 2001 Nies proved that the free group in two generators, $F_2$, is homogeneous, i.e. for every two finite tuples of elements in $F_2$ that share the same first order properties, there is an automorphism that takes one to the other. In this talk we will first give some basic model theoretic properties of non abelian free groups and then we will move on and extend Nies' result to all non abelian free groups. In the last part of the talk we will show that most surface groups are not homogeneous. We note that although the questions we answer are motivated by model theory, the techniques used are purely geometric, thus continuing the nice interplay between the two disciplines. (joint work with C. Perin.) DATE: 1/2/2012 Speaker: Uriya First (Bar-Ilan University) Title: Rings of invariants under endomorphisms Abstract. The classical scenario in the algebraic theory of invariants is where a group G of automorphisms acts on a ring R. Working in a more general setting, where G need not be a group, I will discuss properties of R which are inherited by the ring of invariants R^G, focusing on cases when R is "almost" semisimple Artinian. In particular, if R is semiprimary (resp. left/right perfect; semilocal complete) then so is the invariant ring R^G for any set G of endomorphisms of R. However, that R is artinian or semiperfect need not imply this property for R^G, even when G is a finite group with an inner action. (Examples will be presented if time permits.) The former result actually holds in a more general context: Let S be a ring containing R and let G be a set of endomorphisms of S, then the ring R^G of G-invariant elements inside R inherits from R the properties: being semiprimary, being left (resp. right) perfect.As easy corollaries, we get that if R is a subring of a ring S, then the centralizer in R of any subset of S inherits the property of being semiprimary or left perfect from R. Better still, the centralizer in R of a set of invertible elements in R inherits the property of being semilocal-complete.Similarly, assume S is a ring containing R and let M be a right S-module. Then, that End_R(M) is semiprimary (resp. left/right perfect) implies that End_S(M) is. All ring-theoretic notions will be defined. DATE: 25/1/2012 Speaker: Prof. Sara Westreich (Bar-Ilan University) Title: Conjugacy classes and character tables for semisimple Hopf algebras Abstract. We extend the notion of conjugacy classes and class sums from finite groups to semisimple Hopf algebras. For a large family of Hopf algebras H we prove that the product of two class sums is an integral combination of the class sums up to 1/d^2 where d = dimH. We define character tables and show how Hopf subalgebras and quotients of semisimple Hopf algebras H can be characterized via their character table. DATE: 18/1/2012 Speaker: Prof. Harvey Blau (Northern Illinois University) Title: A subgroup configuration Abstract. We prove an elementary, useful, and apparently new result on a family of subgroups of a group, where each pair from the family has a relatively large intersection. The theorem is stated and proved more generally, in the language of partially ordered sets. DATE: 11/1/2012 Speaker: Dr. Lior Bary-Soroker (Tel Aviv University) Title: Hilbert's irreducibility theorem and Galois representations Abstract. Hilbert's irreducibility theorem asserts: if f is a polynomial in two variables X,Y with integral coefficients that is irreducible and of degree at least 1 in Y, then there exists an irreducible specialization, i.e. a rational number a such that f(a,Y) is irreducible. A field with irreducible specializations is called Hilbertian. The numerous applications of this theorem make the question of under what conditions an extension of a Hilbertian field is again Hilbertian intersting. It turns out the the most difficult part is separable algebraic extensions. Jarden conjectured that if K is Hilbertian, A an abelian variety over K, and E/K is an extension of K that is contained in the field generated by all torsion points of A, then E is Hilbertian.In this talk I shall discuss a solution of the conjecture using Galois representations. DATE: 4/1/2012 Speaker: Dr. Dmitry Kerner (University of Toronto) Title: Discriminants, old and new Abstract. Let f be a homogeneous form in (n+1) (complex) variables. The classical discriminant is a polynomial in the coefficients of f that vanishes precisely when the form f is degenerate. Geometrically, the discriminant parameterizes singular hypersurfaces of a given degree in P^n. More generally one considers the discriminant of (compact) complete intersections.I will recall some classical properties of the discriminant, for example, will give a geometric explanation of why the discriminant of the form ax^2+bxy+cy^2 has no terms like a^2 or bc.Then I will describe one of the modern reincarnations, in the study of compact varieties with non-isolated singularities. The discriminant of the transversal singularity type consists of the points of singular locus where the transversal type degenerates. In many cases its cohomology class can be computed by reduction to the classical discriminant. DATE: 28/12/2011, 10:00 Speaker: Prof. Leonid Makar-Limanov (Wayne State University) Title: A Bavula conjecture Abstract. As is well known and easy to prove the Weyl algebras $A_n$ over a field of characteristic zero are simple. Hence any non-zero homomorphism from $A_n$ to $A_m$ is an imbedding and $m \geq n$. Vladimir Bavula conjectured that the same is true over the fields with finite characteristic. It turned out that exactly one half of his conjecture is correct. DATE: 28/12/2011, 11:00 Speaker: Prof. Yuri Zarhin (Pennsylvania State University) Title: Abelian varieties with and without homotheties Abstract. We discuss variants of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of a finitely generated field $K$. DATE: 21/12 Speaker: Prof. Uri Onn Title: Introduction to representation zeta functions Abstract. This is an introductory talk to the subject of representation growth and representation zeta functions of groups. The main focus will be on arithmetic groups and p-adic analytic groups. I will describe some recent results in the area and open problems. DATE: 14/12 Speaker: Prof. Ami Braun (Haifa University) Title: The Gorenstein property for modular invariants Abstract. Let V be a finite dimensional vector space over a field and G < GL(V) a finite group. G acts by automorphisms on the polynomial ring S(V). The ring of polynomial invariants S(V)^G is a classical object and the investigation of its properties is the main topic of (algebraic) invariant theory. We shall consider the Gorenstein property of S(V)^G in light of a classical theorem of Watanabe. A modular conjecture due to Kemper-Kording-Malle-Matzat-Vogel-Wisse will be also considered and, if time permits, also similar results by Benson-Carlson regarding the cohomology ring of finite groups. DATE: 7/12 Speaker: Prof. Alexander S. Sivatski (St. Petersburg Electrotechnical University) Title: Central simple algebras of exponent 2 and index 8, and divided power operations Abstract. This talk can be viewed as an extension of the colloquium talk on December 4. We show that if char F is not 2 and F has square root of -1, D is a 2-torsion element of the Brauer group of index 8, and a in F is such that ind(D_{F[\sqrt{a}]})=4, then $\gamma_3(D) = \{a,s\}\gamma_2(D)$ for some s \in F, where $\gamma_i$ are the divided power operations on ${}_2Br(F)$. Consequently, we prove that if $D$ has a subfield of degree 4 of certain type, then $\gamma_3(D) = 0$. In particular, if $D$ is a crossed product for a group of order 8 different from (Z/2Z)^3, then $\gamma_3(D) = 0$. A few open related questions are posed. DATE: 30/11 Speaker: Prof. Alexander S. Sivatski (St. Petersburg Electrotechnical University) Title: The chain lemma for biquaternion algebras Abstract. Let A be a biquaternion algebra (a tensor product of two quaternion algebras)over a field F of characteristic dierent from 2. A decomposition of A into a tensor product of two quaternion algebras is not unique, and there is no canonical one. However, it turns out that any two decompositions of A can be connected by a chain of decompositions in which neighboring ones do not dier "too much". In fact there is an analogue of the chain lemma for a quaternion algebra. Theorem. Any two biquaternion decompositions of A are equivalent to one another, and can be connected by a chain of length 3. Moreover, this bound is strict, i.e. in general two decompositions of A cannot be connected by a chain of length 2. DATE: 23/11 Speaker: Prof. Louis Rowen Title: Polynomial identities of an affine PI-algebra over a commutative Noetherian ring Abstract. We will outline the structure of the full proof of Belov's theorem (2002) that the polynomial identities of an affine PI-algebra over a commutative Noetherian ring are finitely based. More details of the proof are to be given in Prof. Margolis' seminar on representation theory. (joint work with Alexei Belov and Uzi Vishne) DATE: 16/11/2011 Speaker: Dr. Claude Marion (Hebrew University of Jerusalem) Title: Triangle generation of finite groups of Lie type and rigidity Abstract. This talk is about the (p_1, p_2, p_3)-generation problem for finite groups of Lie type, where we say that a finite group is (p_1, p_2, p_3)-generated if it is generated by two elements of orders p_1 and p_2 whose product has order p_3. Given a triple (p_1, p_2, p_3) of primes, we say that (p_1, p_2, p_3) is rigid for a simple algebraic group G if the sum of the dimensions of the subvarieties of elements of orders dividing p_1, p_2, p_3 in G is equal to 2 dim G. We conjecture that if (p_1, p_2, p_3) is a rigid triple for G then, given a prime p, there are only finitely many positive integers r such that the finite group G(p^r) is a (p_1, p_2, p_3)-group. We discuss this conjecture, classify the rigid triples of primes for simple algebraic groups, and present a result stating that the conjecture holds in many cases. The conjecture, together with this classification, puts into context many results on Hurwitz (2,3,7)-generation in the literature and motivates a new study of the (p_1, p_2, p_3)-generation problem for certain finite groups of Lie type of low rank. DATE: 9/11/2011 Speaker: Prof. Eli Aljadeff (The Technion) Title: Graded polynomial identities and generic constructions Abstract. In the early 70's Amitsur proved the existence of noncrossed products via generic constructions, namely as a central localization of the relatively free algebra which corresponds to the algebra of $n \times n$-matrices. Similar construction exist for certain gradings on $n \times n$-matrices. A key necessary condition for such graded algebra to exist in general is that graded identities on matrices determine the algebra up to $G$-graded isomorphism. This was established in 2010 by Koshlukov and Zaicev in case $G$ is abelian and recently, in a joint work with Darrell Haile, for general groups. DATE: 2/11/2011, 11:00-13:00 -- (final lecture in Gasarch minicourse) Speaker: Prof. Bill Gasarch (University of Maryland, Dept of Computer Science.) Title: van Der Waerden's Theorem: Variants and ''Applications'' - IV Abstract. No matter how you color the natural numbers RED and BLUE there will be arithmetic sequence (that is, numbers equally spaced) of length 5771 that is all the same color. WOW! In fact, you can BOUND how much you need to color. We rephrase and generalize to arrive at Van Der Waerden's Theorem: For all k, for all c, there exists W=W(k,c) such that, for all c-colorings of {1,...,W} there is an a,d such that a, a+d, a+2d, ..., a+(k-1)d are all the same color. This theorem is part of Ramsey Theory whose basic idea is COMPLETE DISORDER IS IMPOSSIBLE. Note that NO MATTER HOW you color {1,...,W} there will be a nice monochromatic subset. In a series of lectures I will present the following: I) Warmup: No matter how you 2-color the lattice points of the plane there will be a monochromatic square. (This is folklore.) II) VDW's theorem, upper and lower bounds on the VDW numbers The bounds on W(k,c) in the original proof are quite large. They have been reduced quite a bit over the years- we will discuss this. III) No matter you you color the natural numbers RED and BLUE there will x_1,x_2,...,x_{5771} that are the same color such that x_1 + x_2 + ... + x_{5000} = x_{5001} + ... + x_{5771}. Rado's theorem generalizes this and gives a condition about which types of equations it holds for.IV) APPLICATIONS! of Rado's theorem to NUMBER THEORY. V) Multidim VDW theoremVI) APPLICTION of Multidim VDW theorem to Communication Complexity. If three people all have a number on their forehead that is n-bits long, they want to know if x+y+z is == 0 mod 2^n. Everyone sees all numbers but their own. They want to communicate as few bits as possible. How well can they do? Come and find out!VII) The POLYNOMIAL VDW theorem. Note that in VDW's theorem we have a, a+d, a+2d, ..., a+(k-1)d are all the same color. Why this sequence? Can we replace d, 2d, 3d,..., (k-1)d with other functions? YES: POLY VDW: For all polynomials p_1,...,p_k (integer coeffs) with zero constant term, for all c, there exists W=W(p_1,...,p_k;c) such that, for all c-colorings of {1,...,W} there exists a,d such that a, a+p_1(d), ..., a+p_k(d) are all the same color.VIII) APPLICATION of Poly VDW- to graph theory. We use it to get graphs of large chromatic number AND large girth. IX) The SANE BOUNDS program of GASARCH (thats me!). The proofs of VDW's theorem, Rado's theorem, PolyVDW theorem all yield really INSANE bounds. I have been looking at ways to cut them down OR look at variants where the bounds are more reasonable. Have I made progress? Come and find out!

## Academic year   2010-11

### Organizers:   L.H. Rowen and A. Kanel-Belov

 DATE: 15/6/2011 -- final week of this academic year, no seminar due to IMU meeting DATE: 8/6/2011 -- Shavuot DATE: 1/6/2011 -- Yom Yerushalaim DATE: 25/5/2011 -- Amitsur Symposium DATE: 18/5/2011 Speaker: Prof. Daniel Lenz Title: Order based constructions of groupoids from inverse semigroups Abstract. We discuss how the universal groupoid of an inverse semigroup introduced by Paterson can be obtained by a simple order based construction. Along the way one obtains canonically a reduction of this groupoid. In the case of inverse semigroups arising from graphs (respectively, tilings) this reduction is the graph groupoid introduced by Kumjian et al (respectively, the tiling groupoid of Kellendonk). We discuss some topological features of this reduction as well as the structure of its open invariant sets. This can be used to investigate the ideal structure of an associated reduced $C^*t$-algebra. DATE: 18/5/2011 Speaker: Prof. David Guiraud (Universite Paris 7, visiting Bar-Ilan) Title: On semisimple l-modular Bernstein blocks of a p-adic general linear group Abstract. Let G_n = GL_n(F), where F is a local non-archimedean field with residue characteristic p. We are concerned with the Bernstein decomposition into blocks of the representation category of G_n over an algebraically closed field of characteristic l \neq p. In level zero, we construct a replacement for the Iwahori-Hecke algebras, which gives rise to a description of the G_n-blocks associated to semisimple supercuspidal pairs in terms of G_m-blocks associated to simple supercuspidal pairs (with m< n), paralleling the approach of Bushnell and Kutzko in the complex setting. DATE: 11/5/2011 Speaker: Dr. Luda Markus-Epstein Title: Word Problem for Inverse Monoids Presented by a Single Relator Abstract. Since Magnus it has been well known that one-relator groups have a decidable word problem. However, solvability of the word problem in one- relator monoids is far from being completely studied. Only few examples of inverse monoids with solvable word problem are known. Recently, the solvability of the word problem in inverse monoids with a single sparse relator has been announced by Hermiller, Lindblad and Meakin.We consider certain one-relator inverse monoids. In our attempt to solve the word problem, we rely on the result of Ivanov, Margolis and Meakin which states that the word problem for the inverse one-relator monoid is decidable if the membership problem for the corresponding prefix monoid is decidable. Thus, we first solve the membership problem for the prefix monoid and then apply the theorem to solve the word problem. Our methods involve van Kampen diagrams and word combinatorics. DATE: 4/5/2011 Speaker: Adam Chapman Title: Generalized Clifford algebras Abstract. Given a field $F$ containing a primitive $p$th root of unity $\rho_p$ and a homogenous form of degree $p$ with $n$ variables $f(u_1,\dots,u_n)$, the Clifford algebra of this form is defined to be $C_f=F[x_1,\dots,x_n : (u_1 x_1+\dots+u_n x_n)^p= f(u_1,\dots,u_n) \forall u_1,\dots,u_n \in F]$. For $p=2$, $f$ is a quadratic form, and it is well-known that its underlying Clifford algebra is a tensor product of $\lfloor \frac{n}{2} \rfloor$ quaternion algebras.For any odd prime $p$ and $n=2$, we know that in $C_f=F[x,y : \dots]$, $y=z_1+\dots+z_{p-1}$ where for all $1 \leq i \leq p-1$, $z_i x=\rho_p^i x z_i$. We prove that the algebra $C_f/$ is an Azumaya algebra whose center is the affine algebra of a hyperelliptic curve of genus $\lfloor \frac{p-1}{2} \rfloor$, and that every simple homomorphic image of $C_f$ is a cyclic algebra of degree $p$.This generalizes the main result of the paper “On the Clifford algebra of a binary cubic form”/D. Haile.If $p=5$ we also prove that every division image of $C_f/$ is either a tensor product of one or two cyclic algebras of degree $5$ and we calculate the center of its ring of quotients explicitly. (joint work with Uzi Vishne) DATE: 27/4/2011 Speaker: Prof. Boris Kunyavski Title: Equations in simple Lie algebras: variations on a theme of A. Borel Abstract. Given an element P(X,Y,...,Z) of a free Lie K-algebra, for any Lie algebra g we can consider the induced polynomial map P: g x g x...x g ---> g. Assuming that K is an arbitrary field of characteristic different from 2, we prove that if P is not an identity in sl(2,K), then this map is dominant for any Chevalley algebra g. This result can be viewed as a weak infinitesimal counterpart of Borel's theorem on the dominance of the word map on connected semisimple algebraic groups. As in the group case, the proof is based on a construction of division subalgebras due to Deligne and Sullivan. We also prove that for the Engel monomials [[[X,Y],Y],...,Y] and, more generally, for their linear combinations, the map P is, moreover, surjective onto the set of noncentral elements of g provided that the ground field K is big enough, and show that for monomials of large degree the image of this map contains no nonzero central elements. We also discuss consequences of these results for polynomial maps of associative matrix algebras. (joint work with T. Bandman, N. Gordeev, and E. Plotkin.) DATE: 6/4/2011 Speaker: Dr. Shoham Shamir (University of Bergen) Title: Complete intersection rings in algebraic topology Abstract. In commutative algebra, complete intersection rings are the next best thing after regular rings. Such rings have a structural description, but they also have two homological characterizations: in the seventies Gulliksen characterized complete intersection local rings by the growth rate of their homology; more recently Benson and Greenlees characterized such rings by the existence of a certain structure on their derived categories.These homological characterizations can be easily adapted for the cochain-algebras of connected spaces, where the coefficients are in some prime field. It turns out that for nice cochain-algebras both homological conditions are equivalent, and that both are implied by a structural condition reminiscent of the structure of a complete intersection ring. I will explain how the concepts translate from commutative algebra to topology, the equivalence of the homological conditions and the possible interest in such "complete intersection spaces". (joint work with Dave Benson and John Greenlees.) DATE: 30/3/2011 Speaker: Dr. Pooja Singla (Ben-Gurion) Title: Representations of general linear groups over finite local rings Abstract. The general linear groups over finite local rings generalize the well studied general linear groups over finite fields. In this talk we shall discuss methods of constructing complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field. Typical examples of these groups include $GL_n(Z/p^2 Z)$ and $GL_n(F_p[t]/t^2)$ for a prime $p$. We shall construct a dimension preserving canonical correspondence between the irreducible representations of all such groups. This gives a positive reply to Uri Onn's conjecture (2008) that the representation zeta function of these groups depends only on the residue field. At the end we shall discuss similar result for other classical groups as well. DATE: 23/3 Speaker: Prof. John Fountain Title: Reflection monoids Abstract. A partial isomorphism of a vector space V is an isomorphism between two subspaces of V. The collection of all partial isomorphisms of V forms an inverse monoid ML(V) under composition of partial functions.A partial reflection is defined to be the restriction of reflection to a subspace of V, and a reflection monoid is a factorisable-inverse submonoid of ML(V) generated by partial reflections. A reflection monoid can be characterised by two pieces of data: a reflection group W and a collection of subspaces of V that forms a W-invariant semilattice. The talk will be a report of work in progress (joint with Brent Everitt) outlining the basic properties of reflection monoids, providing examples, and giving connections with Renner monoids and hyperplane arrangements. DATE: 16/3/2011 Speaker: Prof. Manfred Knebusch Title: Tropical and supertropical degenerations of a commutative ring Abstract. If R is a commutative ring, then degeneration of R to a "simpler" commutative ring usually means taking the factor ring by an ideal. In particular, a field does not have such a degeneration. Things become more interesting if we allow degeneration to semirings. The simplest such degenerations are provided by m-valuations (monoid valuations). They can be interpreted as a modest generalization of the valuations on R in the sense of Bourbaki. An m-valuation is a multiplicative and subadditive map v: R --> M to a totally ordered semiring of a very special kind, a so-called bipotent semiring. An m-valuation v can be covered by a supervaluation f: R --> U in various ways. This means degenerating R to a multiplicative submonoid of a supertropical semiring U. Applying a supervaluation f to the coordinates of R-valued points of an affine scheme V over R means degenerating V(R) in a less coarse way than by applying v. The various supertropical degenerations of V(R) provide a refinement of tropical geometry.If time allows I will give natural examples of m-valuations and supervaluations in the talk. (joint work with Zur Izhakian and Louis Rowen) DATE: 9/3/2011, 10:00 Speaker: Prof. Ilya Ivanov-Pogodaev ((Moscow State University)) Title: Construction of finitely-presented semigroups with non-integer Gelfand-Kirillov dimension Abstract. Examples of semigroups with arbitrary Gelfand-Kirillov dimension $\gamma>2$ are well known. However, all such semigroups with non-integer Gelfand-Kirillov dimension are not finitely presented, i.e. have infinite set of defining relations. We construct finitely-presented semigroups with non-integer Gelfand-Kirillov dimension $\gamma>2$ for a large class of recursive numbers. DATE: 9/3/2011, 11:00 Speaker: Prof. Ilya Ivanov-Pogodaev Title: Algebras with finite Grobner basis but algorithmically unsolvable zero-divisors problem Abstract. We construct an algebra $A$ presented by a set of relations with finite Gr\"oebner basis such that the following problems are algorithmically unsolvable. Problem 1. Given an element $a\in A$, does there exist $b\in A$ such that $ab=0$. Problem 2. Given an element $a\in A$, does there exist $n\in N$ such that $a^n=0$. Note that in the case of finite Gr\"oebner basis the equality problem is effectively algorithmically solvable. Note that for finitely presented monomial algebras these problems are algorithmically solvable. The construction is based on the Minsky Machine. DATE: 2/3/2011 Speaker: Adina Heilbrunn (Cohen) (Hebrew University of Jerusalem) Title: Dedekind symbols and modular forms Abstract. We generalize the correspondence between modular forms on $SL_2(\mathbb{Z})$ and Dedekind symbols (Fukuhara, 1998) for modular forms on congruence subgroups. In 2005 Fukuhara and Yui associated the classical Dedekind sums with the Eisenstein Series $G_2$ that is a quasi modular form. We show there is another way to get this correspondence, which explains why this correspondence is natural in some sense. As an another interesting example we examine the Dedekind symbols associated with the logarithmic derivatives of an important family of modular forms called the Siegel functions. This work is my master thesis made under the supervision of Prof. Ron Livne. DATE: 5/1/2011 Speaker: Prof. Leonid Makar-Limanov Title: The Freiheitssatz for Poisson algebras Abstract. In my talk I recall what is a Freiheitssatz-type theorem, recall in which situations FT is proved, outline the recent proof (with Umirbaev) of FT for Poisson algebras, and state some open problems related to FT. DATE: 29/12/2010, 10:00 Speaker: Dr. Lior Bary-Soroker (Essen) Title: Irreducible values of polynomials Abstract. Does there exist a polynomial f(X) such that all polynomials f(X), f(X)+1, f(X)+2, ..., f(X)+285 are irreducible? Clearly the answer depends on the field the coefficients are taken from. We will discuss the connection of the above question with the twin prime conjecture. More generally we will explain the general number theoretic Schinzel hypothesis H and its quantitative version the Bateman-Horn conjecture and the corresponding analogs over finite fields. DATE: 29/12/2010, 11:00 Speaker: Prof. Andrzej Zuk Title: On a problem of Atiyah Abstract. In 1976, Michael Atiyah defined L2-Betti numbers for manifolds and asked a question about their rationality. These numbers arise as the von Neumann dimensions of kernels of certain operators acting on the L2-space of the fundamental group of a manifold. The problem concerning their values is closely related to the Kaplansky zero-divisor question. We present constructions of closed manifolds with irrational L2-Betti numbers. DATE: 22/12/2010 Speaker: Prof. Dmitry Kerner (Toronto) Title: On the decomposability of maximally Cohen-Macaulay modules over hypersurface singularities Abstract. Maximally Cohen-Macaulay modules (MCM's) over hypersurface singularities appear naturally in various fields, for example under the names of matrix factorizations, local determinantal representations, and many others.MCM's are well studied in the case of ADE singularities, but for higher singularities the classification problem is wild. Still, one can treat some reasonable questions.If the hypersurface is locally reducible (or non-reduced) it is natural to ask about decomposability of the module. Or at least, when is the module an extension?I will start from a general introduction, then will formulate some criteria. The decomposable modules (or extensions) fall into two non-disjoint classes: those with many generators (e.g. maximally generated) and those that descendfrom modifications of the hypersurface by pushforwards. (joint work with This is joint work with V. Vinnikov.) DATE: 15/12/2010 Speaker: Tomer Schlank (Hebrew Univ.) Title: Homotopy theory and solubility of Diophantine equations Abstract. classical problem in number theory is to determine whether or not a system of polynomial equations E has a rational solution. If there is such a solution one can always present it. But to prove that no solution exists might be a more delicate issue. For this one uses the notion of obstructions.In the talk I would present a way to construct such obstructions based on exploring some kind of topological realization of E called "The \'{e}tale homotopy type", which was defined by Artin and Mazur. It turns out that this method of constructing obstructions can recover many of previously known obstructions (e.g the the Brauer-Manin, the \'{e}tale-Brauer and certain descent obstructions.) and thus give those obstructions a topological interpretation and shed light on the relationships between them. (joint work with Y.Harpaz.) DATE: 8/12/2010 Speaker: Prof. Alexander Luzgarev (Hebrew Univ.) Title: Characteristic free invariants of exceptional groups Abstract. The membership of an individual matrix to the exceptional Chevalley group is traditionally described by equations of degrees 3 and 4. These equations can be deduced from the multilinear invariants of the group in a given representation. Such invariants are classically known in characteristic 0, but it requires some work to make them characteristic free. We discuss some recent results that allow to view the exceptional group of type $E_7$ as the group of symplectic transformations (of a 56-dimensional space) stabilising a certain fourlinear non-symmetric form. DATE: 1/12/2010 Speaker: Prof. Michael Finkelberg (Independent University of Moscow) Title: Gelfand-Tsetlin bases for representations of the affine Lie algebra \hat{gl}_n Abstract. We introduce affine Gelfand-Tsetlin patterns and write down the explicit formulas for the action of generators of \hat{gl}_n in the Gelfand-Tsetlin basis of an irreducible integrable \hat{gl}_n-module. DATE: 24/11/2010 Speaker: Prof. Darrell Haile Title: The Clifford algebra of a quartic curve of genus one Abstract. This is joint work with Ilseop Han. For each irreducible quartic f over a field k, we construct a k-algebra A_f associated to the hyperelliptic affine curve C:y^2=f(x). We prove that A_f has many interesting properties. For example it is an Azumaya algebra of rank 4 over its center and its center is the coordinate ring of the affine elliptic curve E related to the Jacobian of C. Each simple image of A_f is a quaternion algebra. The simple images with center k then come from the rational points on E and the resulting function from the group of rational points on E to the Brauer group of k is a group homomorphism. We also prove that A_f is split if and only if the curve C has a k-rational point. DATE: 17/11/2010 Speaker: Dr. Roland Knevel Title: Super Teichmuller spaces Abstract. The topic of my talk will be a current research project, which can be roughly expressed as 'Classify all families of compact super Riemann surfaces'. I will explain the theory of complex super manifolds and how one can use classical deformation theory and sheaf cohomology for this purpose. Finally I will discuss the uniformization problem. DATE: 10/11/2010 Speaker: Dr. Sefi Ladkani Title: Derived equivalence, mutations and applications Abstract. An important homological invariant of a ring is its derived category of modules. In particular, it is interesting to know when two rings have equivalent derived categories. In principle, Rickard Theorem gives an answer in terms of tilting complexes, but this is far from satisfactory as it does not give a decision process nor does it give an algorithmic approach to construct these complexes.For path algebras of acyclic quivers, however, it is possible to decide on derived equivalence through the use of Bernstein-Gelfand-Ponomarev reflection functors. These are local operations, carrying both combinatorial and algebraic meaning, defined only at vertices which are sinks or sources in the quiver.During the years various generalizations to arbitrary vertices have been introduced, such as Brenner-Butler tilting on the algebraic side and Fomin-Zelevinsky quiver mutation on the combinatorial side. We show how deeply these generalizations are related for certain wide classes of finite-dimensional algebras, including algebras of global dimension two and endomorphism algebras of cluster-tilting objects in 2-Calabi-Yau triangulated categories which play important role in the additive categorification of Fomin-Zelevinsky cluster algebras. In particular this allows us to devise effective algorithms solving the derived equivalence question for various classes of cluster-tilted algebras. DATE: 10/11/2010, 12:00, joint with the algebraic geometry seminar Speaker: Dr. Shelly Garion Title: Triangle groups, finite simple groups and applications to Beauville surfaces Abstract. In this talk we will discuss the following question: Given a triple of integers (k,m,n), which finite simple groups are quotients of the triangle group T(k,m,n)? This question, originally arising in group theory, has found applications in the classification of certain algebraic surfaces, known as Beauville surfaces, providing solutions to conjectures of Bauer, Catanese and Grunewald. DATE: 3/11/2010 Speaker: Prof. Uzi Vishne Title: A solution to the Roquette problem Abstract. When the base field contains roots of unity, every cyclic field extension is generated by a radical; without roots of unity cyclic extensions can be far more complicated. The delicate role played by roots of unity leads to Albert's characterization of cyclic algebras of prime degree p, as those containing a radical element. In search for a generalization, Albert constructed in 1938 a simple algebra of degree 4 with a radical element, which is nevertheless not cyclic.The analogous problem for odd prime powers, which became known as Roquette's problem, remained open until recently. I will discuss the context and solution of this problem. (joint work with Louis Rowen and Eliyahu Matzri.) DATE: 27/10/2010 Speaker: Prof. Alexei Kanel-Belov Title: Construction of finitely presented infinite nill-semigroup. Abstract. We use geometric methods for the construction.We assign elements of the semigroup by paths on special metric space. This space can be considered as aperiodic tiling by finite number of tiles. The relations in the semigroup can be assigned by flips on this tiling. Using this assignments we can transform a given word and obtain some area in which this word's path can be situated. Using some monomial relations we obtain that all words with big powers can be reduced to nil.Unlike classical group situation, our complex is non-planar. DATE: 20/10/2010 Speaker: Prof. Alexei Kanel-Belov Title: Burnise-like problems in semigroups, algebras and groups DATE: 13/10/2010 Speaker: Dr. David Guiraud Title: Mod l representations of p-adic groups, Bernstein blocks, and the Schur algebra Abstract. This talk gives an overview of my PhD project (supervised by Marie-France Vigneras).The starting point is the Bernstein-decomposition of the category of smooth representations of a reductive $p$-adic group $G$ (where I focus on the case $G=GL_n$). I will shortly explain how this gives rise to a good understanding of the representation category in terms of Hecke algebras in the complex case. In the mod-l case (with $l\neq p$), new difficulties arise and the approach of the complex case can only partially be adopted: The majority of blocks in the Bernstein decomposition may fail to be Morita-equivalent to suitable Hecke-algebras. Following Vigneras, I will introduce the Schur-algebra and explain how this algebra is used to give a partial answer for the unipotent block, i. e. the subcategory in the Bernstein-decomposition which contains the trivial representation. I am working on a generalization of this proof, which should give a similar description for any block, using Schur- and Hecke-algebras. I will present recent progress and ideas.

## Academic year   2009-10

### Organizers:   M. Schein and A.Sh. Dahari

 DATE: 7/7 Speaker: Prof. Alexander Lashkhi (Georgian Technical University) Title: Geometry of classical groups over rings DATE: 23/6 (10:00-11:00) Speaker: Shifra Reif (Weizmann Institute of Science) Title: Denominator identities for Lie superalgebras Abstract. In 1972 Macdonald generalized the Weyl denominator identity to the case of affine root system. The simplest example of these identities turned out to be the famous Jacobi triple product identity. In 1994 V. G. Kac and M. Wakimoto conjectured an analog for affine Lie superalgebras and showed that it has applications in Number theory. In this lecture we discuss the progress made on these problems. A proof for the exceptional affine case D(2,1,a) will be given. From this case we conclude a formula for counting the number of representations of an integer as a sum of 8 squares. DATE: 23/6 (11:00-12:00) Speaker: Prof. Doron Shafrir (Hebrew University of Jerusalem) Title: Properties of the ring of invariant polynomials under an algebraic action Abstract. Given an algebraic linear representation, an important tool to study its orbits are the invariant ploynomials. We will see how properties of this ring can be calculated. For example, in case we have some invariants and we want to prove they generate the entire ring. We will also show a sum rule for the degrees of generators in case the ring of invariants is polynomial. (joint work with Anthony Joseph.) DATE: 9/6 --- Amitsur conference DATE: 2/6 --- IMU meeting DATE: 26/5 --- Mia Cohen's conference DATE: 19/5 --- SHAVUOT DATE: 12/5 --- YOM YERUSHALAIM DATE: 5/5 [joint meeting with the CGC and combinatorics seminars] Speaker: Dr. Tali Kaufman (Bar-Ilan University) Title: Symmetric LDPC codes and local testing Abstract. Local computation tasks (as local testing, correcting, decoding) are possible in codes based on polynomials. This is related to the fact that such codes are highly symmetric, yet they are defined by short linear equations. Codes defined by short linear equations are called LDPC. In the heart of this work is the following question: Could we have high rate codes which are highly symmetric, yet are defined by short linear equations? For example, polynomial codes which are LDPC have poor rate. In this work we construct codes with rate better than polynomial codes that are defined by short equations. Moreover we obtain bounds on the best rate of such symmetric codes. (joint work with Avi Wigderson) DATE: 28/4 Speaker: Prof. Stuart Margolis (Bar-Ilan University) Title: Building groups from idempotents: the algebra, geometry, and topology of idempotent-generated semigroups Abstract. Semigroups generated by idempotents are ubiquitous. Every semigroup embeds into one generated semigroup and a (finite) countable semigroup embeds into a (finite) semigroup generated by 3 idempotents. More importantly, idempotents of an arbitrary semigroup have a geometric structure called a biordered set. The Tits building of a group with (B,N) pair has a natural structure of biordered set, for example. There is a notion of a free idempotent generated semigroup on a biordered set and there has been some recent activity on calculating the maximal subgroups of free idempotent generated semigroups. Such groups arise as the fundamental groups of a certain 2 complex associated to a biordered set. We give some examples and an outline of the recent proof of Gray and Ruskuc that every (finitely presented) group is isomorphic to a maximal subgroup of some (finite) free idempotent generated semigroup. DATE: 21/4 Speaker: Prof. Lenny Makar-Limanov Title: The Jacobian conjecture in two variables - III Abstract. (see below) DATE: 14/4 Speaker: Prof. Lenny Makar-Limanov Title: A new approach to the two-dimensional Jacobian conjecture - II Abstract. (see below) DATE: 7/4 Speaker: Prof. Lenny Makar-Limanov Title: A new approach to the two-dimensional Jacobian conjecture - I Abstract. In my talks I will introduce and discuss some properties of a three-dimensional polytope which can be attached to a pair of polynomials with the constant Jacobian. This approach gives new restrictions on a potential counterexample to the Jacobian conjecture. DATE: 31/3 --- PESACH BREAK DATE: 24/3 --- PESACH BREAK DATE: 17/3 Speaker: Sergey Malev Title: Evaluation of non-commutative polynomials Abstract. Let p be a multilinear polynomial in several non-commuting variables with coefficients in an algebraically closed field K of arbitrary characteristic. It is conjectured that for any n, the image of p evaluated on the set M_n(K) of n by n matrices is either zero, or the set of scalar matrices, or the set sl_n(K) of traceless matrices, or the whole M_n(K). We prove the conjecture for n=2 and discuss the cases n=3 and n=4. (joint work with Belov and Rowen.) DATE: 10/3 Speaker: Dr. Crystal Hoyt (Weizmann Institute of Science) Title: TBA Abstract. Good Z-gradings of finite dimensional simple Lie algebras were classified by V.G. Kac and A.G. Elashvili in 2005. This problem arose in connection to W-algebras. We will discuss the classification of good Z-gradings for the basic Lie superalgebras: sl(m|n) : m\neq n, psl(n|n), F(4), G(3) and D(2,1,\alpha). The problem remains open for osp(m|2n). Â A finite dimensional simple Lie superalgebra g = g_0 \oplus g_1 is called basic if the action of g_0 on g_1 is completely reducible and there exists a nondegenerate invariant supersymmetric bilinear form on g. It follows that g_0 is a reductive Lie algebra. Basic Lie superalgebras were classified by V.G. Kac in 1977. A Z-grading of g, g = \oplus_{j in Z} g(j), is called good if there exists e in g_0(2) such that the map ad e from g(j) to g(j+2) is injective for j\leq -1 and surjective for j\geq -1. If e in g_0 belongs to an sl(2)-triple {e,f,h} where [e,f]=h, [h,e]=2e and [h,f]=-2f, then the Z-grading of g given by the eigenspaces of ad h is a good Z-grading for e. It is called a Dynkin grading. We find that all good gradings of the exceptional Lie superalgebras: F(4), G(3), and D(2,1,\alpha) are Dynkin gradings, and the good Z-gradings of sl(m|n) : m\neq n and psl(n|n) are classified using pyramids for gl(m|n), analogously to the case gl(m). DATE: 3/3 Speaker: Prof. Amitai Regev (Weizmann Institute of Science) Title: Standard polynomials are characterized by their degree and exponent Abstract. Given a p.i. algebra A, it has a sequence of codimensions c_n(A). A deep theorem of Giambruno and Zaicev says that as n goes to infinity, the limit of the n-th root of c_n(A) always exists, AND IS AN INTEGER. That integer, denoted exp(A), is called the exponent of the algebra A. Given a polynomial f, one considers U(f), the relatively free p.i. algebra satisfying f=0, then denotes exp(f):= exp(U(f)). Given n >= 6, we show that if f=St_n is the standard polynomial of degree n, and g is any polynomial of degree n which is not a multiple of St_n, then exp(g) < exp(St_n). (joint work with A. Giambruno) DATE: Jan. 27, 2010 Speaker: Dr. Noam Solomon Title: p-adic elliptic polylogarithms and the Leray spectral sequence for syntomic cohomology with coefficients Abstract. In a fundamental paper, Beilinson and Levin defined the elliptic polylogarithm, as a certain element in absolute or $l$-adic cohomology of an elliptic curve minus the identity. This element is motivic, in the sense that the consturction works for any reasonable'' cohomology theory. In this talk, we explain its realization in syntomic cohomology, which is a $p$-adic analog of Beilinson-Deligne cohomology. After a proper introduction, we recall the definition of syntomic cohomlogy (Besser, Bannai, Solomon). We then define $p$-adic analogs of variation of mixed Hodge structures, and state a Leray spectral sequence theorem in syntomic cohomology, which was used extensively in the realization of the $p$-adic elliptic polylogarithms.<\br>Finally, we define these $p$-adic elliptic polylogarithms. We remark here that one motivation for defining these elements is the formulation of a $p$-adic elliptic Zagier conjecture, which is a $p$-adic analog of the conjecture that certain values of $L$-functions of symmetric powers of elliptic curves is expressed using determinants of values of Eisenstein-Kronecker series. DATE: Jan. 20, 2010 Speaker: Dr. Eitan Sayag (Ben-Gurion University) Title: Distinction and functoriality Abstract. Let H be a subgroup of G and let (\pi,V) be a representation of G. We say that (\pi,V) is *H-distinguished *if the space of H-linear functionals Hom_{H}(V,\cc) is non-zero. We will review some results known about this space and the notion of distinction and state some results that indicate that this notion behaves nicely in regards to Langlands functoriality.(The lecture is based on my joint works with Omer Offen on Klyachko models, with A. Aizenbud on Gelfand pairs and on works in progress with Joe Hundley on Langlands functoriality.) DATE: Jan. 13, 2010 Speaker: Dr. Elad Paran Title: Hilbertianity of fields of power series Abstract. Let R[[X]] be the ring of formal power series over a domain R, and let F be the quotient field of R[[X]]. We prove that F is Hilbertian whenever R is contained in a rank-1 valuation ring of its quotient field. This gives a positive solution to an open problem of Jarden. As a corollary, we strengthen previous Galois theoretic results over such fields, obtained by Lefcourt, Harbater-Stevenson, Pop, and the speaker. The talk will include a background to these notions - no prior knowledge is needed. DATE: Dec. 30, 2009 Speaker: Prof. Yuval Flicker (Ohio State University) Title: The tame Hecke algebra Abstract. Let G be a reductive p-adic group. We introduce the tame subgroup I_t of the Iwahori subgroup I and the tame Hecke algebra H_t=C_c(I_t\G/I_t). We show that the tame algebra has a presentation by means of generators and relations, similar to that of the Iwahori-Hecke algebra H=C_c(I\G/I). From this we deduce that each of the generators of the tame algebra is invertible. This has an application to irreducible admissible representations pi of an unramified reductive p-adic group G, and it permits giving a Bernstein-type presentation of H_t. DATE: Dec. 23, 2009 Speaker: Prof. Elena Aladov Title: Isotyped algebras Abstract. Let $\Theta$ be an arbitrary variety of algebras and let $H$ be an algebra in $\Theta$.For every algebra $H$ one can consider its algebraic structure, its geometry and its logic. The interaction of these three components is the main idea of the theory under consideration.For the algebra $H$ we use the concept of a type from Model Theory, and we define isotyped algebras. Isotyped algebras are elementary equivalent but not necessarily isomorphic. An algebra $H\in\Theta$ is called separable in $\Theta$ if each $H' \in \Theta$ isotyped to $H$ is isomorphic to $H$. In particular, it means that such algebra can be distinguished from the other algebras using the logic of types. The main problems under consideration are related to separability of the free algebras from $\Theta$. All necessary definitions will be given. (joint work with B.I. Plotkin) DATE: Dec. 16, 2009 -- (Hanukka vacation) DATE: Dec. 9, 2009, 9:30 Speaker: Dr. Dmitry Kerner (Ben Gurion University) Title: On some new reflection groups appearing in singularity theory Abstract. It was a remarkable discovery by Arnol'd and many others that the Weyl groups (ADE) appear naturally in singularity theory of the simple types ADE. The reflection groups are realized as the groups of monodromy acting on the (co)homology lattice. Their space of orbits is naturally isomorphic to the miniversal deformation of the singular germs, the subset of irregular orbits is isomorphic to the discriminant, etc.Later, the correspondence has been extended to the groups $B_k,C_k,F_4$ and finally to all the Coxeter groups. Recently the game was continued by Goryunov, who generalized this to the groups of complex reflections. Many finite groups (classified by Shephard-Todd) and crystallographic groups (classified by Popov) have been shown to appear in singularities (in the sense above).I will give a general overview of the topic and summarize the current situation with the recent results. DATE: Dec. 2, 2009 Speaker: Prof. Andy Magid (University of Oklahoma) Title: The category of differential vector spaces over a differential field Abstract. Let F be a differential field with algebraically closed characteristic zero field of constants C. The category of F finite dimensional vector spaces with differentiations extending that of F is (anti) equivalent to the category of C finite dimensional modules for a certain proalgebraic group, the so-called absolute differential Galois group of F. This equivalence is an example of Tannaka duality; we give it explicitly as the functor co-represented (in both directions) by the ring of all solutions of all linear differential equations over F. DATE: Nov. 25, 2009 Speaker: Amichai Eisenmann (Hebrew University of Jerusalem) Title: Counting arithmetic subgroups in PSL_2 Abstract. In recent years there is much interest in giving estimates for the number of lattices in Lie groups of co-volume (with respect to the Haar measure) bounded by x \in \mathbb{R}. In the case of PSL_2 over a local field one has infinitely many lattices of a given co-volume, but it is known that the number AL(x) of arithmetic subgroups of co-volume bounded by x is finite. In the talk I intend to discuss two recent results giving a precise limit to the expression \frac{AL(x)}{x log(x)} as x goes to infinity. The first result concerns PSL_2 over the reals and is due to Belolipetsky, Gelander, Lubotzky and Shalev. The second concerns PSL_2 over a large family of p-adic fields including Q_p for p>3. (A.E.). DATE: Nov. 18, 2009 Speaker: Prof. Mikhail Borovoi (Tel-Aviv University) Title: Homogeneous spaces over number fields with finitely many rational orbits Abstract. Let G be a connected linear algebraic group over a number field K, let H be a connected K-subgroup of G, and set X=H\G. We give a convenient criterion to check whether the set of rational orbits X(K)/G(K) is finite, in terms of the Galois actions on \pi_1(H) and on \pi_1(G). Using this criterion, we classify symmetric homogeneous spaces of absolutely simple K-groups with finitely many rational orbits. DATE: Nov. 11, 2009 Speaker: Dr. Meirav Amram (Bar-Ilan) Title: Artin covers of the braid groups Abstract. Computation of fundamental groups of Galois covers recently led to the construction and analysis of Coxeter covers of the symmetric groups. In this talk we consider analogous covers of Artin braid groups. We conclude that there is a geometric extension of the Artin braid group, which is a semidirect product of it with a known group. At the end of the talk we mention another analogue of the theorem in the direction of classical Coxeter and Artin groups (B and D type). DATE: Nov. 4, 2009 Speaker: Prof. John Meakin (University of Nebraska) Title: Subgroups of monoids Abstract. One obtains significant information about the structure of large classes of monoids by studying their embedded subgroups. For example, a great deal of information about the full linear monoid of n by n matrices over a field (or more generally of a linear algebraic monoid) is determined by the subgroup structure of its group of units. As another example, Bass-Serre theory may be used to study the subgroup structure of certain amalgams of inverse monoids and this may be exploited to understand the structure of amalgams of some well known C*-algebras. The set of idempotents of a monoid carries the structure of a biordered set, first elucidated by Nambooripad in the 1970's. Topological techniques may be used to study the subgroup structure of monoids freely generated by biordered sets. In this talk, I will discuss some recent work and unsolved problems. DATE: Oct. 28, 2009 Speaker: Prof. Michael Schein (Bar Ilan University) Title: On irreducible supersingular mod p representations of GL_2(F) Abstract. Let F be a finite extension of Q_p. The mod p local Langlands correspondence should be a natural bijection between n-dimensional mod p representations r of the absolute Galois group of F and certain irreducible mod p representations L(r) of GL_n(F). Irreducible Galois representations correspond to supersingular representations of GL_n(F). The mod p representation theory of GL_n(F) is poorly understood, and, apart from some special cases, few irreducible supersingular representations have been constructed. One can use generalizations of Serre's conjecture to specify what the socle of the restriction of L(r) to a maximal compact subgroup should be. We will show that supersingular representations of GL_2(F) with such socles are generically irreducible and discuss work in progress to construct families of such representations. The relevant notions will be defined. DATE: Oct. 21, 2009 Speaker: Danny Neftin (Technion) Title: On semiabelian groups and the minimal ramification problem Abstract. Let p be a prime number and G a p-group of rank r, i.e. G can be generated by r elements and not less. It is conjectured that G can be realized as a Galois group over Q with exactly r ramified primes. Kisilevsky and Sonn showed the conjecture holds for a certain family of semiabelian groups and asked whether this family is the family of all semiabelian groups. This question was answered positively. We shall discuss the proof and how it can be used to simply the proof of Kisilevsky and Sonn and extend it. (joint work with Hershy Kisilevsky and Jack Sonn)

## Academic year   2008-9

### Organizers:   L.H. Rowen and M. Schein

 DATE: 8/9/2009, 10:30 Speaker: Prof. Agnieszka Bier (Silesian University of Technology, Poland) Title: Lattices of verbal subgroups in groups of triangular matrices Abstract. Let K be a field, T_n(K) the group of all invertible upper triangular matrices of size n x n over K, and UT_n(K) the subgroup of T_n(K) consisting of all unitriangular matrices. Denote by T_{\infty}(K) and UT_{\infty}(K) the inverse limits of T_n(K) and UT_n(K), respectively.In the talk we describe verbal subgroups in UT_n(K), T_n(K), UT_{\infty}(K), and T_{\infty}(K).Let L_n denote the chain of n integers with natural ordering. For an arbitrary field K, we prove that the lattice of verbal subgroups of UT_n(K) is isomorphic to L_{n - 1}. We also determine the width of certain verbal subgroups and discover interesting examples of verbal subgroups with finite width greater than 1. If K has characteristic zero, we prove that the lattice of verbal subgroups of T_n(K) is the join of the lattices of verbal subgroups of UT_n(K) and of K^x, where K^x is the multiplicative group of K, and that these two results also hold for n = \infty. DATE: 8/9/2009, 12:00 Speaker: Prof. Agnieszka Bier (Silesian University of Technology, Poland) Title: Finitely generated nilpotent groups with poor verbal structure Abstract. The talk concerns the problem of characterization of nilpotent groups of some type having poor verbal structure. A residually nilpotent group G is called verbally poor if every verbal subgroup coincides with a term of the lower central series in G. The lattice of verbal subgroups of such a group is isomorphic to a chain L_n, i.e. to the set of natural numbers of cardinality n with the natural order.One of the known examples of a verbally poor group is the group UT_n(K) of unitriangular matrices of size n x n over an arbitrary field K. This group may be finitely or infinitely generated, depending on K.Our goal is to provide necessary conditions for a finitely generated nilpotent group of nilpotency class c to have a lattice of verbal subgroups isomorphic to L_{c + 1}. We will sketch the proof of the following theorem: A verbally poor finitely generated nilpotent group is a finite p-group whose lower central series is a p-central series. DATE: 26/7/2009 (Sunday), 13:00 Speaker: Dr. A. Shmuel Dahari (Bar-Ilan) Title: A simple critertion for a quadratic field with negative discriminant to have class number 1 DATE: 24/6/2009 Speaker: Dr. Zur Izhakian Title: TBA DATE: 17/6/2009 Speaker: Prof. Eli Aljadeff (Technion) Title: On the Specht problem for G-graded algebras Abstract. In this lecture I will present some basic terminology and examples of G-gradings on finite dimensional algebras. Then I will present one idea which appears in the proof of the corresponding Specht problem for PI-affine G-graded algebras (finite generation of T-ideal of G-graded identities) where G is a finite group. DATE: 10/6/2009 Speaker: Dr. Siddhartha Sarkar (Hebrew University of Jerusalem) Title: Genus spectrum of finite groups Abstract. Let G be a finite group. A non-negative integer $g$ is called a genus of $G$ if $G$ acts faithfully on a compact orientable surface $\Sigma_g$ of genus $g$ preserving orientation. The set of all such possible genera $g \geq 2$ for a finite group $G$ is called the genus spectrum of $G$; after re-scaling it is called the reduced genus spectrum for $G$. The reduced genus spectrum of a given finite group $G$ contains all sufficiently large numbers. We will discuss the questions related to the problem of determining genus spectrum of finite groups. DATE: 3/6/2009 Speaker: Prof. Yakov Krasnov Title: Elements of spectral theory in algebras Abstract. The structure of the totality of a finite dimensional real (in general, non-associative) $m$-ary algebras (up to isomorphism) will be studied. The results, we obtain, are based on characterization the idempotents and/or nilpotents set in the algebras as well as on properties of theirs Peirce numbers. We prove using Atiyah-Bott fixed point theorem existence of main syzygy between idempotents and theirs Peirce numbers and show that this is an essential property in order to classify algebras. One of the purposes of this talk is to highlight some aspects of the "spectral theory for multilinear operators" mostly via parallelism with linear theory as well as to demonstrate how one can use such techniques in real $m$-ary algebras. Some of the results, like the bringing multiplication tensor in $m$-ary algebra to their canonical form, diagonalizing as well as using the notion of "quadratic dependence/independence" of vectors have not been well known. DATE: 27/5/2009 Speaker: Dr. Jonathan Beck (Bar-Ilan University) Title: Quantum algebras and the derived category of quiver modules Abstract. The nilpotent subalgebra of the quantum algebra is isomorphic to the Ringel-Hall algebra on the category C of modules over the corresponding Dynkin diagram. Various attempts have been made to realize the entire quantum algebra as a subcategory of the derived category of C. We discuss some recent work in this direction. DATE: 20/5/2009 Speaker: Dr. Arkady Tsurkov Title: The problem of the classification of the quasi-varieties of the nilpotent class 2 torsion free finitely generated groups is wild. Abstract. The varieties of nilpotent groups are Notherian, so the classification of the quasi-varieties generated by single nilpotent groups can be considered as the classification of these groups up to the geometric equivalence. The classification of the nilpotent class 2 torsion free finitely generated groups up to the geometric equivalence can be reduced to the classification up to the geometric equivalence of the finitely dimensional nilpotent class 2 Lie algebras over the field of the rational numbers. We can construct for every finitely dimensional nilpotent class 2 Lie algebra it's geometrically indecomposable envelopment and reduce the classification up to the geometric equivalence to the classification up to the isomorphism. After this it was be proved that this problem is wild. DATE: 13/5/2009 Speaker: Prof. Boris M. Schein (University of Arkansas) Title: Semigroups of cosets Abstract. Consider the set of all possible cosets of an arbitrary group G. The product AB of two cosets A and B is a subset of G that is not necessarily a coset modulo some subgroup, so we define a new product that is the smallest coset containing AB. Under this product the set of all cosets becomes an inverse monoid. This monoid and some of its properties serve as an introduction to the talk. Next we consider generalizations of the concepts of coset and semigroups of cosets from groups to wider classes of semigroups. As always in algebra, an immediate question is "WHY?" What is our motivation? Indeed we can define "cosets" in semigroups in many non-equivalent ways merely by aping certain properties of group cosets. So what? Who needs that?In this talk these questions are approached from a unified point of view for groups and semigroups. What are the most important groups? Groups of transformations. What are cosets from this point of view? If a group $G$ acts on a set A and a and b are two points in A, then all elements g in G that move a to b form a coset. This simple observation, if viewed in a proper context, is our starting point. DATE: 6/5/2009 Speaker: Dr. Mark Berman (Ben-Gurion University of the Negev) Title: Counting conjugacy classes in congruence quotients of GL_n(Z_p) Abstract. Let p be a prime and G a closed subgroup of GL_n. In this talk, I will consider the problem of determining the number of conjugacy classes b_k of the kth congruence quotient of G(Z_p) for each k. The sequence b_k was shown by du Sautoy to satisfy a linear recurrence relation. He achieved this by expressing the zeta function for (b_k) as a definable p-adic integral.I will show how to modify this approach to associate to the zeta function a more explicit p-adic integral. I will present several examples and show how this offers a new means to study the sequence (b_k). One goal is to relate properties of the sequence to questions about the representation growth of arithmetic groups. (joint work with Klopsch, Onn, Paajanen and Voll) DATE: 22/4/2009 Speaker: Prof. David Harari (Univ. Paris-Sud) Title: Rational points and Grothendieck's abelianized fundamental exact sequence Abstract. I will discuss the relationship between the existence of a rational point (for an algebraic variety defined over a p-adic field or a number field) and the existence of a section for an exact sequence of profinite groups related to Grothendieck's fundamental exact sequence. (joint work with Tamas Szamuely) DATE: 1/4/2009 Speaker: Prof. Alexander Kleschev (Univ. of Oregon) Title: Graded representation theory of symmetric groups and cyclotomic Hecke algebras Abstract. We explain how to grade the blocks of the group algebra of symmetric groups and related Hecke algebras. This opens up a prospect for studying graded representation theory of these objects and connects them to the recently defined Khovanov-Lauda-Rouquier algebras. Some of the applications are graded analogues of Ariki's categorification theorem and a conjecture of Lascoux-Leclerc-Thibon. DATE: 25/3/2009 Speaker: Prof. Stuart Margolis (Bar-Ilan) Title: Embedding monoids in groups: a classical and challenging problem Abstract. The problem of embedding a monoid in a group can be considered to be one of the most classical problems in mathematics, motivating, for example, the construction of the integers from the natural numbers. It is known that it is undecidable if a finitely presented monoid embeds in a group, so we can only hope for partial answers. It is clear that a necessary condition for a monoid to embed in a group is that it satisfy the cancellation laws. At one time this was thought to be sufficient, but Malcev constructed counterexamples and proved that there is no finite basis of first order implications that defines the class of submonoids of groups.We look at the Ore conditions that guarantee that a cancellative monoid has a group of fractions. This important class includes all cancellative monoids of polynomial growth, all those that satisfy some non-trivial identity, the Braid Monoids and, more generally, the class of Garside Monoids. We give a lovely proof of Ore's Theorem due to Rees using the theory of inverse monoids.There are analogous problems for embedding categories into groupoids that have important applications. Here we note that even though it is trivial that a submonoid of a finite group is itself a group, it is undecidable if a finite category embeds into a finite groupoid.We close by looking at other classes of embeddable cancellative monoids and raise a number of challenging open problems that the speaker would certainly like to work on when he grows up. DATE: 18/3/2009 (first talk in the second semester) Speaker: Prof. Manfred Knebusch Title: itle: Semirings with bounds Abstract. We call a (commutative) semiring R a semiring with upper bounds (or ub-semiring for short) if the addition on R gives a partial ordering on R such that, for any two elements x,y of R the sum x+y is an upper bound of x and y. (It may be bigger than the maximum of x and y which perhaps does not even exist). This new notion in semiring theory generalizes the notion of an upper bound group invented recently by Niels Schwartz. Ub-semirings are often highly noncancellative. Every ub-semiring can be degenerated to a semiring with 1+1 = 1, called an idempotent semiring.'' This degeneration means killing all archimedean phenomena in R. An important class of ub-semirings are the supertropical semirings invented by Zur Izhakian. There the archimedean properties are only nearly killed.'' DATE: 28/1/2009 (final talk for the first semester) Speaker: Dr. Chloe Perin Title: Elementary embeddings in free groups Abstract. We say that a subgroup H is elementarily embedded in a group G if first-order logic formulas with coefficients in H cannot distinguish between H and G. Studying first-order logic formulas over a group G can be seen as a generalisation of the study of equations over G. In these terms, an elementarily embedded subgroup can be thought of as the analogue of an 'algebraically closed' subgroup, for which every equation with coefficients in H which has a solution in G also has a solution in H.Sela showed that if F is a finitely generated free group, a non-abelian free factor F' of F is elementarily embedded in F. In my PhD thesis, I proved the converse, namely that the only elementarily embedded subgroups of F are its non-abelian free factors. After giving some background and examples on first-order logic and elementary embeddings, I will try to give some ideas of a proof of this result. The techniques used include Sela's analysis of the set Hom(G,F) of homomorphisms from a finitely generated group to a free group. DATE: 21/1/2009 Speaker: Prof. Boris Plotkin (Hebrew U.) Title: Unityped algebras Abstract. In the talk we give an extension of the ideas developed in B.Plotkin, G.Zhitomirski, "Some logical invariants of algebras and logical relations between algebras", St.Peterburg Math. J., {19:5}, (2008) 859 -- 879, whose main notion is that of logic-geometrical equivalence of algebras (LG-equivalence of algebras). This equivalence of algebras is more strict than elementary equivalence. We introduce the notion of unityped algebras and relate it to LG-equivalence. We show that these notions coincide. The idea of the type is one of the central ideas in Model Theory. The correspondence between types and LG-equivalence stimulates a bunch of problems which connect universal algebraic geometry and Model Theory. We touch the following topics: 1. General look 2.Logical noetherianity 3. Unitypeness and isomorphism 4. Logically perfect algebras 5. Some facts from algebraic logic. We provide a new general view on the subject, arising "on the territory" of universal algebraic geometry, which yield applications of algebraic logic and universal algebraic geometry in Model Theory. We give a list of new unsolved problems. DATE: 14/1/2009 Speaker: Prof. Daniel Wise (McGill, Visiting Hebrew U.) Title: The W-cycle Conjecture Abstract. I will describe an elementary problem which arises from consideration of a one-relator group < a, b | W >. The problem is about bounds on the number of cycles of a certain type in a labeled oriented graph. The problem seems destined for a few puzzlists in a Putnam competition. But while it wouldn't surprise me if someone came up with a gem of a proof, after 5 years, I've given up on finding a (direct) solution. I will present some partial results, and seek help from the audience towards a resolution or a counterexample... DATE: 7/1/2009, 10:00-11:00 Speaker: Prof. Mark Lawson (Department of Mathematics) Title: Representations of the polycyclic monoids Abstract. The polycyclic (or Cuntz) inverse monoids are amongst the most interesting classes of inverse monoids. They arise in settings as diverse as the syntactic monoids of correct bracketing languages in formal language theory and in the construction of the Cuntz C*-algebras. In this talk, I shall introduce these monoids from scratch and discuss their representations by means of partial permutations. This work has connections with papers by Bratteli and Jorgensen and of Kawamura on iterated function systems and representations of the Cuntz C*-algebras, and with constructions of certain of the Thompson groups. DATE: 7/1/2009, 11:00-12:00 Speaker: Prof. Steve Miller (Rutgers) Title: Rounding in Matrix Groups Abstract. I will describe the problem of rounding elements of Lie groups to finitely generated subgroups, which is a nonabelian generalization of lattice rounding in Euclidean space. Unlike that situation, one can prove results which rule out approximation algorithms as uniformly effective as LLL (unless P=NP). (joint work with Evgeni Begelfor and Ramarathnam Venkatesan) DATE: 31/12/2008 Speaker: Prof. Yuval Flicker (Ohio State University) Title: Counting Cuspidal Automorphic Representations Abstract. We compute the number of nowhere rami?ed cuspidal automorphic representations of the multiplicative group of a division algebra of prime degree over a global ?eld of positive characteristic in terms of the zeta function of the underlying curve. DATE: 24/12/2008 Speaker: Dr. Zak Mesyan Title: Minimal and Ideal Extensions of Rings Abstract. A ring S is said to be a minimal extension of a ring R if R is a subring of S and there are no subrings strictly between R and S. I will discuss minimal extensions of an arbitrary ring R and give a classification of all minimal extensions of R, in the case where R is prime. Further, a ring S is said to be an ideal extension of a ring R if S=R+I, where I is an ideal of S and the sum is direct. Ideal extensions play an important role in the classification mentioned above, and I will also present some results about the ideal extensions of an arbitrary ring. DATE: 17/12/2008 Speaker: Dr. Oren Ben-Bassat Title: Gerbes and the Holomorphic Brauer Group of Complex Tori Abstract. Holomorphic gerbes are certain geometric objects whose isomorphism classes form the second cohomology group of the sheaf of nowhere vanishing holomorphic functions. Locally a gerbe on a small open set should be thought of as something isomorphic to the collection of all line bundles. Actually the line bundles act on gerbes similarly to the way to functions act on sections of line bundles. In this talk, we will present some aspects of the study of gerbes on complex tori. This study is analogous to the classical study of line bundles on complex tori. Concepts such as the Appell-Humbert theorem, and the Poincare bundle and more will be presented in this new setting.Although this study is not strictly speaking algebraic geometry due to the use of the classical topology, it should be noted that there is a theorem due to Serre which says that the torsion part of the holomorphic Brauer group and the torsion part of the cohomological Brauer group in the etale topology are isomorphic.I will include an introduction to gerbes and also the other words in this abstract will be explained. DATE: 10/12/2008 Speaker: Prof. Eric Leichtnam (Centre de Mathematiques de Chevaleret) Title: Asymptotic equivariant index of Toeplitz operators and relative index of CR structures Abstract. We recall (for non-specialists) the statement of the Atiyah-Weinstein conjectures of the relative index of CR structures. Then we outline the main ideas of a new proof of it, using the so called Toeplitz operators. DATE: 3/12/2008 -- no meeting due to the HU Midrasha and the ENL2008 meeting. DATE: 26/11/2008, 10:00-11:00 Speaker: Dr. Igor Goldberg Title: The Finite Basis Problem for Certain Semigroups of Transformations Abstract. A finite semigroup S is said to be finitely based if there exists a finite set I of its identities such that every identity of the semigroup S can be deduced from identities in I. Given a class $\Sigma$ of finite semigroups, the finite basis problem for $\Sigma$ consists of determining which semigroups in $\Sigma$ are finitely based and which are not. The finite basis problem for finite semigroups has been studied intensively since the end of the 1960's. There is a number of classes of finite semigroups for which it remains open. We address the following: - the semigroups of all triangular n x n matrices over finite fields- the semigroups of all Boolean triangular n x n matrices and all Boolean unitriangular n x n matrices- some important classes of transformation semigroups of an n-element chain. DATE: 26/11/2008, 11:00-12:00 Speaker: Dr. Svetlana Goldberg Title: The Identity Checking Problem for Finite Semigroups Abstract. The identity checking problem Check-Id(A) for a finite algebra A is a combinatorial decision problem. Given a pair of terms (p,q) in the type of the algebra A, the question is whether the identity p = q is satisfied in A. Clearly the question is decidable, but the running time of the straightforward algorithm, in the worst case, depends exponentially on the size of the input identity. One the other hand, for any finite algebra A the problem Check-Id(A) belongs to the complexity class co-NP. In the talk I will give an overview of the computational complexity of checking identities for some kinds of finite algebras. The main part will be devoted to the Check-Id problem for finite semigroups, including 0-simple semigroups, transformation semigroups of a finite set, and the semigroup of all matrices over a finite field. DATE: 19/11/2008 Speaker: Prof. Alexey Zykin Title: On the generalized Brauer-Siegel theorem and limit zeta functions Abstract. The classical Brauer-Siegel theorem states that if $k$ runs through the sequence of normal extensions of $\mathbb{Q}$ such that $n_k/\log|D_k|\to 0,$ then $\log h_k R_k/\log \sqrt{|D_k|}\to 1.$ In this talk we will discuss possible generalizations of this result to the case of higher dimensional varieties over global fields. We will present several different versions of the Brauer-Siegel theorem depending on which special values of L-functions we are interested in. First, the one dealing with the asymptotic properties of the residue at $s=d$ of the zeta function in a family of varieties over finite fields. Second, following the track laid independently by M. Hindry and B. Kunyavskii together with M. Tsfasman, we will present certain results and open problems concerning the asymptotic behaviour of the first coefficient of the Taylor expansion at $s=1$ of the L-functions of elliptic curves. It turns out that in all the cases the technique of limit zeta functions is quite useful and gives a better understanding of what stands behind these results. DATE: 12/11/2008 Speaker: Prof. Jack Sonn (Technion) Title: On the minimal ramification problem for p-groups Abstract. Let $\ell$ be a prime number. It is not known if every finite $\ell$-group of rank $n>1$ can be realized as a Galois group over $\dQ$ with no more than $n$ ramified primes. We prove that this can be done for the family of finite $\ell$-groups which contains all the cyclic groups of $\ell$-power order, and is closed under direct products, wreath products, and rank preserving homomorphic images. This family contains the Sylow $\ell$-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not $\ell$. On the other hand, it does not contain all finite $\ell$-groups. DATE: 5/11/2008 Speaker: Dr. Lior Bary-Soroker (Hebrew U.) Title: Pseudo algebraically closed extensions of fields Abstract. The notion of Pseudo Algebraically Closed (PAC) extensions' is a generalization of the classical notion of PAC fields. It was originally motivated by Hilbert's tenth problem, and recently had several new applications. In this talk we shall introduce this notion and describe an approach to study the Galois structure of such PAC extensions. This approach is based on a generalization of embedding problems to field extensions.

## Academic year   2007-8

### Organizers:   L.H. Rowen and B. Kunyavskii

 DATE: 24/8/2008 (Sunday), at 12:00 -- special colloquium Speaker: Prof. Leonid Makar-Limanov Title: On strangeness of Weyl algebras in finite characteristic Abstract. Bavula recently conjectured that a homomorphism of A_n into A_n is an injection even in the finite characteristic case. Surprisingly it is very far from being true: Gelfand-Kirillov dimension of the image can be as small as n+1. So the new conjecture is that Gelfand-Kirillov dimension of the image cannot be n. DATE: 13/8/2008 Speaker: Prof. Lance Small Title: Old and New Problems in Noetherian Rings Abstract. A discussion of problems proposed at various meeting, etc. and what, if any, progress has been made. DATE: 30/7/2008 Speaker: Prof. Benjamin Steinberg Title: The Cerny Conjecture and Group Representations Abstract. Cerny conjectured in 1964 that if X is a set of maps on n letters so that some product from X is a constant map, then there is a product of length at most (n-1)^2 which is a constant map. This conjecture has received a lot of attention from people working in automata theory, but despite the intensive work on the subject the conjecture remains open. In this talk, we make some progress on this problem in the case some subset of X generates a regular permutation group G using the representation theory of G over the rational field. DATE: 9/7/2008 -- no seminar due to Amitsur Symposium at the Technion DATE: 25/6/2008 Speaker: Dr. Alexei Belov Title: Non-associative algebras DATE: 11/6/2008 Speaker: Prof. M. Borovoi (Tel-Aviv University) Title: Extended Picard complexes and linear algebraic groups Abstract. We are motivated by calculations of the Picard group and the Brauer group of a linear algebraic group. For a geometrically irreducible variety X over a field k of characteristic 0, we define a certain complex of Galois modules of length 2 UPic(X), which is related to the Picard group of X over an algebraic closure of k. We compute UPic(X) up to an isomorphism in the derived category when X is a linear algebraic group or a homogeneous space of such a group. (joint work with the late Joost van Hamel.) DATE: 4/6/2008 Speaker: Prof. B. Kunyavski Title: Rationality problems for the adjoint action of a reductive group Abstract. Given an action of a group $G$ on an integral $k$-variety $X$, one can ask whether the field extension $k(X)/k(X)^G$ is purely transcendental. In the case where $G$ is a connected reductive algebraic group defined over a field $k$ of characteristic zero, we give a nearly complete answer to this question in the following cases: a) $X=Lie(G)$ with the adjoint action of $G$; b) $X=G$ with the action of $G$ by conjugation. (joint work with J.-L. Colliot-Th\'el\ene, V.L. Popov and Z. Reichstein) DATE: 28/5/2008 Speaker: Prof. L. Rowen Title: tropical ring theory DATE: 21/5/2008 -- no seminar: Hirzebruch conference DATE: 14/5/2008 Speaker: Prof. Dmitry Piontkovsky Title: TBA DATE: 7/5/2008 -- no seminar: Yom Hazikaron DATE: 9/4/2008 Speaker: Prof. D. Gurevich (Valenciennes University, France) Title: Analyse and dynamical systems on q-Minkowski space Abstract. q-Minkowski space algebra is treated to be a special case of the so-called Reflection Equation Algebra (of A_n type). This algebra has many remarkable properties. In particular, it has a representation category looking like that of the super-Lie algebra gl(m|n). Besides, there exist some matricies with entries belonging to it subject to a noncommutative version of Cayley-Hamilton identity. All these properties enable us to write analogs of some dynamical equations. In parrticular, a q-analog of the Maxwell equation wil be discussed. DATE: 2/4/2008 Speaker: Prof. Roman Mikhailov Title: An inverse limit approach to homology theories DATE: 26/3/2008 Speaker: Eli Matzri (Bar-Ilan University) Title: Composition algebras and applications to central simple algebras Abstract. We will explain the relation between central simple algebras of degree 3 and composition algebras of degree 8, and use the theory of maximal isotropic subspaces of the latter, to study elements whose cubic power is in central in the former. DATE: 19/3/2008 Speaker: Prof. Roman Mikhailov (Steklov Institute) Title: Homotopical aspects of group theory DATE: 12/3/2008 Speaker: Prof. Manfred Knebusch Title: Prufer extensions - continuation DATE: 5/3/2008 Speaker: Prof. Manfred Knebusch Title: Prufer extensions; a new chapter in commutative algebra DATE: 27/2/2008 Speaker: Prof. Mikhail Muzychuk (Netanya College) Title: An application of Schur rings to a solution of polynomial moment problem Abstract. A polynomial moment problem is formulated as follows: given a complex polynomial P(z) and distinct complex numbers a and b, describe polynomials q(z) such that $\int_a^b {P(z)^i q(z)}dz = 0$ for all $i\geq 0$. It was realized that a solution of this problem depends essentially on the following question about permutation groups. Given a permutation group G containing a full cycle (1,2,..,n), find all G-invariant subspaces of the permutation module $\mathbb{Q}^n$. In my talk I'll show how this problem was solved using Schur rings technique. (joint work with F. Pakovitch) DATE: 24/2/2008, NOTE: Sunday, 14:00 [joint with the CGC seminar] Speaker: Fabienne Chouraqui (Technion) Title: Rewriting systems and embedding of monoids in groups Abstract. A connection between rewriting systems and embedding of monoids in groups is presented. We consider monoids and groups with the same presentation and we show that if the group admits a complete rewriting system, which satisfies the condition that each rule with positive left-hand side has a positive right-hand side, then the monoid admits also a complete rewriting system and it embeds in the group. As an example, we give a very simple proof that right angled Artin monoids, embed in the corresponding right angled Artin groups. This is a special case of a celebrated result of Paris that Artin monoids embed in their groups. DATE: 23/1/2008 Speaker: Prof. Vladimir Shchigolev (Moscow University) Title: Modular branching problem for the general linear group. Abstract. Let K be an algebraically closed field of characteristic p>0. Consider the natural embedding GL_{n-1}(K)\le GL_n(K) (in the top left corner). Let L(lambda) denote the rational simple GL_n(K)-module with highest weight lambda. We describe a combinatorial criterion for the existence of a nonzero vector in L(lambda) having weight lambda-d alpha, where 0 \le d \le p and alpha is a positive root, that is primitive with respect to GL_{n-1}(K). For this purpose, we introduce new lowering operators generalizing those introduced by A.S.Kleshchev. DATE: 16/1/2008 Speaker: Prof. Lev M. Shneerson (Hunter College of the City University of New York) Title: Polynomial growth in semigroup varieties Abstract. In 1989 M. Sapir posed a problem of describing all semigroup varieties in which every finitely generated semigroup has polynomial growth. We will discuss some new results following from the solution of this problem for an arbitrary nonperiodic variety defined by a system of identities over a finite set of variables. DATE: 9/1/2008 Speaker: Prof. Martin Kassabov (Cornell) Title: Presentations of symmetric groups Abstract. We study presentations of symmetric groups $Sym(n)$ from a quantitative point of view. Our main result provides somewhat unexpected answers to the following questions: how many relations are needed to define $Sym(n)$, and how short can these relations be? I will describe the main idea which allows us to produce presentations with a bounded number of relations (independent on $n$). I will also explain several tricks which further allows us to shorten the lengths of the relations and decrease their number. (joint work with R. Guralnik, W. Kantor and A. Lubotzky) DATE: 2/1/2008 ---No meeting: Representation Theory Conference. DATE: 26/12/2007 Speaker: Prof. Uri Bader (The Technion) Title: Unitary representations of fundamental groups Abstract. Consider the fundamental group of a compact negatively curved manifold. There is a class of unitary representations of that group arising in geometry, called boundary representations. It turns out that these representations are all irreducible, and one can assoicate with each a certain character in one to one manner. These characters are given by lengths of geodesics. In my talk I will describe these representations and characters, and will give the main ideas of the proof. (joint work with Roman Muchnik) DATE: 19/12/2007 ---No meeting. DATE: 12/12/2007 Speaker: Prof. Boris Kunyavski (Bar-Ilan) Title: The Bogomolov multiplier of finite simple groups DATE: 5/12/2007 Speaker: Prof. Grigory Mashevitzky (Ben-Gurion University) Title: The Finite Basis Problem and identities of transformation semigroups Abstract. We discuss the following theorem: the semigroup T(k,X) of rank at most k transformations on a set X has no finite basis of identities, if and only if k is finite, and either (k=2 and (|X|=3 or |X|=4)), or k>2. We also discuss possible applications of the developed methods to Tarski's finite basis problem and to the complexity of the Check Identity problem. DATE: 28/11/2007 Speaker: Dr. Eitan Sayag (Hebrew University) Title: Bernstein-Gelfand models for unitary representations of GL(n,F) where F is a p-adic field Abstract. Bernstein and Gelfand introduced the notion of a model for the representation theory of a compact group. Roughly speaking a model is a "natural" representation of $G$ which contains with multiplicity one all the irreducible representations of the group $G$. In the case of the finite groups $GL(n,F_{q})$ (or $S_{n}$) such models were studied in the 80s by Klaychko and Inglis-Saxl and found significant applications to combinatorics. Recently, we provided a model for the unitary representations of $GL(n,F)$ where $F$ is a p-adic field. Our initial motivation is the study of periods of automorphic forms (special values of $L$ functions) and of representations distinguished by subgroups. Our method use the classification (due to Moeglin and Waldspurger) of the discrete automorphic spectrum via residues of Eisenstein series and the classification (due to Tadic) of the unitary dual of $GL(n,F)$. The lecture will be of expository nature: we will overview the basic notions needed from the theory of representations of $p$-adic groups as well as from the theory of automorphic forms. (joint work with Omer Offen) DATE: 14/11/2007 Speaker: Shai Saroussi (Bar-Ilan) Title: Quasy-valuations of fields Abstract. Suppose $F$ is a field with valuation $v$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study quasi-valuations extending $v$; in particular, their corresponding rings and their prime spectrums. We prove that these rings satisfy INC, GU and GD over $O_{v}$; in particular, they have the same Krull Dimension and the size of the prime spectrum is bounded. We also prove that every such quasi-valuation is dominated by some valuation extending $v$. In addition, a 1:1 correspondence is obtained between exponential quasi-valuations and integrally closed quasi-valuation rings. DATE: 7/11/2007 Speaker: Prof. Eugene Plotkin Title: How we spent our summer vacation: Solvability and Burnside problems Abstract. In the talk we give a survey of Burnside-type problems and relate them to solvability property (joint work with B.Guralnick and A.Shalev, and with N.Gordeev, F.Grunewald, B.Kunyavski) DATE: 31/10/2007 Speaker: Prof. Stuart W. Margolis (Bar-Ilan) Title: How I Spent My Summer Vacation: Multiplying Idempotents on the Great Plains Abstract. Semigroups generated by their idempotents play a crucial role in semigroup theory. Every (finite) countable semigroup is embedded into a (finite) semigroup generated by 3 idempotents for example. A finite semigroup has a semisimple complex algebra if and only if it is Von Neumann regular and its idempotent generated subsemigroup has a semisimple complex algebra, in some sense a generalization of Maschke's Theorem to finite semigroups. The collection of idempotents of an arbitrary semigroup has the structure of a so called biordered set. For example, the collection of pairs of opposite parabolic subgroups of an algebraic group or a finite group of Lie type have this structure. There is a notion of a free idempotent generated semigroup on a biordered set and it was conjectured that the maximal subgroups of such a semigroup are all free. We give counterexamples to this conjecture that arise both in topology, by looking at a certain graph embedding on the surface of a torus and identifying maximal subgroups with the fundamental group of the torus and also look at the biordered set of full matrix monoids. The general technique is to identify the maximal subgroup as the fundamental group of a 2-complex associated naturally with the biordered set. No background in semigroup theory is needed, just the belief that perhaps multiplying idempotents in semigroups is a bit more interesting than the corresponding problem in group theory. (joint work with John Meakin.) DATE: 24/10/2007 Speaker: Prof. Erez Lapid Title: An algebra of relations for reduced decompositions of the longest element of the Weyl group Abstract. The algebra of piecewise polynomial functions with respect to a cone decomposition is a well-studied object. We will discuss an analogous algebra pertaining to reduced decompositions of the longest element of a finite Coxeter group. (joint work with Tobias Finis.)

## Academic year   2006-7

### Organizers:   L.H. Rowen and U. Vishne

 DATE: 4/7/2007 Speaker: Dr. Alexei Belov (Bar-Ilan) Title: Open problems for PI-algebras DATE: 27/6/2007 Speaker: Prof. Boris Plotkin (Hebrew U.) Title: Some open problems for associative algebras DATE: 20/6/2007 -- no meeting (Amitsur Symposium) DATE: 13/6/2007 Speaker: Prof. Jack Sonn (Technion) Title: Polynomials with roots mod n for all n Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q_p for all p, or equivalently, with roots mod n for all n. It is known that f(x) cannot be irreducible but can be a product of two or more irreducible polynomials, and that if f(x) is a product of m>1 irreducible polynomials, then its Galois group must be a union of conjugates of m proper subgroups. We prove that for any m>1, every finite solvable group which is a union of conjugates of m proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with m=2) holds for all nonsolvable Frobenius groups. For the full article see arxiv. DATE: 6/6/2007, 10:45 Speaker: Prof. Michael Larsen (Indiana and the Hebrew U.) Title: Images of word maps DATE: 30/5/2007 Speaker: Michael Schein (Hebrew University) Title: The Bernstein-Gelfand-Gelfand complex and weights in Serre's conjecture for GL_n Abstract. We will discuss the classical Bernstein-Gelfand-Gelfand resolution of representations of GL_n and how it can be adapted to obtain results about weights of n-dimensional Galois representations. DATE: 16/5/2007 -- no meeting (Jerusalem day) DATE: 9/5/2007 Speaker: Dr. Emanuelle Rodaro Title: Sch\"{u}tzenberger automata in the study of amalgams of finite inverse semigroups Abstract. Schutzenberger graphs and automata play the same role in inverse semigroup theory that Cayley graphs play in group theory. We give a description of the the Sch\"{u}tzenberger automata for the case of an amalgam of finite inverse semigroups. This gives also a proof that the word problem for an amalgam of finite inverse semigroups is decidible. In contrast the word problem for an amalgam of finite semigroups is undecidable in general Moreover the particular structure of the underlying graph of these automata allows one to deduce many properties of the maximal subgroups of the amalgams of finite inverse semigroups. DATE: 2/5/2007 Speaker: Prof. Harry Dym (Weizmann Institute) Title: One analyst's thoughts on teaching linear algebra DATE: 25/4/2007 -- cancelled, due to EckmannFest at the Technion. DATE: 18/4/2007 Speaker: Elad Paran (Tel-Aviv University) Title: Embedding problems over complete domains Abstract. We prove that every finite split embedding problem over the field K((x_1,...,x_n)) of formal power series in n\geq 2 variables (over an arbitrary field K) is solvable. This generalizes a theorem of David Harbater and Kate Stevenson, who settled the case K((x_1,x_2)). The methods developed give new insights into Galois theory over several large families of fields. DATE: 11/4/2007 Speaker: Prof. Sara Westreich (Bar-Ilan University) Title: Fourier transforms and the Verlinde algebra for Hopf algebras DATE: 21/3/2007, 10:30 Speaker: Dr. Elena Aladova Title: Polynomial identities in nil-algebras over a field of a prime characteristic Abstract. Let $\mathbb F$ be a field, let $A$ be a free associative algebra (without 1) over $\mathbb F$ on free generators $x_1, x_2, \dots$ and let $R$ be an associative $\mathbb F$-algebra. Let $f(x_1,\dots,x_n) \in A$. We say that $f(x_1,\ldots ,x_n)=0$ is a {\it polynomial identity} in $R$ if $f(r_1,\ldots,r_n)=0$ for all $r_1,\dots,r_n\in R$. Two systems of polynomial identities are {\it equivalent} if every associative $\mathbb F$-algebra satisfying all the identities of the first system satisfies all the identities of the second system and vice versa. If a system of polynomial identities is equivalent to some finite system we say that the system is {\it finitely based} or {\it has a finite basis}. We study the following {\bf Problem.} {\it Find the smallest positive integer $n=n(\mathbb F)$ such that the identity $x^{n}=0$ can be included in the non-finitely based system of polynomial identities in associative algebras over a field $\mathbb F$ of a prime characteristic.} Let $\mathbb F$ be a field of characteristic $p\ge 5$, let $[x,y] = xy - yx$, $f(x_1,x_2)=x_1^{p-1}x_2^{p-1}[x_1,x_2]$ and let $$w_n = [[x_1,x_2],x_3] f(x_3,y_3)\cdots f(x_{n},y_{n}) [[y_1,y_2],y_3] ([[x_3,x_1],x_2] [[y_3,y_1],y_2])^{p-1}.$$ Our main result (with A.Krasilnikov) is as follows: Over a field $\mathbb F$ of characteristic $p\ge 3$ the system of polynomial identities $$\{ x^{2p} = 0 \} \cup \{ w_n = 0 \mid n = 3,4, \dots \}$$ is not equivalent to any finite system of identities in associative $\mathbb F$-algebras. DATE: 14/3/2007, 10:00 - note special time Speaker: Prof. Moshe Jarden (Tel-Aviv University) Title: The absolute Galois group of the field of totally $S$-adic numbers Abstract. For a finite set $S$ of primes of a number field $K$ and for $\sigma_1,\ldots,\sigma_e\in\Gal(K)$ we denote the field of totally $S$-adic numbers by $K_{\tot,S}$ and the fixed field of $\sigma_1,\ldots,\sigma_e$ in $K_{\tot,S}$ by $K_{\tot,S}(\bfsig)$. We prove that for almost all $\bfsig\in\Gal(K)^e$ the absolute Galois group of $K_{\tot,S}(\bfsig)$ is the free product of $\Fhat_e$ and a free product of local factors over $S$. DATE: 7/3/2007, 10:30 Speaker: Prof. Louis Rowen (Bar-Ilan University) Title: Zariski closure of algebras DATE: 28/2/2007, 10:30 Speaker: Prof. Boris Plotkin (Hebrew University) Title: Algebraic logic and some logical invariants of algebras Abstract. We consider algebras from a fixed variety of algebras. In the talk we define different geometrical and logical invariants of algebras. We also consider geometrical and logical relations between algebras. All these notions rely on the structures of algebraic logic. All necessary definitions will be given. DATE: 17/1/2007, 10:00 Speaker: Prof. Lev M. Shneersohn (Hunter College, NY, NY) Title: Types of Growth in Semigroup Varieties Abstract. We study the growth, Gelfand-Kirillov dimension and superdimension of a finitely generated semigroup satisfying a given system of identities. DATE: 17/1/2007, 11:00 Speaker: Prof. Elena Klimenko Title: The geometry and a parameter space of Kleinian groups. Abstract. I will talk about PSL(2,C), which can be identified with the full group of orientation preserving isometries of hyperbolic 3-space. The discrete subgroups of this group are called Kleinian groups, and their orbit spaces are Kleinian orbifolds. Bianchi groups are examples of finite co-volume Kleinian groups which show the interplay between the number theory and geometry. We will concentrate mainly on the geometry of Kleinian groups and give a taste of how complicated the structure of the parameter space of 2-generator Kleinian groups is by showing a slice through this space. (joint work with Natalia Kopteva.) DATE: 3/1/2007 -- cancelled Speaker: Prof. Victor Kac (MIT) Title: Quantization and chiralization Abstract. I will discuss algebraic structures arising in four fundamental physical theories and relations between them DATE: 27/12/2006 Speaker: Dr. Elena Kireeva (Weizmann Institute) Title: On T-spaces in associative algebras. Abstract. Let K be a commutative and associative ring with 1, A be a free or a relatively free associative algebra over K. A K-submodule U in A is called a T-space if U is a fully characteristic submodule, that is U is closed under all endomorphisms of the algebra A. We plan to give an overview of the results related to finitely generated and non-finitely generated T-spaces in free algebras of varieties of associative algebras. DATE: 20/12/2006 Speaker: Prof. Eugene Plotkin (Bar Ilan) Title: On the solvable radical of a finite group Abstract. Let $G$ be a finite (linear) group. In the talk we give an overview of new results related to descriptions of the solvable radical $R(G)$ of the group $G$. In particular we will discuss theorems which characterize $R(G)$ via commutators and conjugates.These theorems are similar to the Baer-Suzuki Theorem which characterizes the nilpotent radical of a finite group via conjugates. (joint work with N.Gordeev, F.Grunewald and B.Kunyavskii) DATE: 13/12/2006 Speaker: Prof. Jean-Marie Bois Title: Generators for simple Lie algebras in arbitrary characteristic Abstract. Around the year 2000, Guralnick and Kantor proved a theorem stating that finite simple groups are generated by one and a half elements'': in other words, for any nontrivial elements x in a finite group G, there exists another element y such that $\{x,y\}$ generates G. In particular, and finite simple group is generated by 2 elements. In this talk we will expose some partial results towards the analogue question in the Lie-algebraic setting. We will show that classical simple Lie algebras are indeed generated by one and a half elements. We will then proceed to show that graded Cartan type Lie algebras are generated by 2 elements; we will also give a necessary and sufficient condition for a simple algebra of type W to be generated by one and a half elements. DATE: 6/12/2006 Speaker: Prof. Leonid Makar-Limanov (Wayne State University) Title: An algebraic proof of the Abhyankar-Moh-Suzuki theorem. Abstract. What can be said about two polynomials in one variable with complex coefficients if the subalgebra which they generate is all polynomial ring? The answer is well-known from the mid-seventies: the smaller degree should divide the larger degree. In my talk I'll give a new algebraic proof of this fact which is, at least from my point of view, is simpler than the known proofs. DATE: 29/11/2006, 11:00 Speaker: Luda Markus Title: Stallings' Foldings and Subgroups of Amalgams of Finite Groups. Abstract. A well known result of J.Stallings states that every finitely generated subgroup of a free group can be canonically represented by a finite labeled graph (a finite minimal immersion of a bouquet of circles = a minimal finite inverse automaton). This object can be constructed algorithmically by the process of Stallings' foldings. It turns out that the same happens for finitely generated subgroups of amalgams of finite groups. Namely, they can be effectively represented by finite canonical graphs. These graphs posses all the essential information about the subgroups, which enables one to use them in order to solve various algorithmic problems: the membership problem, the finite index problem, the conjugacy problem, the freeness problem, the separability problem (M.Hall theorem) and others. We'll discuss the construction of such subgroup graphs and their applications for the solutions of some algorithmic problems from the above list. DATE: 22/11/2006 Speaker: Prof. Pierre Koseleff (Universite Paris 6) Title: On polynomial Torus Knots. Abstract. We show that no torus knot of type $(2,n)$ (n odd) can be obtained from a polynomial embedding $t \mapsto ( f(t), g(t), h(t) )$ where $(\deg(f),\deg(g))\leq (3,n+1)$. Eventually, we give explicit examples with minimal lexicographic degree. Our method is based on the fact that all plane projections of this torus $K_n$ knot with the minimal number $n$ of crossings have essentially the same diagram. This is a consequence of the now solved classical Tait's conjectures. This allows us to transform our problem into a problem of real polynomial algebra. You can find a related paper on http://arxiv.org/pdf/math.AG/0610663 (joint work with D. Pecker.) DATE: 15/11/2006 Speaker: Prof. Stuart Margolis (Bar-Ilan) Title: On Idempotent and Idempotent Generated Semigroups Abstract. Semigroups consisting of or generated by idempotents arise naturally in many parts of semigroup theory and its applications. In some sense they are as far away from groups as possible and thus require tools of study that are particular to semigroup theory.In this talk, we begin with a survey on idempotent semigroups. The main result due to Green and Rees is that a finitely generated idempotent semigroup is finite. The free idempotent semigroup on a set X can be identified with the set of permutation trees on subsets of X- that is the collection of binary trees labelled by elements of X and such that every path from root to leaf is a permutation. This allows us to count the number of elements in this semigroup.Semigroups generated by idempotents are much more complex.For example, every (finite) countable semigroup embeds into a (finite) semigroup generated by 3 idempotents. Nambooripad characterized the idempotents of a (Von Neumann) regular semigroup as a biordered set. Remarkably, classical results of the theory of groups with BN-pairs were used by Putcha to show that the set of pairs of opposite parabolic subgroups of such a group has the structure of a biordered set and thus are isomorphic to the collection of idempotents of a regular semigroup. There is a free idempotent semigroup on a biordered set and we show how to compute its maximal subgroups by identifying these as the fundamental group of a certain two-complex associated with the idempotents. DATE: 8/11/2006 Speaker: Dr. Timo Hanke (The Technion) Title: Galois covers of cyclic extensions with full local degree Abstract. Let $K/k$ be a finite cyclic $p$-extension of global fields, $p$ prime, and let $m$ be a non-negative integer. By a $p^m$-Galois cover of $K/k$ we mean a Galois extension $M/k$ that contains $K$ and has degree $[M:K]=p^m$. The term "full local degree" refers to $M/K$ having local degree as large as possible at a given prime of $K$. We ask whether for any finite set $S$ of primes of $K$ one can find a $p^m$-Galois cover of $K/k$ which has full local degree at all primes in $S$. The question is answered by a suitably defined height-function of $K/k$ for which we can give a formula in terms of (non-arithmetic!) invariants of the extension $K/k$. Applications lie in characterizing which division algebras over function fields over global fields are crossed products. (joint work with Jack Sonn) DATE: 1/11/2006 Speaker: Prof. Boris M. Schein (Arkansas) Title: Semigroups of reflexive or transitive binary relations Abstract. A product of two reflexive binary relations between the elements of a set is a reflexive binary relation. Thus, the set R(A) of all reflexive relations on a set A is a semigroup. We consider abstract semigroups S that can be isomorphically embedded into R(A) for an appropriate set A. It turns out that the class of such abstract semigroups forms a quasi variety (but not a variety) of semigroups described by a simple scheme of quasi-identities. However, this classof semigroups is not finitely axiomatizable. \\ A product of two transitive binary relations is not necessarily transitive. Thus the set T(A) of all transitive binary relations on a set A does not form a semigroup. Yet certain subsets of T(A) may be semigrouops closed under the ordinary product of relations. We consider abstract semigroups S that can be isomorphically mapped onto a subsemigroup of T(A) for an appropriate set A. It turns out that, as in the previous case, the class of such abstract semigroups forms a quasi variety (but not a variety) of semigroups described by a simple scheme of quasi-identities. However, this class of semigroups is not finitely axiomatizable.\\ The results are based on research of the speaker but some early results of K.A. Zaretsky obtained in the end of the 1950-ies will be mentioned. DATE: 25/10/2006 Speaker: Eli Matzri Title: All dihedral algebras of degree 5 are cyclic

## Academic year   2004-5

### Organizers:   L.H. Rowen

 DATE: 9/11/2005 Speaker: Prof. Stuart W. Margolis (Department of Mathematics, Bar-Ilan University) Title: Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory Abstract. In this talk we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature involving triangularizability of finite semigroups, characterizing finite semigroups that have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids, unambiguous products of rational formal languages, products of rational languages with counter and Cerny's conjecture. (joint work with Jorge Almeida, Benjamin Steinberg and Mikhail Volkov.) DATE: 6/7/2005 Speaker: Prof. Yuri Bazlov Title: Braided Heisenberg doubles and Cherednik algebras Abstract. Rational Cherednik algebras (= rational double affine Hecke algebras) were formally introduced by Cherednik, but in fact appeared in an earlier work of Drinfeld. Their representation theory was systematically studied by Etingof, Ginzburg, Opdam and others. We propose a new construction which we call braided Heisenberg double', and show that Cherednik algebras are obtained by deforming suchdoubles. This provides a new proof of properties of Cherednik algebras such as PBW decomposition, and suggests a more general version of the Cherednik algebra. (joint work with Arkady Berenstein) DATE: 15/6/2005 Speaker: Prof. Inna Korchagina (Birmingham) Title: Characterization and Identification of some simple groups DATE: 8/6/2005 Speaker: Dr. Michael Natapov (Technion) Title: On Projective Representations of Nilpotent Groups Abstract. Let $k$ be a field. For each finite group $G$ and cohomology class $\alpha$ in $H^2(G,k^\times)$ (with trivial action), one can form the twisted group algebra $k^\alpha G$. We refer to $G$ as a {projective basis} of $k^\alpha G$. There is a complete classification of groups which are projective bases of division algebras. Namely, there is a short list of groups $\Lambda$ such that $G$ is a projective basis of some $k$--central division algebra if and only if $G$ in $\Lambda$ (Aljadeff, Haile, N). In particular, $G$ is nilpotent and its commutator subgroup is cyclic. Let $A$ be a $k$--central simple homomorphic image of $k^\alpha G$, where $G$ is nilpotent. We show that the structure of $A$ depends on subquotients of $G$ which have cyclic commutator subgroups or are in the list $\Lambda$. In particular, we obtain a bound on index of $A$ in terms of such subquotients. DATE: 1/6/2005 Speaker: Prof. Ron Adin (BIU) Title: Permutation Statistics and Group Actions Abstract. Length (inversion number) and major index are two important statistics on permutations. By a classical result of MacMahon, they have the same generating function ("equi-distributed") over the symmetric group. In joint work with Y. Roichman (answering a question of Foata), we found a natural parameter ("flag major index") which is equi-distributed with length on the symmetry group of the cube. It also serves to extend a more refined result of Carlitz (joint work with F. Brenti and Y. Roichman). In the talk we shall review these and other related results, such as analogues for other groups, algebraic interpretations, and bijective proofs (in works of Bagno, Biagioli, Bernstein, Foata, Han, and Regev). DATE: 25/5/2005 Speaker: Dr. Andrei Reznikov (BIU) Title: Uniqueness of invariant functionals and bounds on automorphic L-functions Abstract. We show how the uniqueness of certain invariant functionals on irreducible representations of GL(2,R) could be used in order to obtain non-trivial bounds onL-functions, periods and Fourier coefficients of cusp forms. Joint with J. Bernstein. DATE: 18/5/2005 Speaker: Prof. Nikolai Gordeev (St. Perersburg) Title: Sums of orbits of algebraic groups Abstract. In this talk we deal with sums of orbits of simple algebraic groups acting linearly on finitely dimensional vector spaces. We consider such problems as estimates of covering numbers, an analog of J.Thompson problem for conjugacy classes of simple groups, and some related questions. DATE: 13/4/2005 Speaker: Prof. Moshe Jarden (TAU) Title: Unit-Difference Sequences Abstract. A sequence $u_1,u_2,u_3,\ldots$ of elements in a commutative ring $R$ is said to be a {\bf unit-difference sequence} if $u_j-u_i$ is a unit of $R$ for all $j>i$. We prove that for almost all $\sigma_1,\ldots,\sigma_e\in\Gal(\bbQ)^e$ the ring $\tilde{\mathbb Z}(\sigma_1,\ldots,\sigma_e)$ has an infinite unit-difference sequence. DATE: 13/4/2005 Speaker: Prof. Michael Tsfasman (Poncelet Russian-French laboratory (CNRS and the Independent University of Moscow) and IML (CNRS, Marseille)) Title: Euler-Kronecker constant for number fields Abstract. The number theory has long ago passed from the study of rational numbers to that of algebraic numbers, i.e., to the study of finite extensions of the field of rational numbers Q. Almost all notions find interesting generalizations here. We shall discuss the analogue of the Euler constant gamma. Then we shall make another step forward, passing to infinite extensions of number fields. This will not only create a new theory but also help us to understand the asymptotic behaviour of the Euler constant when the number field grows. DATE: 6/4/2005 Speaker: Prof. Arkady Tsurkov Title: Weak geometric equivalence and action type weak geometric equivalence of representations. Abstract. Recently B. Plotkin and G. Zhitomirski described all the automorphisms of the category of free representation. By using of this description the reducing of the problem of the weak geometric equivalence to the problem of geometric equivalence (both in the regular sense and in the action type sense) will be proved in my lecture. DATE: 30/3/2005 Speaker: Dr. Uzi Vishne (BIU) Title: Characters and solution to equations in finite groups Abstract. A classical result of Frobenius (1896) states that the number of solutions to the equation [x,y]=g in a finite group G, considered as a function of g, is a character (namely an integral combination of irreducible characters, with positive coefficients). More generally, we study the number N_w(g) of solutions to the equation w(x_1,...,x_t)=g for x_1,...,x_t in G, where w() is an arbitrary word on t letters. A typical problem: when is N_w a character? a virtual character? Our main results concern N_w for w=[x_1,...,[x_{t-1},x_t],...], a generalize commutator. Generalizing Frobenius' theorem, we show that in this case N_w is indeed a character (computed rather explicitly). Applying probabilistic arguments (which can be phrased as saying that all finite groups are `probabilistically nilpotent') we compute the operator norm of a certain matrix derived from the character table of G. Moreover, we show how better bounds on certain character sums can provide new types of subgroup growth. (joint work with Alon Amit) DATE: 23/3/2005 Speaker: Prof. Boris Kunyavski (Bar Ilan) Title: Picard and Brauer groups of smooth compactifications of homogeneous spaces Abstract. Let k be a field of characteristic zero. Let Y=G/H, where G is a semisimple simply connected algebraic group over k and H is a connected closed k-subgroup of G. Let T be the maximal k-torus quotient of H. Let X be a smooth compactification of Y over k. We give a formula for the Brauer group of X in terms of the Galois cohomology of the character group of T. The geometric Picard group of X is a lattice equipped with an action of the absolute Galois group of k. We conjecture that this Galois lattice is flasque. We prove partial results in this direction, and we reduce the general case to a conjecture on the bad reduction of certain homogeneous spaces. Our proofs involve detours over local and global fields. The results extend to homogeneous spaces under G which need not have a rational point. (joint work with J.-L. Colliot-Thelene) DATE: 16/3/2005 Speaker: Prof. Inna Korchagina (Hebrew U.) Title: Classification of Finite Simple Groups: Aspects of Second Generation Proof Abstract. The classification of finite simple groups is widely acknowledged to be one of the major results in modern mathematics. The successful completion ofits proof was announced in the early 1980's by Daniel Gorenstein. The original proof occupied somewhere around 15,000 journal pages spread across more than 500 separate articles written by more than 100 mathematicians. Shortly thereafter, a "revision" project has been started by Gorenstein, Lyons and Solomon. Its goal is to produce a new unified correct proof of the Classification Theorem of less than 5,000 pages in length. The strategy of the revision proof differs from the original one. In this talk we will outline the "Generation 2"-proof of the Classification, and discuss a specific part of it, in which the speaker is involved. DATE: 9/3/05 Speaker: Prof. Louis Rowen (BIU) Title: On the multiplicative group of quaterinion algebras (joint work with Yoav Segev) DATE: 2/3/2005 Speaker: Prof. Eugene Plotkin (BIU) Title: On solvable radical of a finite group Abstract. We will discuss the recent progress in the description of the solvable radical of a finite group. We will also consider the case of finite dimensional Lie algebras. DATE: 12/1/2005 Speaker: Prof. Eli Aljadeff Title: The Schur and projective Schur subgroups of the Brauer group Abstract. In the first part of the lecture I'll give the necessary definitions and basic results obtained in joint work with J. Sonn. In the second part I'll explain some new results obtained with J. Sonn and A. Wadsworth. DATE: 12/1/2005 Speaker: Dr. Yair Glasner Title: Maximal subgroups of lattices in SL_2(C) are either finite index or infinitely generated Abstract. Let G be a lattice in SL_2(C), (i.e. a fundamental group of a finite volume 3-dimensional hyperbolic orbifold). Let H < G be a maximal subgroup of infinite index, (or alternatively a pro-dense subgroup), we prove that H cannot be finitely generated. A pro-dense subgroup is one that maps onto every proper quotient of the group. The theorem about pro-dense subgroups was conjectured by myself and Tsachik Gelander, in the more general setting of hyperbolic groups. The theorem about maximal subgroups was was suggested as a question about lattices in simple (but not semi-simple) Lie groups by Margulis and Soifer. So our work answers a special case of both these questions.The proof splits into two parts. We first prove the theorem under the additional assumption that H is geometrically finite (= quasi-convex) and then we prove that H has to be geometrically finite. The first part involves some nice hyperbolic geometry. For the second part we appeal to Ian Agol's recent solution of the Marden conjecture. This result basically classifies all subgroups that are finitely generated and not geometrically finite. (joint work with Pete Storm and Juan Souto) DATE: 5/1/2005 Speaker: Prof. Issai Kantor Title: Peirce decomposition of Jordan triple systems DATE: 29/12/2004 Speaker: Prof. Alexander Shapiro (Bar Ilan) Title: Explicit formulas for applications of Bezout matrices Abstract. There are three basic applications of Bezout matrices:- Jacobi-Darboux theorem determines the number of common zeroes of two polynomials.- Hermite theorem gives the number of zeroes of a polynomial in the upper half-plane of the complex plane. - Kravitsky theorem presents the explicit formula for the polynomial that defines the image of the complex plane under a rational transformation. We give new proofs of two first classical results and present explicit formulas for the coefficients of the inverse of Bezout matrix. DATE: 15/12/2004 Speaker: Prof. A.S. Sivatski (St. Petersburg) Title: Nonexcellence of the function field of the product of two conics Abstract. Let $k_0$ be a field, $\chr k_0\not= 2$, $\a ,\b$ $2$-fold Pfister forms over $k_0$. Denote by $[\a ]$, $[\b ]$ the classes of the corresponding quaternion algebras in $_2 Br (k_0)$, and by $X_{\a}$, $X_{\b}$ the corresponding projective $k_0$-conics. Suppose that $\ind\ ([\a ] + [\b ])=4$. We construct a field $F$ over $k_0$ such that the field extension $F(X_{\a}\times X_{\b})/F$ is not excellent. Moreover, we find a $2$-fold Pfister form $\gamma$ over $F$ such that $\ind\ ([\a ] +[\b ] + [\gamma ] )=4$ and the homology group of the complex $$F^*/{F^*}^2\o_{\zz} U\rightarrow H^3(F,\Bbb Z/2\Bbb Z )\rightarrow H^3(F(X_{\a}\times X_{\b}\times X_{\gamma} ),\Bbb Z/2\Bbb Z )$$ is $\Bbb Z/2\Bbb Z$, where $U$ is the subgroup of $_2 Br (F)$ generated by $\a$, $\b$, $\gamma$, the first map is induced by the cup product and the second by the inclusion of the fields. In particular, this implies that for any odd $m$ the forms $\a$, $\b$ and $\gamma$ have no common splitting field of degree $4m$ over $F$. Also it follows that $Tors\ CH^2 (X_{\a}\times X_{\b}\times X_{\gamma})=\Bbb Z/2\Bbb Z$. DATE: 1/12/2004 Speaker: Prof. Shmuel Rosset (Tel Aviv) Title: An interesting subgroup of the quaternion algebra DATE: 24/11/2004 Speaker: Dr. Robert Schwartz (Technion) Title: Certain equations of length 6 over one relator free products. DATE: 17/11/2004 Speaker: Eli Matzri Title: Azumaya algebras over semilocal rings DATE: 10/11/2004 Speaker: Prof. Martin Markl Title: Variations on Deligne Conjecture Abstract. One of formulations of the Deligne conjecture states the existence of a natural action of a chain version of the little discs operad on the Hochschild cochain complex of an associative algebra. This conjecture, which certainly does not sound very attractive, has many interesting and surprizing applications, for example in Kontsevich's formality theorem. I will sketch out a proof of this conjecture proposed by Tamarkin that uses the quantization procedure by Etingof and Kazhdan. I will also discuss variations on the theme of this conjecture that emphasize the problem of understanding natural operations on chain complexes, and mention some open problems.Though the nature of the above topics is obviously technical, I will try to avoid details and stress the conceptual part of the story as much as possible. DATE: 3/11/2004 Speaker: Prof. Sara Westreich (Bar-Ilan) Title: Old/new constructions of quasitriangular quantum groups of type A_n Abstract. We study the pointed Hopf algebras U(R_Q) obtained by the FRT construction. We show that Hopf algebras arising as U(R_Q) are of type A_n. Two such Hopf algebras are twists of each other if and only if they possess the same groups of grouplike elements. For n=2 we compute the groups arising as G(U(R_Q)). DATE: 27/10/2004 Speaker: Prof. Malka Schaps (Bar-Ilan) Title: The Chuang-Rouquier theorem on derived equivalence for blocks of the symmetric group, using quantum group methods. Abstract. (no prior knowledge of quantum groups or representations of the symmetric group will be assumed.) DATE: 20/10/2004 Speaker: Dr. A.J. Kanel-Belov (Hebrew university) Title: On the generalized cancellation conjecture Abstract. Zariski posed the following question: Suppose $K_1(t_1)\equiv K_2(t_2)$, where $K_i$ are fields. Does it follow that $K_1\equiv K_2$? A special case is when $K_2$ is field of rational functions. The similar questions can be posed for rings. A counterexample to the Zariski cancellation conjecture was found even in a special case; a 3-dimensional field $K$ over $\Bf C$ such that $K(t_1,t_2,t_3)\eqiuv {\bf C(t_1,\dots,t_6)$ but $K\not\equiv {\bf C}(t_1,t_2,t_3)$; also an example was found of two non isomorphic rings $A$ and $B$ such that $A[t]\eqiuv B[t]$. Recently together with L.Makar-Limanov we proved the following result: If $A[t]$ embeds to $B[\tau$ then $A$ embeds in $B$. This result answers a question of Abhyankar: if $A[t_1,\dots,t_n]\equiv B[t_1,\dots,t_n]$ then $A$ embeds to $B$ and $B$ embeds to $A$. Another corollary of this result is: If $V$ is affine algebraic variety over algebraic closed field $k$, $char(k)=0$ such that $V\times k\equiv k^4$ then $V$ is birationally equivalent to $k^3$, Also it implies a positive answer to the cancellation conjecture for 2-dimentional fields in the case of characteristic 0. To transfer all this results to positive characteristic, we need a notion of nice embedding. An embedding $\varphi: K_1\to K_2$ is nice, if $K_2$ is separable extension of $\varphi(K_1)$. The notion of nice embedding can be easily transferred to rings.Theorem. If there is a nice embedding of $K_1(t_1)$ to $K_2(t_2)$ there is a nice embedding of $K_1$ to $K_2$. The similar fact is true for rings.Note that the 2-dimensional cancellation conjecture for algebraic closed fields of positive characteristic was not known before. (joint work with Yu Jie-tai (HK university))

## Academic year   2003-4

### Organizers:   L.H. Rowen

 DATE: 16/6/2004 Speaker: Prof. Tamar Seeman (Weizmann Institute) Title: Z_2 graded tensor products and polynomial identities of matrices. DATE: 16/6/2004 Speaker: Luda Markus Epstein (BI) Title: Automata and inverse semigroup theoretic algorithms for subgroups of free groups with amalgamation Abstract. In the 1980's Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In the first part of the talk, we review the theory for the free group and discuss a number of algorithmic problems solved by these methods including the membership problem, the finite index problem and the computation of closures of subgroups in various profinite topologies. In the second part of the talk, we look at applying the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for "sewing" on relations of non-free groups. We look at the class of groups that are amalgams of finite groups or amalgams of free groups over a maximal cyclic subgroup. It is known that these groups are locally quasiconvex and thus all finitely generated subgroups are represented by finite automata. We give an algorithm to compute such a finite automaton and use it to solve various algorithmic problems. DATE: 9/6/2004 Speaker: Prof. Professor Joseph Ponizovsky Title: On matrix representations of semigroups, a survey Abstract. Matrix representation theory of semigroups is one of the classical areas of semigroup theory. The speaker contributed some of the most important and basic results in this theory, including the correct generalization of Maschke's Theorem to finite semigroups in the 1950's. Over his career, Professor Ponizovsky has developed the theory of representations and matrix semigroups, sometimes single handedly, into a powerful and important branch of modern algebra. Today, there are applications of his work in the theory of algebraic groups and monoids, in the theory of finite dimensional algebras and in formal language theory and the theory of automata, as well as the theory of semigroups itself.The lecture is a self contained survey of the most important results of the representation theory of semigroups. DATE: 2/6/2004 Speaker: Prof. Gregory Soifer Title: Euclidean boundary of an affine group. DATE: 12/5/2004 Speaker: Prof. Alex Lubotzky Title: Finite groups and hyperbolic manifolds Abstract. The isometry group of a compact hyperbolic manifold is finite. In 1974, Greenberg proved that for every finite group G there exists a 2-dimensional closed hyperbolic manifold whose isometric group is isomorphic to G. A similar result was shown by Kojima in 1988 for n=3 who also conjectured that the same is true for every fixed n. We prove this conjecture. Unlike prevous results which used low dimensional geometry methods, our proof is mainly group theoretical. It uses counting results from subgroup growth theory and it it thus non constructive. (joint work with M. Belolipetsky) DATE: 5/5/2004 Speaker: Prof. Eugene Plotkin Title: Radicals in finite groups and finite dimensional Lie algebras (joint work with T. Bandman, M.Borovoj, F.Grunewald, B. Kunyavskii) DATE: 21/4/2004 Speaker: Prof. Boris Plotkin (Hebrew University) Title: PI-groups DATE: 14/4/2004 Speaker: Prof. Aharon Razon Title: Structure of symmetric tensor products of a simple algebra of prime degree Abstract. Let $A$ be a symbol algebra of prime degree $p$over a field $F$ of characteristic $0$. Consider a positive integer $n$. The symmetric group $S_n$ acts faithfully on $A^{\otimes n}$. We show there is an embedding $\tet\colon F[S_n]\rightarrow A^{\otimes n}$ suchthat $\sig(a)=\tet(\sig)\cdot a\cdot\tet(\sig)^{-1}$ foreach $\sig\in S_n$ and each $a\in A^{\otimes n}$. Let $R^{(n)}$ be the fixed ring of $A^{\otimes n}$ under the action of $S_n$, let$z^{(n)}\in F[S_n]$ be the sum of all transpositions, and let $w^{(n)}={1\over n!}\sum_{\sig\in S_n}\sig$. Then $A^{\otimes n}=A^{\otimes n}\cdot\tet(z^{(n)}-{n\choose 2})\oplus A^{\otimes n}\cdot\tet(w^{(n)})$. Moreover, $R^{(n)}\cdot\tet(w^{(n)})$ is a central simple algebra of degree ${n+p-1\choose n}$ over $F$. Let $\calL$ be the set of all $e\in A$ such that $[F[e]:F]=p$ and $e^p\in F$. For each $e\in\calL$ let $\fra_e^{(n)}$ be the left ideal of $A^{\otimes n}$ generated by the $n-1$ elements: $e\otimes 1\otimes\cdots\otimes 1-1\otimes e\otimes 1\otimes\cdots\otimes 1,\dots, e\otimes 1\otimes\cdots\otimes 1-1\otimes\cdots\otimes 1\otimes e$. It is shown that $R^{(n)}\cap\bigcap_{e\in\calL}\fra_e^{(n)}=R^{(n)}\cdot\tet(z^{(n)}-{n\choose 2})$. DATE: 14/4/2004 Speaker: Prof. Eli Aljadeff (Technion) Title: Profinite groups and Moore's Conjecture DATE: 24/3/2004 Speaker: Dr. Yair Glasner (University of Illinois) Title: New geometric methods in groups generated by finite automata Abstract. We introduce a geometric method to analyze groups generated by finite automata (a la Grigorchuk). With a finite automaton we associate a two dimensional square complex. We deal mainly with bi-reversible automata, or with automata whose associated square complex is covered by a product of trees. Using methods coming from group actions on products of trees we prove some new and some known results about bi-reversible automata: 1. On bi-reversible automata and the commensurator of a tree, after Macedonska, Nekrashevych and Sushchansky. 2. First examples of free and of Kazhdan groups generated by finite automata. (joint work with Shahar Mozes) DATE: 24/3/2004 Speaker: Prof. Darrell Haile (U. of Indiana) Title: TBA DATE: 17/3/2004 Speaker: Dr. Alexei Belov (Hebrew University) Title: Automorphisms of Weyl algebras and affine space DATE: 3/3/2004 Speaker: Prof. Wolfgang Herfort (Tech. Univ. Vienna) Title: Classes of groups with CC subgroups DATE: 25/2/2004 Speaker: Prof. Yoav Segev (BGU) Title: Normal subgroups of quaternion algebras DATE: 14/1/2004 Speaker: Prof. Avinoam Mann (Hebrew U) Title: Positively finitely generated groups, probabilistic zeta-functions, and arithmetic groups DATE: 7/1/2004 Speaker: Prof. Elena Perelman Title: A projection from domino tableaux to Young tableaux and its applications DATE: 10/12/2003 Speaker: Prof. Amitai Regev (Weizmann Inst) Title: permutation statistics on the Alternating groups Abstract. By choosing generators and canonical presentations of elements, we define various statistics on the alternating groups $A_n$, statistics which are analogues of those on the symmetric groups $S_n$. We extend MacMahon's equi-distribution theorem on $S_n$, then prove the analogue theorem for $A_n$.

List of speakers and topics for 2002-2003 (second semester) can be found here.

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