Colloquium Talks in Mathematics Department

All lectures are in Mathematics Building (No. 216), Colloquium Room (no. 201), Sunday at 12:00 noon. Light refreshments are served at 11:30.

December 6, 2009

Speaker:  Eran Nevo , Technion

Title: Face enumeration: from spheres to flag spheres

ABSTRACT: A major open problem in algebraic combinatorics is the g-conjecture, which suggests a characterization of the face-vectors of simplicial spheres. It was conjectured by McMullen (1970) for boundary complexes of simlpicial polytopes, and posed as a question for simplicial spheres. The simplicial polytopes case was proved by Billera and Lee (sufficiency) and by Stanley (necessity) in 1980. 

Flag spheres are simplicial spheres whose faces correspond to the complete subgraphs of a graph.
Gal's Conjecture, which implies an important case of a conjecture by Hopf in Riemannian geometry, gives strong lower bounds on face-vectors of flag spheres.

We will discuss several approaches as well as recent progress on these 2 conjectures.

Moreover, by introducing ``new g-vectors", we will present new conjectured lower bounds on face-vectors of spheres with bounded dimension of minimal non-faces. These interpolate between McMullen's generalized lower bound conjecture for spheres and Gal's conjecture for flag-spheres.

Some of the new results reflect joint works with Eric Babson, Martina Kubitzke, and Kyle Petersen.

December 20, 2009

Speaker: Gabi Ben Simon, ETH .

Title: From a new charecterization of Teichmuller spaces to positive representations.

ABSTRACT: Based on a new  study  of the relations of two structures on Lie groups and groups in general, developed in a three joint works with T.Hartnick, I will give a new geometrical characterization of classical Teichmuller spaces. We will see how this new study is used when we will survey the main steps of the proof ( Interestingly enough this study has been motivated by questions that came from contact geometry!).
Then we will generalize the above charecterization-theorem in 2 steps
1. We will suggest a higher Teichmuller type-moduli space as a generalization of the classical Teichmuller space. We call it "Positive representations" ( closely related to the maximal representations space, which is due to Burger-Iozzi-Wienhard).
2. We will explain, briefly, how the geometrical description of the classical space is generalized.

If time will allow, I will explain why is the positive representations space is a good candidate for being a generalization of the classical case. (This will be based on joint results with Burger-Hartnick-Iozzi-Wienhard).

January 3, 2010

Double Session

12:00:

Title Victor Kac, MIT

Title: Classifying simple linearly compact algebras and n-algebras

ABSTRACT: A linearly compact algebra is a topological algebra, whose underlying vector space is linearly compact. Killing-Cartan classification of simple finite-dimensional Lie algebras and Cartan's classification of simple infinite-dimensional Lie algebras of vector fields on a finite-dimensional manifold is the first one in a series of classifications of simple linearly compact algebras and superalgebras, which I am going to explain. As an afterthought, this has lead also to a classification of n-Lie algebras, which have been playing an important role in recent developments in the mysterious M-theory.

13:05:

Title: Michaela Vancliff (Univ of Texas at Arlington)

Title: Generalizing Graded Clifford Algebras and their Associated Geometry

ABSTRACT: Graded Clifford algebras are non-commutative algebras that have been known since at least the 1980s, and one can read off certain properties of such an algebra from certain commutative geometric data associated to the algebra.  Recently, T. Cassidy and the speaker introduced a generalization of such an algebra, called a graded skew Clifford algebra, and they found that most of the results concerning graded Clifford algebras can be extended to graded skew Clifford algebras, provided the appropriate non-commutative geometric data is defined.  In this talk, the main focus will be this algebra-geometry correspondence, main results, examples and some background history on the subject.

January 10, 2010

Speaker: Michael Deza  (Ecole Normale Superieure, Paris, and JAIST,Ishikawa)

Title: Geometry of Virus Structure

ABSTRACT: Since the discovery of molecule of C60 (truncated icosahedron), fullerenes, i.e., simple polyhedra with only pentagonal and hexagonal faces, became the main object in Organic Chemistry; the synthesis of C60 was marked by the Nobel prize 1996. Crick and Watson.s article in Nature, 10-3-1956, starts: "It is a striking  fact that almost all small viruses are either rods or spheres". In fact, all virions, except most complex, as brick-like pox virus, and some enveloped ones, are helical or (about half of all and almost all  human) icosahedral: dual fullerenes with Ih or chiral I symmetry.

Caspar and Klug, Nobel prize 1982, gave quasi-equivalence principle: virion minimizes by organizing capsomers in minimal number of locations ith noneqvuivalent bonding, resulting in icosadeltahedral (dual cosahedral fullerene) structure.

We give an up to date survey on geometries of virion capsids and related
mathematics. It will be an expositary lecture.

January 17, 2010

Speaker: Yoel Shkolnisky, Tel-Aviv University

Title: Cryo-EM Structure Determination through Eigenvectors of Center of Mass Operators

ABSTRACT: The goal in Cryo-EM structure determination is to reconstruct the 3D structure of molecules from their noisy projections, taken by an electron microscope at unknown random orientations. Resolving the Cryo-EM problem is of great scientific importance, as the method is applicable to essentially all macromolecules, as opposed to other existing methods such as crystallography. Since almost all large proteins have not yet been crystallized for 3D X-ray crystallography, Cryo-EM seems the most promising alternative, once its associated mathematical challenges are solved. A central challenge is devising a reconstruction algorithm that uses only the acquired data and makes no assumptions on the underlying structure.

In this talk, we present an extremely efficient and robust algorithm that successfully recovers the acquisition direction of each projection image. Once all acquisition directions are known, the molecule is reconstructed using standard tomography techniques. The key idea of the algorithm is designing an operator defined on the projection data, whose eigenvectors reveal the orientation of each projection. Such an operator is constructed by utilizing the geometry induced on Fourier space by the projection-slice theorem. The presented algorithm does not require any prior model for the reconstructed molecule, shown to have favorable computational and numerical properties, and does not impose any assumption on the distribution of the projection orientations, thus applicable to molecules that have unknown spatial preference.

Although the presented algorithm was originally derived in the context of Cryo-EM, it turns out that its underlying ideas are applicable to a large number of problems. This lays the foundations to a new paradigm in machine learning, in which relations between pairs of data points are used to construct a problem-dependent operator, whose eigenvectors encode the coordinates of each data point.

Joint work with Amit Singer, Ronald Coifman and Fred Sigworth.

January 24, 2010

Speaker: Dmitry Novikov, Weizmann Institute of Science

Title: Recent progress in Infinitesimal Hilbert 16th problem

ABSTRACT: Hilbert 16th problem asks about an upper bound for the number of limit cycles of a polynomial vector field on the real plane. Despite continuous efforts for more than one hundred years, existence of such a bound is not established even for quadratic vector fields. Infinitesimal Hilbert 16th problem asks the same questions but for vector fields appearing as perturbations of integrable ones (which do not have limit cycles at all). For an important particular case of perturbations of Hamiltonian vector fields the problem reduces, in first approximation, to the question about the number of zeros of the so-called Abelian integrals. Our recent result settles this long standing question. In the more general case of Darboux systems almost all good algebro-geometric properties are gone, and almost nothing is known. I will try to outline the main problems and indicate the recent progress.

March 7, 2010

Speaker:   Harry Dym , Weizmann Institute

Title: Recent and not so recent developments in interpolation problems of the Nevanlinna-Pick type

ABSTRACT: A leisurely tour of some extensions of the classical Nevanlinna-Pick problem based primarily on reproducing kernel Hilbert spaces of a type introduced by L. de Branges.

Connections with matrix Lyapunov equations and Riccati equations will be discussed. The talk will be expository.

March 14,2010

Speaker:  Boris Khesin, University of Toronto

Title: Real-complex correspondence for links and gauge theories

Abstract: In the talk we present two manifestations of the parallelism between the Stokes formula (the real case) and the Cauchy residue formula (the complex case).

The first example is the definition of  the holomorphic linking number for complex curves in complex three-folds, a complex counterpart of the Gauss linking number of two curves in the three-space.  The second example is the gauge-theoretic version of this correspondence, the parallelism between symplectic structures on moduli of flat connections on real surfaces and holomorphic bundles on complex surfaces.

March 21, 2010

Speaker:  Herbert Kurke, Humboldt-University Berlin

Title: COMMUTING LINEAR DIFFERENTIAL OPERATOPRS AND KP-HIERARCHY, ONE AND SEVERAL VARIABLES

ABSTRACT: I will focus on the relation of the subject to algebraic geometry and report on  some new attempts in the multivariable case. It is well understood in one variable, that one can associate a projective algebraic curve (=compact Riemann surface) to each commutative ring of (germs  of) LDO (lin. differential operators) (they encode spectral data of the  operators). Isospectral deformations of the operators are described by solutions of a system of nonlinear differential equations,  the "KP-Hierarchy" (containing the classical KP- and KdV equation, originally  coming from physics). Deforming the associated geometric data gives solutions of the KP-Hierarchy.

There are several attempts to study multivariable analogues and to relate them  to higher dimensional varieties. I will report on an approach proposed by Parshin, and recent results on this (joint work with D. Osipov and A. Zheglov)

April 11, 2010

Speaker:  Eitan Bachmat (Ben Gurion University)

Title:  Mathematical adventures in the supermarket

ABSTRACT: We will discuss some experiences that are related to supermarket lines. Things like express lines and "I just want to buy Shoko, while you have a truck load of items, please let me through". While it is generally difficult to analyze such lines in some cases it is easy. We will produce some super(market) friendly examples using the (proven) Taniyama-Shimura conjecture.

The talk will be self contained, and you can bring your groceries with you.

April 18, 2010

Speaker:   Anatolij Antonevich (Belarussian State University, Minsk, Belarus)

Title:  ON MULTIPLICATION OF DISTRIBUTIONS

ABSTRACT: The distributions were   introduced by S.L. Sobolev and L.Schwartz and the distribution theory is now an attractive  instrument of investigation of numerous problems in  mathematical physics and differential equations theory. The space of distributions is a generalization of the space of  functions. For example, the celebrated Dirac delta- function, defined by Dirac, in a not quite mathematically correct way,  as a function, is in fact a  well-defined object of  the distributions space. An essential  obstacle  for applications is that it is impossible to define a natural multiplication on  the distributions space.In order to solve the multiplication problem new objects (new generalized functions) were introduced that form the algebra, i.e. allow well-defined multiplication, and at the same time  preserve general properties of distributions.

We will present the main ideas of these constructions as well as  some examples and applications.

April 25, 2010

Speaker: Stuart Margolis, BIU

Title: Dealing Cards, Markov Chains and Random Walks on Semigroups and Hyperplane Arrangements

ABSTRACT: In the last ten years work of Bidigare, Hanlon, Rockmore, K. Brown and Diaconis and others have shown that there is a remarkable connection between the behavior of classical Markov chains such as the Tsetlin Library can be approached via the representation theory of a class of finite semigroups called left regular bands. Every hyperplane arrangement can be given the structure of a left regular band and this brings a geometrical side of the theory as well. We show that the transition matrices of such chains are diagonalizable with real eigenvalues whose multiplicities  are easy to compute.

May 2, 2010

Speaker: Boaz Tsaban (BIU)

Title: Characters of topological groups: From Pontryagin-van Kampen duality to Shelah's pcf theory

ABSTRACT: I will outline the main results in a more than two years long project, which included two extended visits to my coauthors from Spain (Salvador Hernandez, M. Vincenta Ferrer, and their student C. Chis). Being sensitive to accessibility, I will make special effort to define everything needed to understand the lecture. The minimal cardinality of a base at the identity in a topological group G, denoted \chi(G), is one of the major topological invariants of G. A celebrated 1936 result of Kakutani and Birkhoff asserts that G is metrizable if, and only if, \chi(G) is countable. We consider the case where G is the dual group of a metrizable group. Using Pontryagin-van Kampen duality and pcf theory, we show that also in this case,

\chi(G) is well behaved, and that it is determined by the density and the local density of the base, metrizable group. We apply our result to compute the character of free abelian topological groups, extending a number of results of Nickolas and Tkachenko. This phenomenon is also reformulated in an inner language, not referring to duality theory. Here, the compact subgroups and compact subsets of G determine its character.

For G dual to a metrizable group, \chi(G) is especially well behaved in the absence of large cardinals (so that a weak hypothesis of Shelah holds). On the other hand, when large cardinals are available, some limitations on the well behaveness of \chi(G) are demonstrated using Cohen's "forcing" method. The results of this part answer a question of Bonanzinga and Matveev which arose in an entirely different context.

The talk will be self contained. Students are also welcome.

May 2, 2010

Speaker: Dmitry Kerner (Ben-Gurion University)

Title: On the determinantal representations of plane curves.

(joint with Vinnikov, based on arXiv:0906.3012)

ABSTRACT: Let M be a non-degenerate d by d matrix whose entries are linear forms in homogeneous coordinates on CP^n.

M is called a global determinantal representation of the hypersurface  {det(M)=0}. A local determinantal representation is a matrix whose entries are locally analytic functions of n variables.

Such representations appear constantly in various areas (we restrict mostly to the case n=2).

* in the global case (linear systems of quadrics in P^{d-1}, theta characteristics, matrix problems and quiver representations)

* in the local case (families of matrices, matrix factorizations, maximally Cohen Macaulay modules,...).

I will start from a short intro and explain various points of view on determinantal representations, how do they appear in various fields.Then I proceed to the recent work on det.reps of singular curves.

We obtain various new results on both local and global det.reps for curves with arbitrary singularities, possibly reducible and non-reduced.

May 16, 2010

Speaker: Yehuda Pinchover, Technion

Title: Large time behavior of the heat kernel

ABSTRACT:  In the study of heat conduction and diffusion, the minimal positive heat kernel is the fundamental solution to a second-order parabolic initial value problem on a particular domain with Dirichlet boundary condition. It is also one of the main tools in the study of the spectrum of second-order elliptic operators.

In this talk we will discuss large time behaviors of the heat kernel of a general time-independent second-order linear parabolic operator which is defined on a noncompact manifold.  In particular, we will present strong ratio limit properties of the quotients of the heat kernels of subcritical and critical operators.

May 23, 2010

Speaker: Amir Leshem (School of Engineering, BIU)

Title: Application of game theory to resource allocation in wireless communication networks

ABSTRACT: In the talk I will describe some recent results regarding cooperative solutions to the problem of frequency allocation and admission control in wireless networks. Among the solutions I will describe the Nash bargaining as well as use of the stable marriage theorem to resource allocation in cognitive radio networks. I will not assume familiarity with game theoretic concepts.

May 30, 2010

Speaker: Richard Tsai (University of Texas)

Title: Optimal trajectories for curvature constrained motion

ABSTRACT: We propose a Hamilton-Jacobi equation approach for computing time- optimal trajectories of a vehicle which travels under curvature constraints. We derive a class of Hamilton-Jacobi equations which models such motions; it unifies some well-known vehicular models, the Dubins' and Reeds- Shepp's cars,and gives further generalizations.

June 6, 2010

Speaker: Alexander Olevskii (Tel-Aviv University)

Title: SAMPLING OF SIGNALS WITH BOUNDED SPECTRUM

ABSTRACT: How often one should measure a signal with given spectrum in order to be able to recover it ? Does a "universal" sampling exist, which works well for any spectrum of fixed size, independently of its structure and localization ?  I will survey classical background and discuss some recent results (joint with A.Ulanovskii).

Sunday, June 20, 2010

First lecture: 12:00-12:55

Speaker: Michael Larsen (Indiana University)

Title: STRUCTURE THEOREMS FOR LINEAR GROUPS

ABSTRACT: A group is linear if it can be realized as a subgroup of a matrix group over a field.  There are two obvious ways of constructing subgroups of GL(n,K):

by imposing polynomial relations on the entries or by restricting to a subring of K.  There are a number of theorems asserting that under various hypotheses,

linear groups can be obtained (or are closely related to groups which can be obtained) by combining these operations.  I will discuss various examples, classical and modern.

Second lecture: 13:05-14:00

Speaker: Michael Entov (Technion) 

Title: ALMOST LINEAR  FUNCTIONALS  ON LIE ALGEBRAS

ABSTRACT: Assume we are given a real-valued functional on a real Lie algebra and suppose that its restriction to any abelian subalgebra is linear. Does it imply that the
functional is linear? If not, can one describe all the non-linear functionals of this sort on the given Lie algebra? I will discuss how these questions are related to a mathematical model of quantum mechanics, symplectic topology and the structure of classical Lie algebras.

The talk is based on a joint work with Leonid Polterovich.

Sunday, October 17, 2010

Speaker: Professor Benjamin Weiss (Hebrew University)

Title: Sofic groups and their actions

ABSTRACT: Sofic groups are a common generalization of amenable groups and residually finite groups. I will describe all of these classes and then explain some exciting new developments in the study of actions of these groups. We will consider both topological actions as well as measure preserving ones.

Sunday, October 24, 2010

Speaker: Prof. Andre Reznikov (Bar-Ilan)

Title: The recent impact of representation theory on PDE theory

ABSTRACT: I will discuss some recent interactions between representation theory (of SL(2)) and the regularity of traces for some natural hyperbolic equations over a compact Riemann surface.

Sunday, October 31, 2010

Speaker: Hershel Farkas (Hebrew University)

Title: The Möbius Function, Fermat's Theorem and the Goldbach Conjecture

ABSTRACT: I will discuss how the Möbius function and some of its variants enter into these problems. The solutions will be reduced to questions in linear algebra and will involve only elementary ideas.

Sunday, November 7, 2010

Time: 12 noon

Speaker: Prof. Uzi Vishne (Bar-Ilan)

Title: A solution to Roquette's 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.

This is a joint work with Louis Rowen and Eliyahu Matzri.

Sunday, November 14, 2010

Time: 12 noon

Speaker: Prof. Michael Cwikel (Technion)

Title: The John-Nirenberg and John-Stromberg theorems for BMO. A new approach.

ABSTRACT: We develop some techniques for studying various versions of the function space BMO. Special cases of one of our results give alternative proofs of the celebrated John-Nirenberg inequality and of related inequalities due to John and to Wik. Our approach enables us to pose a simply formulated geometric question, for which an affirmative answer would lead to a version of the John-Nirenberg inequality with dimension free constants.

A more extensive version of this abstract, and also a preview, more or less, of the slides that will be used for this lecture, are available at http://www.math.technion.ac.il/~mcwikel/bmo
The most recent version of our detailed paper about all this is at http://arxiv4.library.cornell.edu/abs/1011.0766

Joint work with Yoram Sagher and Pavel Shvartsman

Sunday, November 21, 2010

Time: 12 noon

Speaker: Dr. Michael Schein (Bar-Ilan)

Title: Representation theory and statistics

ABSTRACT: We present several applications of representation theory to statistics, from classical results in the theory of variance to recent work in optimal designs.

Sunday, November 28, 2010

Time: 12 noon

Speaker: Prof. Avraham Feintuch (Ben-Gurion University)

Title: Stable Rank and Linear Systems

ABSTRACT: The notion of stable rank was introduced as a ring invariant by H. Bass in the 1960's. For certain Banach algebras of operators, having stable rank 1 means that appropriate classes of linear systems can be stabilized by means of a stable system using output feedback. We discuss these issues. No knowledge of linear systems will be assumed, just a basic acquaintance with linear operators on Hilbert space.

Sunday, December 12, 2010

Time: 12 noon

Speaker: Jozef Dodziuk (CUNY)

Title: Harmonic differentials on hyperbolic surfaces with cusps and funnels

ABSTRACT: Celebrated theorems of de Rham and Hodge assert that every cohomology class over the reals on a compact, oriented Riemannian manifold is represented uniquely by a harmonic exterior differential form. This fails completely for non-compact manifolds. In certain cases the statement above may be salvaged by imposing appropriate "boundary conditions" at infinity. I will discuss one such situation where the manifold is topologically a compact surface with finitely many points removed and is equipped with a complete metric of constant negative curvature. Geometrically, the ends of such a manifold are either expanding (funnels) or contracting (cusps); and different boundary conditions have to be imposed on different ends, depending on whether the end is a cusp or a funnel. This is joint work with Jeff McGowan and Peter Perry.

Sunday, December 19, 2010

There will be 2 speakers, as follows:

Time: 12 noon

Speaker: Benny Sudakov (UCLA)

Title: Expanders, Ramanujan graphs and random lifts

ABSTRACT: Expansion of a graph is one of the most fundamental concepts in modern combinatorics, which has numerous applications in many mathematical areas. It is well known that expansion is closely relates to the spectral properties of graph. The celebrated Alon-Boppana bound says that all eigenvalues of a d-regular graph must be at least 2sqrt(d-1) - o(1) and graphs that meet this bound are called Ramanujan Graphs. There are still many unresolved questions about the existence of such graphs. In this talk we survey this background material, then we explain what lifts of graphs are and how the above questions can be approached using random lifts of graphs.

Joint work with Lubetzky and Vu

Time: 13:10

Speaker: Dan Kushnir (Yale)

Title: Anisotropic Diffusion Maps of Sub-Manifolds with Applications

ABSTRACT: We introduce a method for computing extendable independent components of stochastic data sets generated by nonlinear mixing. The method relies on the spectral decomposition of a particular type of anisotropic diffusion kernels constructed from observed data. The independent components are computed for a surprisingly small sub-sample of the data, and can be efficiently extended to a much larger set of new observations. We demonstrate our method performance for the empirical solution of inverse problems in electro- magnetic measurements of geological formations, and in Acoustics. This method suggests a general tool for solving empirically inverse problems.

Sunday, December 26, 2010

Time: 12 noon

Speaker: Dmitry Kerner (University of Toronto)

Title: Newton diagrams and applications

ABSTRACT: One basic result of Calculus 1 can be stated as follows. Suppose the Taylor expansion of a locally analytic function near the origin is f(x)=x^p+(higher order terms). Then in some new (local) coordinates the function is $f=x^p$.

The moral in higher dimension is: for a given power series near the origin only a very few monomials are important. To formulate the precise criteria one uses the Newton diagram associated to
f(x_1,...,x_n). This helps to study degenerate minima/maxima, types of saddle points etc.

Today, the Newton diagram is an important tool in various areas, e.g., understanding the local embedded geometry/topology of a hypersurface. In particular, if the curve/hypersurface is "generic enough" then the local topology is completely determined by its Newton diagrams.

So, various hard topological questions are translated into combinatorial ones. Such hypersurface singularities (non-degenerate with respect to their diagrams) are easy to work with and serve as "baby models".

I will give a short introduction to this topic, then report on some recent advances.  No preliminary knowledge of the subject is assumed; the talk should be accessible to graduate students.

Sunday, January 2, 2011

There will be 2 speakers, as follows:

Time: 12 noon

Speaker: Nathan Keller (Weizmann Institute)

Title: Discrete Harmonic Analysis of Boolean functions and its applications

ABSTRACT: Boolean functions are a central object of study in combinatorics, complexity theory, probability theory and other areas of mathematics and computer science. In a paper from 1988, Kahn, Kalai and Linial (KKL) introduced the use of tools from harmonic analysis in the study of Boolean functions. The paper of KKL was the starting point of an entire area of research. In the last two decades the results of KKL were greatly expanded, the analytic tools were developed significantly, and the techniques were applied in numerous fields.

In this talk, we present the basic objects of study (e.g., influences, Fourier-Walsh expansion etc.) and the basic analytic tools (e.g. hyper- contractive inequalities), and then show several new applications, focusing on applications to correlation inequalities, percolation theory, and social choice theory. Some of the results are based on joint work
with Guy Kindler and with Ehud Friedgut, Gil Kalai, and Noam Nisan.

Time: 13:10

Speaker: Alexander Fish (University of Wisconsin)

Title: Ergodic Plunnecke inequalities with applications in additive number theory

ABSTRACT:  Plunnecke inequalities for sumsets of finite sets in abelian groups are extended to ergodic dynamical systems. As an application, we obtain the following theorem: Given a set B in a countable abelian group such that kB (k-folded sum of B) has full Banach density (BD(B) = 1), for every set A in the group we have BD(A+B) >= BD(A)^{1-1/k}.

Sunday, January 9, 2011

Time: 12 noon

Speaker: Vitali Milman (Tel-Aviv University)

Title: Rigidity of some Classical Constructions in Geometry and Analysis

ABSTRACT: During the last few years it was observed that some very classical constructions and transforms in Geometry and Analysis are uniquely defined by certain elementary conditions which, in some cases, were not expected to be so strongly connected with the transforms they defined. We provide four examples of such phenomena: duality, mixed volumes, Fourier transform, and derivative operations.

Sunday, February 20, 2011

Time: 12 noon

Speaker: Burglind Joricke (Weizmann Institute)

Title: Analytic knots, the 4-ball genus, satellites and holomorphic coverings

ABSTRACT: Starting from a classical paper by Schubert, satellites and companion knots have been considered from various aspects, for instance in connection with knot invariants. After an overview, I will come to a complex variable situation, namely, to knots in the unit sphere in complex affine 2-space which bound smooth complex curves. It turns out that in case both the companion and its satellite have this property and, in addition, the satellite is contained in a small enough tubular neighbourhood of the companion, a cobordism reduces the situation to the study of braided links. The description of these braided links and the estimate of their 4-ball genus is related to problems in another topic: holomorphic coverings of open Riemann surfaces and lifting of coverings to embeddings
into disc bundles.

Sunday, February 27, 2011

Time: 12 noon

Speaker: Yoram Louzoun (Bar-Ilan University)

Title: Viruses selectively mutate their CD8+ T cell epitopes – an optimization framework, a novel machine learning methodology and a large scale genetic analysis

ABSTRACT: The relation between organisms and proteins complexity and between the rate of evolution has been discussed in the context of multiple generic models. The main robust claim from most such models is the negative relation between the organism complexity and the rate of mutation accumulation.  We here validate this conclusion, through the relation between viral gene length and their CD8 T cell epitope density. Viruses mutate their epitopes to avoid detection by CD8 T cells and the following destruction of their host cell. We propose a theoretical model to show that in viruses the epitope density is negatively correlated with the length of each protein and the number of proteins. In order to validate this conclusion, we developed a novel machine learning methodology to combine multiple modalities of peptide-protein docking measurement. We use this methodology and a large amount of genomic data to compute the epitope repertoire presented by over 1,300 viruses in many HLA alleles. We show that such a negative correlation is indeed observed.  This negative correlation is specific to human viruses. The optimization framework also predicts a difference between human and non-human viruses, and an effect of the viral life cycle on the epitope density. Proteins expressed early in the viral life cycle are expected to have a lower epitope density than late proteins. We define the "Size of Immune Repertoire (SIR) score," which represents the ratio between the epitope density within a protein and the expected density.  This score is applied to all sequenced viruses to validate the prediction of the optimization model. The removal of early epitopes and the targeting of the cellular immune response to late viral proteins, allow the virus a time interval to propagate before its host cells are destroyed by T cells. Interestingly, such a selection is also observed in some bacterial proteins. We specifically discuss the cases of Herpesviruses, HIV and HBV showing interesting selection biases.

(joint work with Tal Vider, Yaacov Maman, Alexandra Agaranovich, and Lea Tsaban)

Sunday, March 6, 2011

Time: 12 noon

Speaker: Gregory Soifer (Bar-Ilan University)

Title: Affine (semi)groups acting properly discontinuously on a hyperbolic space

ABSTRACT: The most celebrated questions are the Auslander conjecture asserting that every affine crystallographic group is virtually solvable and the question stated by J. Milnor that a fundamental group of an affine locally flat complete manifold is virtually solvable. The talk will review the development and the recent progress in attempts to prove the Auslander conjecture and the Milnor question in our joint work with H. Abels and G. Margulis. Our approach based on the study of the dynamics of action of free subgroups of an affine group, since by the Tits alternative, to be virtually solvable is equivalent to contain no nonabelian free subgroup. First we will provide an introduction to study the dynamics of the action of an affine group and will outline a general method. Then we will explain some new ideas and recent results which give an answer to several questions arise from our works.

Sunday, March 13, 2011

Time: 12 noon

Speaker: Stuart Margolis (Bar-Ilan University)

Title: Left regular bands arising in geometry, probability and combinatorics

ABSTRACT: A left regular band (LRB) is a semigroup S satisfying the identities, xx = x and xyx = xy for all elements x,y of S. While this definition is completely unmotivated, LRBs arise naturally in a number of seemingly unrelated areas of mathematics.

Every hyperplane arrangement has the natural structure of an LRB. For the arrangement associated to a Coxeter group, this was rediscovered by Tits and plays an integral part in the theory of Buildings and groups with (B,N) pairs.

Brown and Diaconis, following work of Bidigaire, showed that a number of natural Markov chains are modeled by random walks on LRBs. They significantly developed the representation theory of LRBs in order to compute eigenvalues associated to the Markov chains. This has spurred intense interest in the representation theory of LRBs and related monoids.

The purpose of this talk is to give a gentle introduction to the theory of LRBs. We emphasize examples and applications like those noted above.

Sunday, March 27, 2011

Time: 12 noon

Speaker: David Shoikhet (ORT Braude College, Karmiel)

Title: A generalized version of the Earle-Hamilton fixed point theorem for the Hilbert ball

ABSTRACT: Let D be a bounded domain in a complex Banach space. According to the Earle-Hamilton fixed point theorem, if a holomorphic mapping F : D D maps D strictly into itself, then it has a unique fixed point and its iterates converge to this fixed point locally uniformly. Now let B be the open unit ball in a complex Hilbert space and let F : B B be holomorphic. We show that a similar conclusion holds even if the image F(B) is not strictly inside B, but is contained in a horosphere internally tangent to the boundary of B. This geometric condition is equivalent to the fact that F is asymptotically strongly nonexpansive with respect to the hyperbolic metric in B.

Sunday, April 3, 2011

Time: 12 noon

Speaker: Steve Shnider (Bar-Ilan University)

Title: Some remarkable mathematical cuneiform texts from the early second millennium

ABSTRACT: The scribes of the Old Babylonian period (~1850-1600 BCE) in the cities of southern Mesopotamia   showed mathematical skills of surprising sophistication. After a short introduction to cuneiform mathematical texts and sexagesimal notation, I will discuss several examples connected with solutions to quadratic equations and the so called "Pythagorean Theorem".

Sunday, April 10, 2011

Time: 12 noon

Speaker: Mikhail Katz (Bar-Ilan University)

Title: Ten misconceptions from the history of analysis and their debunking

ABSTRACT: The founders of infinitesimal calculus were working in a vacuum caused by an absence of a satisfactory number system.  The incoherence of infinitesimals was effectively criticized by Berkeley as so much hazy metaphysical mysticism.  D'Alembert's visionary anticipation of the rigorisation of analysis was ahead of his time. Cauchy took first steps toward rigor and epsilontics without
infinitesimals, in particular giving a modern definition of continuity.  Cauchy's false 1821 version of his "sum theorem" was corrected by Cauchy in 1853 by adding the hypothesis of uniform convergence.  Weierstrass finally rigorized analysis and eliminated infinitesimals from mathematics.  Dedekind discovered "the essence of continuity," which is captured by his cuts.  One of the spectacular successes of the rigorous analysis was the mathematical justification of "Dirac delta functions''.  Robinson develops a new theory of infinitesimals in the 1960s, but his approach has little to do with historical infinitesimals.  Lakatos pursued an ideological agenda of Popperism and fallibilism, inapplicable to mathematics.  Each of the above ten claims is in error.

Sunday, May 1, 2011

Time: 12 noon

Speaker: Tahl Nowik (Bar-Ilan University)

Title: Random constructions in topology

ABSTRACT: The probabilistic approach in topology is relatively new, following the very well established probabilistic method in combinatorics. I will present some results of N. Dunfield and W. Thurston, of R. Meshulam and N. Linial, and joint results with N. Linial. The talk will assume no prior knowledge, and will mostly focus on describing the topological constructions on which randomness is then applied.

Sunday, May 15, 2011

Time: 12 noon

Speaker: Yuval Roichman (Bar-Ilan University)

Title: Diameter of graphs of reduced words

ABSTRACT: There is a natural graph structure on the set of all reduced words for the longest element in a finite Coxeter group W, where edges are determined by braid relations. A classical theorem of Tits says that this graph is connected. Autord and Dehornoy showed that when W = S_n, the symmetric group on n letters, the diameter of this graph grows asymptotically as O(n⁴).

The main result is that the diameter for W = S_n is exactly 1/24 n(n-1)(n-2)(3n-5). This result is then extended to W = B_n, the hyperoctahedral group. The proof idea is to rephrase the problem as a general question about hyperplane arrangements.

This is joint work with Victor Reiner.

Sunday, May 22, 2011

Time: 12 noon

Speaker: Ely Merzbach (Bar-Ilan University)

Title: What is a Set-Indexed Lévy process?

ABSTRACT: Lévy processes constitute a very natural and fundamental class of stochastic processes, including Brownian motion, Poisson processes and stable processes. On the other hand, set-indexed processes like the set-indexed Brownian motion (also called the white noise) and the spatial Poisson process are very important in several fields of applied probability and spatial statistics. As a general extension of these processes, the aim of this lecture is to present a satisfactory definition of the notion of set-indexed Lévy process.

Our definition is sufficiently broad to also include the set-indexed Brownian motion, the spatial Poisson process, spatial compound Poisson processes and some other stable processes.

One main concept is stationarity and it will be shown that no group structure is needed in order to define the increment stationarity property for Lévy processes.

We discuss links with infinitely divisible distributions and prove the Lévy-Khintchine representation formula.

Work done jointly with Erick Herbin (Ecole Centrale Paris).

Sunday, May 29, 2011

Time: 12 noon

Speaker: Leonid Berlyand (Penn State )

Title: Modeling and homogenization of bacterial suspensions

ABSTRACT: Modeling of bacterial suspensions and, more generally, of suspensions of active microparticles has recently become an increasingly active area of research.  The focus of our work is on the development and   analysis of mathematical PDE models for bacterial suspensions. The ultimate goal is to develop a model which describes dynamics and rhelogy of bacterial suspensions.

We discuss recent results on the effective viscosity of dilute suspensions of swimming bacteria and on non-dilute suspensions when interactions between bacteria are taken into account.

Sunday, June 5, 2011

Time: 12 noon

Speaker: Victor Katsnelson (Weizmann Institute)

Title: Stieltjes Functions and Stable Entire Functions

ABSTRACT: Stable polynomials are polynomials which only have roots in the open left halfplane {z : Re z < 0}. These polynomials are important in automatic control theory.
A Stieltjes function ψ(z) is a function holomorphic in the domain C \ (−∞, 0] which possesses the following properties:
ψ (x) ≥ 0  for  x > 0,   Im ψ(z) ≤ 0  for  Im z > 0,   Im (z) ≥ 0  for Im z < 0.
Stieltjes functions appear as a Cauchy transform of positive measures on [0,+∞):
\psi(z)=\int_{[0,\infty)}\frac{d\sigma(\lambda)}{\lambda+z}, z\in\mathbb{C}\setminus(0,\infty].
They are also related to positive operators in Hilbert spaces. Let \frak H  be a Hilbert space with the scalar product <.,.> , and let A be a positive self-adjoint operator in    \frak H and e \in  H with e ≠  0. The function <(A + zI)^{-1}e,e> is then a Stieltjes function. The simplest of our results can be formulated as follows:
Let P(z)=\sum_{k}p_kz^k  be a polynomial with only negative roots, ψ(z) a Stieltjes function and P_{\psi}(z)  the polynomial P_{\psi}(z)=\sum_{k}p_{k}\psi(k+1)z^k. Then the polynomial P_{\psi}(z) is stable.

Sunday, June 12, 2011

In Honor of Prof. Lawrence Zalcman on his Retirement

Time: 12 noon

Speaker: Lawrence Zalcman (Bar-Ilan)

Title: 100 Years of Normal Families (in 50 minutes)

ABSTRACT: A bird's-eye view of the theory of normal families of meromorphic functions on plane domains, with an emphasis on recent progress in the subject.

 
Sunday, October 30, 2011
 

Time: 12 noon

Speaker: Bill Gasarch (Univ. of Maryland)

Title: When can you color a grid and not have any monochromatic rectangles?
ABSTRACT: We will be looking at colorings of grids. A c-coloring of a grid is an assignment toe every grid point a color. For example, this is a
2-coloring of the 3 x 7 grid using colors R,B,G.
 
rrbbbrr
bbrbrbb
rrrrbrb
 
Note that there is a rectangle with all four corners the same color (we use capital letters to denote it)
 
RrbbbRr
bbrbrbb
RrrrbRb
 
If a grid can be c-colored without a monochromatic rectangle we say that the grid is c-colorable.
 
Which grids can be 2-colored? 3-colored? 4-colored? etc.
1) We have characterized EXACTLY which grids are 2-colorable.
2) We have characterized EXACTLY which grids are 3-colorable.
3) We have made progress on EXACTLY which grids are 4-colorable.
4) We have GIVEN UP on trying to find EXACTLY which grids are 5-colorable.
 
The work is a combination of some clever math and some computer work. The questions has its origins in Ramsey Theory which we will also discuss.
 
Sunday, November 13, 2011
 

Time: 12 noon

Speaker: Ron Livne (Hebrew University)

Title: Greek Mathematics in the Bible
 
ABSTRACT: We will show that a lot of interesting mathematics is implicit in the first books of the Bible, mainly in the books of Genesis and Numbers. The mathematics, which makes a cohesive whole, is encoded by numbers in the text which otherwise appear arbitrary. The mathematics appears to be 3rd century BC Greek Mathematics, based on Euclid's Elements and Eratosthenes' sieve.
 
The talk will concentrate on the numbers and the mathematics: the theory of prime numbers as a main theme, as well as the theory of Musical Harmony and of Platonic Solids. The mathematics suggests a rational explanation of how it came about and what its religious function is. Assuming this, one learns some previously unknown facts
about Greek mathematics and about the Bible.

Sunday, November 20, 2011
 

Time: 12 noon

Speaker: Abe Feintuch (Ben Gurion University)

Title: Spectral properties of multilinear operators

ABSTRACT: A spectral problem and inverse spectral problem for multilinear operators are discussed. The $m$-independence of vectors
based on the symmetric tensor powers performs as a main tool in the study of structure of the spectrum. Possible restrictions on this
structure are described in terms of syzygies provided by the Euler-Jacobi formula.

One of the possible applications of this theory is to show how algebraic formalism can successfully be applied to a remarkably
elegant description of the geometry of curves that are the solutions to homogeneous polynomial ODEs as well as to arouse interest in
algebraic language in PDEs.

Sunday, November 27, 2011
 

Time: 12 noon

Speaker: Hershel Farkas (Hebrew University)

Title: Special theta constant identities for zn curves
ABSTRACT: A zn curve is a compact Riemann surface with algebraic equation wn = (z-λ1)..(z λ r) with r at least 2n and λ i ≠ λ i, i ≠ j. 
We relate the algebraic parameters λ i to certain transcendental parameters of the curve and then show that the theta constants on these curves satisfy special identities.
Sunday, December 4, 2011
 

Time: 12 noon

Speaker: A.S. Sivatski (St.Petersburg Electrotechnical University)

Title: Central simple algebras of prime exponent and divided power 
ABSTRACT: Let F be a field, p a prime number not equal to char F. Suppose that all p-primary roots of unity are contained in F. Denote by p-Br(F) the p-torsion of the Brauer group Br(F). A natural question arises whether the given element in p-Br F can be represented by a cyclic algebra. In this case the element is called cyclic. It turns out that there is a simple necessary condition for cyclicity of an element, which is formulated in terms of divided power operations on p-Br F.

Sunday, December 11, 2011
 

Time: 12 noon

Speaker: Jeremy Schiff (Bar-Ilan University)

Title: Quantum Mechanics without Wavefunctions
 
ABSTRACT: In classical mechanics, a particle, or a collection of particles, is described by a trajectory in configuration space. In the standard formulations of quantum mechanics, the trajectory is replaced by a complex-valued wavefunction. We present a self-contained formulation of (spin-free, nonrelativistic) quantum mechanics that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued trajectories, described by a remarkable PDE obtained from an action principle and obeying a law of conservation of energy.
Joint work with Bill Poirier (Texas Tech).
No prior understanding of quantum mechanics is necessary to understand this talk.
 
 
Sunday, December 18, 2011
 

Time: 12 noon

Speaker: Eran Calderon (Technion)

Title: A New Geometric Framework for Classical Dynamics
 
ABSTRACT: In this talk I will introduce a novel geometric framework for classical dynamical systems. The language of Riemannian geometry (rather than symplectic geometry) underlies this new approach, and we show how it is applicable to the study of dynamics over rather general manifolds, including the cotangent bundle. Some of the implications of this formulation will be discussed, primarily pertaining to the theory of nearly integrable Hamiltonian systems. The lecture is intended for a general audience and will include a presentation of the necessary background.
 
Joint work with Larry Horwitz, Raz Kupferman, and Steve Shnider
 
 
Sunday, January 1, 2012
 

Time: 12 noon

Speaker: Shahar Nevo (Bar-Ilan)

Title: Normal families and Calculus
 
ABSTRACT: In the subject of normality we discuss the connection between differential inequalities to normality, quasi-normality and value sharing problems. We present results concerning the notion of Q_alpha-normality which is a geometrical extension to the notions of normality and quasi-normality that involves arbitrary ordinal numbers. Some open question concerning the first uncountable ordinal will be presented. In the area of calculus, we present a new formula for the natural logarithm of a natural number and a multidimensional extension to Riemann's Theorem on conditionally convergent series.
 
All necessary definitions will be given.
 
Most of the results are parts of joint work with Q.Y. Chen , J. Grahl, S. Gul, X. J. Liu and X.C. Pang.
 
Sunday, January 8, 2012
 
Time: 12 noon



Speaker: Frol Zapolsky (LMU Munich)
 
Title: Quasi-states in classical mechanics and applications
 
ABSTRACT:  Motivated by the axiomatic approach to quantum mechanics, we introduce a certain type of non-linear functionals on the space of observables in classical mechanics, called quasi-states. Quasi-states exist on many symplectic manifolds (these are the phase spaces of classical mechanics). It turns out that a slightly more general kind of functionals, called partial quasi-states, exist on yet more symplectic manifolds and are at the intersection of numerous mathematical disciplines - functional analysis, optimization theory, symplectic geometry, Riemannian geometry, and others. We explain how the methods of symplectic geometry are used in order to construct such functionals
and present a few of their applications.
 
Tim