# Abstracts of talks

**Enrico Arbarello**

*Hyperplane Sections of K3 Surfaces*

K3 surfaces and their hyperplane sections play a central role in algebraic geometry. This is a survey of the work done during the past five years to characterize which smooth curves lie on a K3 surface. Related topics will be discussed. These are joint works with a combination of the following authors: Andrea Bruno, Gavril Farkas, Giulia Saccà and Edoardo Sernesi.

**Ivan Arzhantsev**

*Equivariant version of Hirzebruch's problem on compactifications of affine spaces*

In 1954, Hirzebruch raised the question of classifying algebraic compactifications of complex affine spaces. This question has led to many results and attracts considerable attention up to now. Forty-five years later, Hassett and Tschinkel initiated a systematic study of equivariant compactifications of vector groups. They described such equivariant structures on projective spaces and some other projective varieties, and formulated questions for further research. In this talk we discuss recent results on such structures on generalized flag varieties, toric varieties, and Fano manifolds.

**Werner Ballmann**

*Bottom of spectra under Riemannian coverings*

Robert Brooks showed that the bottom of the spectrum of the universal covering space of a closed Riemannian manifold is zero if and only if its fundamental group is amenable. Later on, he also obtained results for non-compact manifolds, where the situation is more complicated. I will report on recent joint work with Henrik Matthiesen and Panagiotis Polymerakis, in which we extended the work of Brooks.

**Alexander I. Bufetov**

*Patterson-Sullivan measures for point processes and the reconstruction of harmonic functions*

The Patterson-Sullivan construction is proved almost surely to recover every Hardy function from its values on the zero set of a Gaussian analytic function on the disk. The argument relies on the conformal invariance of the point process and the slow growth of variance for linear statistics. Patterson-Sullivan reconstruction of Hardy functions is obtained in real and complex hyperbolic spaces of arbitrary dimension, while reconstruction of continuous functions is established in general CAT(-1) spaces.

Based on joint work with Yanqi Qiu, https://arxiv.org/abs/1806.02306.

**Alex Furman**

*Lyapunov spectrum for some dynamical systems via Boundary theory*

Consider products of matrices in

*G=SL(d,R)*that are chosen using some ergodic dynamical system. The Multiplicative Ergodic Theorem (Oseledets) asserts that the asymptotically such products behave as

*exp(nΛ)*where

*Λ*is a fixed diagonal traceless matrix - the Lyapunov spectrum of the system. The spectrum

*Λ*depends on the system in a mysterious way, and is almost never known explicitly. The best understood case is that of random walks, where by the work of Furstenberg, Guivarc'h-Raugi, and Gol'dsheid-Margulis we know that the spectrum is simple (i.e. all values are distinct) provided the random walk is not trapped in a proper algebraic subgroup. Recently, Avila and Viana proved a conjecture of Kontsevich-Zorich that asserts simplicity of the Lyapunov spectrum for another system related to the Teichmuller flow. In the talk we shall describe an approach to proving simplicity of the spectrum and some continuity properties, based on ideas from boundary theory that were developed to prove rigidity of lattices. This a joint work with Uri Bader.

**Gert-Martin Greuel**

*Equinormalizable deformations of isolated non-normal singularities*

Starting from the correspondence of Hirzebruch with Brieskorn about Brieskorn's discovery of a singular normal 3-fold which is a topological manifold, we turn to new results on equinormalizability of families with isolated non–normal singularities (INNS) of arbitrary dimension. We define a generalized delta invariant and Milnor number for an INNS and prove necessary and sufficient numerical conditions for equinormalizability, using these invariants. For families of generically reduced curves, we investigate the topological behavior of the Milnor fibre and characterize topological triviality of such families.

**Klaus Hulek**

*The geometry and topology of moduli spaces of cubic threefolds*

Cubic threefolds played an important role in algebraic geometry. Indeed, Clemens and Griffiths showed that smooth cubic threefolds are unirational but not rational, thus providing the first example of such varieties. There are several birational models of the moduli space of cubic threefolds. These include the GIT quotient, the ball quotient model due to Allcock, Carlson and Toledo, the Kirwan blow-up and the so-called wonderful compactification. Each of these models plays its own geometric role depending on the angle from which one approaches this moduli problem. In this talk I will discuss the geometry of these moduli spaces and their relation to each other, and, in particular, their topology. This is joint work with S. Casalaina-Martin, S. Grushevsky and R. Laza.

**Jun-Muk Hwang**

*Rigidity of Legendrian singularities*

Let (M, D) be a holomorphic contact manifold, i.e., a complex manifold M of dimension 2m+1 equipped with a holomorphic contact structure D. An m-dimensional complex analytic subvariety Z in M is called a Legendrian subvariety if the smooth locus of Z is tangent to D. A Legendrian singularity means the germ of a Legendrian subvariety at a point. We discuss some rigidity results on Legendrian singularities.

**Ludmil Katzarkov**

*P=W conjecture. New instances and applications*

In this talk we discuss generalization of the P=W conjecture. We connect it with theory of algebraic cycles.

**Dmitry Kleinbock**

*Points with divergent orbits on algebraic hypersurfaces of homogeneous spaces*

Let f be a homogeneous polynomial with rational coefficients in d variables. Jointly with Nikolay Moshcbevitin, we prove several results concerning uniform simultaneous approximation to points on the hypersurface {f(x1,…,xd)=1}. This can be interpreted as bounds for the rates of divergence of the corresponding orbits in the space of lattices.

**Anatoly Libgober**

*A generalization of Landau-Ginzburg/Calabi-Yau correspondence*

Landau-Ginzburg/Calabi-Yau correspondence is a relation between invariants of weighted homogenous singularities and invariants of Calabi-Yau hypersurfaces in weighted projective space. Invariants previously considered relate the Gromov-Witten invariants of singularities and hypersurfaces, categories of matrix factorizations and derived categories, Arnold-Steenbrink spectrum and Hodge numbers. This talk discusses LG/CY correspondence which involves certain Chern numbers of hypersurfaces and which comes from a version of McKay correspondence for elliptic genus. Moreover, this approach allows to obtain LG/CY correspondence as a special case of relation between elliptic genera of GIT quotients, as was suggested by Witten in the context of study of hybrid models.

**Elon Lindenstrauss**

*Joinings of higher rank diagonalizable actions*

Higher rank diagonalizable actions have subtle rigidity properties which are quite hard to understand. One aspect where the current state of knowledge is quite satisfactory is the study of joinings of such actions, where Einsiedler and I have a rather general classification of ergodic joinings. This classification has several striking applications, I will describe two: the work of Aka, Einsiedler, and Shapira studying joint distribution of integer points on a two dimensional sphere and the shape of its orthogonal lattice and recent work of Khayutin on orbits of the class group on pairs of CM points.

**Alex Lubotzky**

*Groups' approximation, stability and high dimensional expanders*

Several well-known open questions, such as: are all groups sofic? hyperlinear? have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm. The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomenon is proven to imply stability, and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenius stable and hence cannot be Frobenius approximated. All notions will be explained. Joint work with M. De Chiffre, L. Glebsky and A. Thom.

**Thomas Peternell**

*The Miyaoka-Yau inequality, non-abelian Hodge correspondence and uniformization of manifolds of general type*

The classical Riemann mapping theorem states in particular that the universal cover of a compact Riemann surface of genus at least 2, i.e, of negatively curved Riemann surfaces, is the unit ball. In higher dimensions, the famous Miyaoka-Yau inequality establishes a Chern class inequality for negatively curved manifolds. In case of equality the manifold is covered by the unit ball. In my talk I will discuss to what extent this picture is still valid for manifolds of "general type“. Manifolds of general type are the most natural generalizations of Riemann surfaces of genus at least 2 from the viewpoint of algebraic geometry. I will explain recent results in this direction with D.Greb, S. Kebekus and B.Taji. A main tool is the Simpson (non-abelian) Hodge correspondence on certain singular spaces.

**Mark Sapir**

*On the R. Thompson group F*

R. Thompson group F was discovered about 50 years ago and has many connections with various areas of mathematics (analysis, knot theory, logic and so on). I will talk about some of these connections and also about some unexpected algebraic properties of F.

**Benjamin Weiss**

*On minimal actions of countable groups*

I will survey some recent results on minimal actions of countable groups. This will include connections between actions and their invariant measures, the recent theory of uniformly recurrent subgroups and disjointness properties of minimal actions.

**Yosef Yomdin**

*Smooth parametrizations, and their applications in Dynamics, Analysis, and Diophantine geometry*

Smooth parametrization consists in a subdivision of a mathematical object under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. Main examples for this talk are C^k or analytic parametrizations of semi-algebraic and o-minimal sets.

We provide an overview of some results, open and recently solved problems on smooth parametrizations, and their applications in several apparently rather separated domains: Smooth Dynamics, Diophantine Geometry, and Analysis. This includes a short report on a remarkable progress, recently achieved in this (large) direction by two independent groups (G. Binyamini, D. Novikov, on one side, and R. Cluckers, J. Pila, A. Wilkie, on the other). The structure of the results, open problems, and conjectures in each of these domains shows in many cases a remarkable similarity, which we plan to stress.

We consider a special case of smooth parametrization: ``doubling coverings” (or “conformal invariant Whitney coverings”), and “Doubling chains”. We present some new results (joint with O. Friedland) on the complexity bounds for doubling coverings, doubling chains, and on the resulting bounds in Kobayashi metric and Doubling inequalities.

**Efim Zelmanov**

*Brackets, superalgebras and spectral gap*

We will discuss Poisson and contact brackets and related superalgebras.

**Tamar Ziegler**

*Concatenating cubic structure and polynomial patterns in primes*

A major difficulty in finding polynomial patterns in primes is the need to understand their distribution properties at short scales. We describe how for some polynomial configurations one can overcome this problem by concatenating short scale behavior in "many directions" to long scale behavior for which tools from additive combinatorics are available.

**Boris Zilber**

*Between model theory and physics*

There are several important issues in physics which model theory have potential to help with. First of all, in our opinion, there is the issue of adequate language and formalism, and closely related to this there is a more specific problem of giving rigorous meanings to limits and integrals used by physicists. I will present a variation of 'positive model theory' and the notion of approximation which addresses these issues and discuss some progress in defining and calculating oscillating integrals of importance in quantum physics.