Let u be a harmonic function on the plane. The Liouville theorem claims that if |u| is bounded on the whole plane, then u is identically constant. It appears that if u is a harmonic function on a lattice Z^2, and |u| < 1 on 99,99% of Z^2, then u is a constant function.   Based on a joint work(in progress) with L.Buhovsky, Eu.Malinnikova and M.Sodin.  

The incompatibility of multiplicative and additive structures in various fields and rings is an important phenomena. In this talk I will talk about a special case of it. Let us consider a finite subset of integers, A. The sum set of A is the set of pairwise sums of elements of A and the product set is the set of pairwise products. Erdős and Szemeredi conjectured that either the sum set or the product set should be large, almost quadratic in size of A. The conjecture is still open. Similar questions can be asked over any ring or field.

Consider the following questions: 1. How does the volume of the set f(x_1,...,x_d) < epsilon behaves when epsilon goes to 0? 2. How does the number of solutions of the equation f(x_1,...,x_d) = 0 (mod n) behaves when n goes to infinity. I will present these and other questions which looks as if they are taken from different areas of mathematics. I'll explain the relation between those questions. Then I'll explain how this relation is used in order to show the following theorem answering a question of Larsen and Lubotzky:

The goal of the talk is to survey the emerging field of integrable probability, whose goal is to identify and analyze exactly solvable probabilistic models. The models and results are often easy to describe, yet difficult to find, and they carry essential information about broad universality classes of stochastic processes.

Abstract: Consider two Lagrangian submanifolds L, L′ in a symplectic manifold (M,ω). A Lagrangian cobordism (W;L,L′) is a smooth cobordism between L and L′ admitting a Lagrangian embedding in (([0,1]×R)×M,(dx∧dy)⊕ω) that looks like [0,ϵ)×{1}×L and (1−ϵ,1]×{1}×L′ near the boundary. In this talk we will show that under some topological constrains, an exact Lagrangian cobordism (W;L,L′) with dim(W)>5 is diffeomorphic to [0,1]×L.

We show that the boundary of a one-ended simply connected at infinity hyperbolic group with enough codimension-1 surface subgroups is homeomorphic to a sphere. By works of Markovic and Kahn-Markovic our result gives a new characterization of groups which are virtually fundamental groups of hyperbolic 3-manifolds. Joint work with B. Beeker.

For almost every real number x, the inequality |x-p/q|<1/q^a has finitely many solutions if and only if a>2. By Roth's theorem, any irrational algebraic number x also satisfies this property, so that from that point of view, algebraic numbers and random numbers behave similarly.

We will study n-dimensional badly approximable points on curves. Given an analytic non-degenerate curve in R^n, we will show that any countable intersection of the sets of weighted badly approximable points on the curve has full Hausdorff dimension. This strengthens a previous result of Beresnevich by removing the condition on weights. Compared with the work of Beresnevich, we study the problem through homogeneous dynamics. It turns out that the problem is closely related to the study of distribution of long pieces of unipotent orbits in homogeneous spaces.

Abstract: The “geometrization" of mechanics (whether classical, relativistic or quantum) is almost as old as modern differential geometry, and it nowadays textbook material. The formulation of a mathematically-sound theory for the mechanics of continuum media is still a subject of ongoing research. In this lecture I will present a geometric formulation of continuum mechanics, starting with the definition of the fundamental physical observables, e.g., force, deformation, stress and traction. The outcome of this formulation is a generalization of Newton’s "F=ma” equation for continuous media.

I’ll talk about the advances and open questions in three dimensional Ricci flow. Topics include the finiteness of the number of surgeries, the long-time behavior and flowing through singularities. No prior knowledge of Ricci flow will be assumed.

A compact Riemann surface gives rise to several families of vector spaces, associated to divisors on the Riemann surface. A finite group G of automorphisms acts on the spaces associated with invariant divisors, and a natural question is to characterize the resulting representations of G. We show how a very simple normalization for the invariant divisors can help in answering this question in a very direct manner, and if time permits present some applications.

Abstract: The development of flexible and rigid sides of symplectic and contact topology towards each other shaped this subject since its inception, and continues shaping its modern development. In the talk I will discuss the history of this struggle and describe recent breakthroughs on the flexible side.

Abstract:

We approach the formalism of quantum mechanics from the logician point of view and treat the canonical commutation relations and the conventional calculus based on it as an algebraic syntax of quantum mechanics. We then aim to establish a geometric semantics of this syntax. This leads us to a geometric model, the space of states with the action of time evolution operators, which is a limit of finite models. The finitary nature of the space allows us to give a precise meaning and calculate various classical quantum mechanical quantities.