Title: Proving conformal invariance using discrete holomorphicity
Abstract:

Billiards in polygons can exhibit some bizarre behavior, some of which can be explained by deep connections to several seemingly unrelated branches of mathematics. These include algebraic geometry (and in particular Hodge theory), Teichmuller theory and ergodic theory on homogeneous spaces. I will attempt to give a gentle introduction to the subject. A large part of this talk will be accessible to undergraduates.

The problem of bounding the number of rational or algebraic points of a given height in a transcendental set has a long history. In 2006 Pila and Wilkie made fundamental progress in this area by establishing a sub-polynomial asymptotic estimate for a very wide class of transcendental sets. This result plays a key role in Pila-Zannier's proof of the Manin-Mumford conjecture, Pila's proof of the Andre-Oort conjecture for modular curves, Masser-Zannier's work on torsion anomalous points in elliptic families, and many more recent developments.

A recent result of Lubetzky and Peres showed that the random walk on a $q+1$-regular Ramanujan graph has $L^{1}$-cutoff, and that its “almost-diameter” is optimal. Similar optimal results were proven by other authors in various contexts, e.g. Parzanchevski-Sarnak for Golden Gates and Ghosh-Gorodnik-Nevo for Diophantine approximations. Those results rely in general on a naive Ramanujan conjecture, which is either very hard, unknown, or even false in some situations. We show that a general version of those results can be proven using the density hypothesis of Sarnak-Xue.

A common theme in many extremal problems in graph theory is the
relation between local and global properties of graphs. We will
consider the following variant of this theme: suppose a graph G
is far (in some well defined sense) from satisfying property P.
Must G contain a small proof of this fact? We will show that
for many natural graph properties the answer is Yes. In particular,
we will show that the answer is Yes whenever P is a semi-algebraic
graph property, thus conforming a conjecture of Alon.
Joint work with L. Gishboliner

A measure on the plane is called self-affine if it is stationary with respect to a finitely supported measure on the affine group of R^2. Under certain randomization, it is known that the Hausdorff dimension of these measures is almost surely equal to the Lyapunov dimension, which is a quantity defined in terms of the linear parts of the affine maps. I will present a result which provides conditions for equality between these two dimensions, and connects the theory of random matrix products with the dimension of self-affine measures.

Abstract: I will report on some old and new classification results
in equivariant symplectic geometry,
expanding on my classification, joint with Sue Tolman,
of Hamiltonian torus actions with two dimensional quotients.

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: Given two permutations A and B which "almost" commute, are they "close" to permutations A' and B' which really commute? This can be seen as a question about a property the equation XY=YX.
Studying analogous problems for more general equations (or sets of equations) leads to the notion of "locally testable groups" (aka "stable groups").

G.D.Birkhoff introduced a mathematical billiard inside of a convex domain as the motion
of a massless particle with elastic reflection at the boundary. A theorem of Poncelet says
that the billiard inside an ellipse is integrable, in the sense that the neighborhood of the
boundary is foliated by smooth closed curves and each billiard orbit near the boundary
is tangent to one and only one such curve (in this particular case, a confocal ellipse).
A famous conjecture by Birkhoff claims that ellipses are the only domains with this

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.