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

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.

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 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.

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.

To a linear differential equation on the projective line with finitely many points of singularities, is associated a monodromy group; when the singularities are "reguar singular", then the monodromy group gives more or less complete information about the (asymptotics of the ) solutions. The cases of interest are the hypergeometric differential equations, and there is much recent work in this area, centred around a question of Peter Sarnak on the arithmeticity/thin-ness of these monodromy groups. I give a survey of these recent results.

Serre's thesis and its aftermath rolled in a golden age of algebraic topology which led to the impression that we can really understand (necessarily highly nonlinear) maps from one space to another. With the work of Thom on cobordism and Smale on immersions and the Poincare conjecture, a paradigm developed where geometric problems would be solved by reduction to algebraic topological ones.

The Globally Valued Fields (GVF) project is a joint effort with E. Hrushovski to understand (standard and) non-standard global fields - namely fields in which a certain abstraction of the product formula holds. One possible motivation is to give a model-theoretic framework for various asymptotic distribution results in global fields. Formally, a GVF is a field together with a "valuation" in the additive group of an L^1 space, such that the integral of v(a) vanishes for every non-zero a .

Abstract: We will discuss the geometry of CAT(0) (non-positively curved) cube complexes and groups which act on them. In particular, we will focus on "hyperbolic-like" behavior in such spaces.