Schedule

Title: Sums and products in finite fields

Abstract: In 2004 J. Bourgain, N. Katz and T. Tao published a celebrated paper on sum-product estimates in finite fields. The paper had numerous significant applications and many extensions in various directions. Our talk is a survey on different sum-product type problems in finite fields (mainly, fields of prime order).

Title: Super-approximation and its applications

Abstract: Let G be a finitely generated subgroup of GL(n, Q). Under certain algebraic conditions, strong approximation describes the closure of G with respect to its congruence topology, Super-approximation essentially tells us how dense G is in its closure!

Here is my plan for this talk:

1. I will start with the precise formulation of this property

2. Some of the main results on this subject will be mentioned

3. Some of the (unexpected) applications of super-approximation will be mentioned, e.g. Banach-Ruziewicz problem, orbit equivalence rigidity, variation of Galois representations

4. Some of the auxiliary results that were needed in the proof of super-approximation will be mentioned: sum-product phenomena, existence of small solutions.

Title: Thin Groups and Applications

Abstract: We will discuss a variety of problems in which “thin groups” appear naturally, as well as some tools used to make partial progress towards said problem.

Title: Concatenating cubic structures

Abstract: We describe concatenation results for uniformity norms in ergodic and combinatorial categories. Roughly speaking we show that if a system exhibits nil-behavior with respect to two different communing actions, then it exhibits nil-behavior (of higher step) with respect to their joint action. These results have applications to counting polynomial progressions in primes. Joint work with T. Tao.

Title: Working with Bourgain

Abstract: The history of my collaboration with Bourgain, how we worked together on four very different papers, and some unsolved questions which arose from our conversations.

Title: The polynomial method and the restriction problem

Abstract: About ten years ago, Zeev Dvir solved the finite field Kakeya problem, giving a short proof of the conjecture using polynomials in an unexpected way. How much can this method help us to understand the original Kakeya problem and the restriction problem in Euclidean space? I will describe an approach to the restriction problem using the polynomial method, which gives small improvements on the best known exponents.

Title: Decouplings and applications: a journey from continuous to discrete

Abstract: I will give an overview of the methods developed with Jean Bourgain and most recently with Larry Guth.

Title: Old-new perspectives on the winding number

Abstract: Where Cauchy, Fourier, Sobolev, Gelfand and Bourgain meet.

Title: Algebraic related structures and the reason behind some classical constructions in Convex Geometry and Analysis.

Abstract: The main goal of the talk is to show how some classical constructions in Geometry and Analysis appear (and in a unique way) from elementary and very simple properties. For example, the polarity relation and support functions are very important and well known constructions in Convex Geometry, but what elementary properties uniquely imply these constructions, and what would be their functional versions, say, in the class of log-concave functions? It turns out that they are uniquely defined also for this class, as well as for many other classes of functions.

In this talk we will use these Geometric results as an introduction to the main topic which involves the analogous results in Analysis. We will start the Analysis part by characterizing the Fourier transform (on the Schwartz class in R^n) as, essentially, the only map which transforms the product to the convolution, and discuss a very surprising rigidity of the Chain Rule Operator equation (which characterizes the derivation operation). There will be more examples pointing to an exciting continuation of this direction in Analysis.

The results of the geometric part are mostly joint work with Shiri Artstein-Avidan, and of the second, Analysis part, are mostly joint work with Hermann Koenig.

Title: On Sidon sets

Abstract: We will survey the classical theory of Sidon sets of characters on compact groups (Abelian or not), in particular for sets of integers, with a review of the main problems left open. We will then give several recent extensions to Sidon sets and randomly Sidon sets in bounded orthonormal systems, initiated by recent work due to Bourgain and Lewko.

Title: Combinatorics of Boolean functions, and some applications

Abstract: I will discuss some questions and results about influence, noise and harmonic analysis. I will mention three important directions that arose from Jean Bourgain’s work in this area and several novel hopes for applications: One representing an ongoing work with Jeff Kahn is for questions on random graphs, and another representing ongoing work with Gadi Kozma is for questions on percolation in three dimensions.

Title: Matrix and operator scaling: Analysis in the service of Algebra, Combinatorics, Geometry and more..

Abstract: I will define what it means to make matrices and tensors “doubly stochastic” via scaling, describe efficient algorithms for doing so, and give several applications of this ability to problems in diverse areas, from elementary incidence geometry to the word problem over free skew fields, from algebraic complexity theory to the Brascamp-Lieb inequalities.

Title: Quasiperiodic Schroedinger operators with multiple frequencies

Abstract: We describe joint work with Michael Goldstein and Mircea Voda on discrete Schroedinger operators on the line with potentials defined on higher-dimensional tori. We describe properties of these operators related to Anderson localization, the separation of the eigenvalues, and the shape of the spectrum.

Title: Long time dynamics of random data nonlinear wave and dispersive equations

Abstract: In this talk we show how certain well-posedness results that are not available using only deterministic techniques (eg. Fourier and harmonic analysis) can be obtained when introducing randomization in the set of initial data and using powerful but still classical tools from probability as well. These ideas go back to seminal work by J.Bourgain on the invariance of Gibbs measures associated to dispersive PDE.

We will explain some of these ideas and describe in more detail a probabilistic propagation of regularity result for certain almost sure globally well-posed dispersive equations. This talk is based on joint work with G.Staffilani.

Title: Soliton resolution along a sequence of times for the energy critical wave equation

Abstract: We will describe the proof of the recent result stated in the title, due to Duyckerts, Jia, Kenig and Merle.

Title: Effective density of unipotent orbits

Abstract: Raghunathan conjectured that if G is a Lie group, Gamma a lattice, p in G/Gamma, and U an (sd)-unipotent group then the closure of U.p is homogeneous (a periodic orbit of a subgroup of G). This conjecture was proved by Ratner in the early 90’s via the classification of invariant measures, significant special cases were proved earlier by Dani and Margulis using a different, topological dynamics approach.

The proof of the Raghunathan conjecture given by Ratner as well as the proofs given by Dani and Margulis are not effective, nor do they provide rates – e.g. if p is generic in the sense that it does not lie on a periodic orbit of any proper subgroup L<G with U<=L, these proofs do not give an estimate (possibly depending on Diophantine-type properties of the pair (p, U) how large a piece of an orbit is needed so that it comes within distance epsilon of any point in a given compact subset of G/Gamma.

I will present joint work with Margulis, Mohammadi and Shah giving an effective and quantitative density theorem for orbits of unipotent groups.

Title: Entropy, Mahler measure and Bernoulli convolutions

Abstract: I will describe past and ongoing joint work with P.Varju in which we relate the Lehmer conjecture about the Mahler measure of polynomials with uniform exponential growth of linear groups via entropy-increasing arguments for Bernoulli convolutions, which in turn yield new insights regarding the regularity of the stationary measure.