Schedule
Analysis and Beyond  Celebrating Jean Bourgain's Work and Impact
Video lectures can be found here: https://www.ias.edu/video/analysisbeyond/2016/052124
Saturday, May 21
9:00 am  Registration and Continental Breakfast, Fuld Hall 
10:00 am  Robbert Dijkgraaf Institute welcome, Wolfensohn Hall 
10:10 am  Peter Sarnak (Session Chair) 
10:15 am  Sergei Konyagin (Moscow State University) 
Title: Sums and products in finite fields Abstract: In 2004 J. Bourgain, N. Katz and T. Tao published a celebrated paper on sumproduct estimates in finite fields. The paper had numerous significant applications and many extensions in various directions. Our talk is a survey on different sumproduct type problems in finite fields (mainly, fields of prime order). 

11:15 am  Coffee Break, Wolfensohn Hall 
11:35 am  Alireza Salehi Golsefidy (UCSD) 
Title: Superapproximation 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, Superapproximation 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 superapproximation will be mentioned, e.g. BanachRuziewicz problem, orbit equivalence rigidity, variation of Galois representations 4. Some of the auxiliary results that were needed in the proof of superapproximation will be mentioned: sumproduct phenomena, existence of small solutions. 

12:35 pm  Lunch, Simons Hall 
1:55 pm  Alex Gamburd (Session Chair) 
2:00 pm  Alex Kontorovich (IAS) 
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. 

3:00 pm  Afternoon Tea Break, Wolfensohn Hall Lobby 
3:30 pm  Tamar Ziegler (Hebrew University of Jerusalem) 
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 nilbehavior with respect to two different communing actions, then it exhibits nilbehavior (of higher step) with respect to their joint action. These results have applications to counting polynomial progressions in primes. Joint work with T. Tao. 

4:30 pm  Enrico Bombieri (IAS) 
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. 

Workshop Buffet Dinner to immediately follow conclusion of program. 
Sunday, May 22
9:30 am  Coffee, Wolfensohn Hall Lobby 
10:00 am  Terence Tao (Session Chair) 
10:05 am  Larry Guth (MIT) 
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. 

11:05 am  Coffee Break, Wolfensohn Hall 
11:30 am  Ciprian Demeter (Indiana University) 
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. 

12:25 pm  Lunch, Simons Hall 
1:55 pm  Assaf Naor (Session Chair) 
2:00 pm  Haim Brezis (Rutgers University) 
Title: Oldnew perspectives on the winding number Abstract: Where Cauchy, Fourier, Sobolev, Gelfand and Bourgain meet. 

3:00 pm  Afternoon Tea Break, Wolfensohn Hall Lobby 
3:30 pm  Vitali Milman (Tel Aviv University) 
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 logconcave 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 ArtsteinAvidan, and of the second, Analysis part, are mostly joint work with Hermann Koenig. 

4:30 pm  Gilles Pisier (Texas A&M) 
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. 

5:45 pm  Program concludes, participants on their own for dinner 
Monday, May 23
9:30 am  Coffee, Wolfensohn Hall Lobby 
10:00 am  Peter Sarnak (Session Chair) 
10:05 am  Gil Kalai (Hebrew University of Jerusalem & Yale Univ.) 
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. 

11:05 am  Coffee Break, Wolfensohn Hall Lobby (20 minutes) 
11:25 am  Avi Wigderson (IAS) 
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 BrascampLieb inequalities. 

12:25 pm  Lunch, Simons Hall 
2:15 pm  Svetlana Jitomirskaya (Session Chair) 
2:30 pm  Wilhelm Schlag (University of Chicago) 
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 higherdimensional tori. We describe properties of these operators related to Anderson localization, the separation of the eigenvalues, and the shape of the spectrum. 

3:30 pm  Afternoon Tea Break, Fuld Hall Common Room 
4:00 pm  Gigliola Staffilani (Session Chair) 
4:05 pm  Andrea Nahmod (University of Massachusetts, Amherst) 
Title: Long time dynamics of random data nonlinear wave and dispersive equations Abstract: In this talk we show how certain wellposedness 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 wellposed dispersive equations. This talk is based on joint work with G.Staffilani. 

5:05 pm  Carlos Kenig (University of Chicago) 
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. 

6:00 pm  Program concludes, participants on their own for dinner 
Tuesday, May 24
9:30 am  Coffee, Wolfensohn Hall Lobby 
10:00 am  Peter Varjú (Session Chair) 
10:05 am  Elon Lindenstrauss (Hebrew University of Jerusalem) 
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 Diophantinetype 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. 

11:05 am  Coffee Break, Wolfenson Hall Lobby 
11:25 am  Emmanuel Breuillard (ParisSud) 
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 entropyincreasing arguments for Bernoulli convolutions, which in turn yield new insights regarding the regularity of the stationary measure. 

12:30 pm  Lunch, Simons Hall; Conference concludes 