This talk relates a well known factorization result by Szego and the quadratic inverse problem of phase retrieval. Modern applications of this include signal recovery from intensity measurements such as X-ray crystallography. After a brief historical overview, we focus on a finite-dimensional model that already exhibits the main difficulties that need to be overcome in X-ray crystallography. A stable recovery strategy is developed for this model with the help of reproducing kernel spaces and convex optimization theory.

The condition number of a polynomial system measures the sensitivity of its roots to perturbations in the coefficients. We study the condition number of random polynomial systems for a broad family of distributions. Our work is motivated by Smale's 17th problem on the complexity of solving polynomial systems.

There is an abstract version of the Gabor systems duality principle for group representations, and it is known that this duality principle has some connection with the classication problem for free group von Neumann algebras. In this talk I will revisit this general duality principle, and discuss some recent both trivial and nontrivial observations that lead to establish a generalization for the well-known Wexler-Raz biorthogonality and the Fundamental Identity for Gabor representations to general group representations. Moreover, I will present some some new connections between "super-frames" and "muti-frames" through a commutant dual pair of group representations.

The talk concerns functions acting on matrix variables and inequalities arising from them. The functions are typically (noncommutative) polynomials or rational functions. So the sets of matrices they define are thought of as (free) noncommutative semialgebraic sets. It is now known that bounded open free semialgebraic sets are convex if they are defined by Linear Matrix Inequalities (LMIs). Fortunately, LMIs constitute one of the main advances in optimization over the previous two decades, so they are intensively studied especially with regard to numerical solution. This convexity issue being settled, a next generation of questions arises: matricial (free) convex hulls, checking if one LMI dominates another and changing variables to produce convexity. The talk will focus on the latter topic, where we study analytic functions in (free) matrix variables and focus on their mapping properties. The goal is to develop free analogs (or anti-analogs) of theorems in classical several complex variables.

Fundamental properties of smooth functions (meaning C¹ or C² or...) in infinite dimensional Banach spaces can be quite different from properties of smooth functions in finite dimensional spaces. The purpose of this talk will be to explain that, by contrast, real or complex analytic functions in a large class of Banach spaces behave in many ways rather similarly to their counterparts in finite dimensional spaces.

A longstanding question asked whether every amenable, norm-closed algebra of operators acting on a Hilbert space is necessarily similar to a C*-algebra. This was answered in the negative by Y. Choi, I. Farah and N. Ozawa. In this talk, we discuss a positive result obtained with Alexey Popov which shows that if A is an abelian, amenable algebra of Hilbert space operators, then A is indeed similar to a C*-algebra.

When does the spectrum of an operator determine the operator uniquely? This question and its many versions have been studied extensively in the field of inverse spectral theory for differential operators. Several notable mathematicians have worked in this area. Recent results have further fueled these studies by relating the completeness problems of families of functions to the inverse spectral problems of the Schrödinger operator. In this talk, we will discuss the role played by the Toeplitz kernel approach in answering some of these questions, as described by Makarov and Poltoratski. We will also describe some new results using this approach. This is joint work with Mishko Mitkovski.

As is well known, a complex m × m matrix A is a commutator (i.e., there are matrices B and C of the same dimensions as A such that A=[B,C]=BC-CB) if and only if A has zero trace. If ∥·∥ is the operator norm from ℓ₂^m to itself and ∣·∣ is any ideal norm on m × m matrices, then clearly for any A,B,C as above ∣A∣≤2∥B∥∣C∣. Does the converse hold? That is, if A has zero trace are there m×m matrices B and C such that A=[B,C] and ∥B∥∣C∣≤K∣A∣ for some absolute constant K? If not, what is the behavior of the best K as a function of m? The talk will concentrate on two recent results on this problem. The first is a couple of years old result of Johnson, Ozawa and myself, which gives some partial answers to this problem for the most interesting case of ∣·∣=∥·∥. The second is a more recent result of Angel and myself which solves the problem for ∣·∣ equal to the Hilbert-Schmidt norm.

We study a convolution operation on polynomials which may be seen as a finite-dimensional analogue of the free convolution of two measures in free probability theory. We show that this operation preserves real-rootedness, and establish bounds on the extreme roots of the convolution of two polynomials via the inverse Cauchy transforms. We use these properties to study the expected characteristic polynomials of random regular graphs, and in particular to establish the existence of bipartite Ramanujan graphs of every degree and every size. Joint work with A. Marcus and D. Spielman.

In this talk we will show that the distortion is at least of order log(h)^1/p when embedding the countably branching tree of height h into a Banach space satisfying property (β) of Rolewicz with modulus of power type p∈(1,∞). The tightness of the lower bound will also be discussed. As an application, we will then show how this unifies and extends a series of results in the nonlinear quotient theory of Banach spaces. This is joint work with Florent Baudier.