Abstract: During the last 20 years there has been a considerable literature on a collection of related mathematical topics: higher degree versions of Poncelet’s Theorem, certain measures associated to some finite Blaschke products and the numerical range of finite dimensional completely non-unitary contractions with defect index 1. I will explain that without realizing it, the authors of these works were discussing OPUC.

Title: A decomposition of the Laplacian on symmetric metric graphs
Abstract
The spectrum of the Laplacian on graphs which have certain symmetry properties can be studied via a decomposition of the operator as a direct sum of one-dimensional operators which are simpler to analyze. In the case of metric graphs, such a decomposition was described by M. Solomyak and K. Naimark when the graphs are radial trees. In the discrete case, there is a result by J. Breuer and M. Keller treating more general graphs.

The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system.

Title:
Regularity via minors and applications to conformal maps.
Abstract:
Let f:\mathbb{R}^n \to \mathbb{R}^n be a Sobolev map; Suppose that the k-minors of df are smooth. What can we say about the regularity of f?
This question arises naturally in the context of Liouville's theorem, which states that every weakly conformal map is smooth. I will explain the connection of the minors question to the conformal regularity problem, and describe a regularity result for maps with regular minors.

The moduli space of curves, first appearing in the work of Riemann in the 19th century, plays an important role in geometry. After an introduction to the moduli space, I will discuss recent directions in the study of tautological classes on the moduli space following ideas and conjectures of Mumford, Faber-Zagier, and Pixton. Cohomological Field Theories (CohFTs) play an important role. The talk is about the search for a cohomology calculus for the moduli space of curves parallel to what is known for better understood geometries.

A ``random stationary signal'', more formally known as a Gaussian stationary function, is a random function f:R-->R whose distribution is invariant under real shifts (hence stationary), and whose evaluation at any finite number of points is a centered Gaussian random vector (hence Gaussian).
The mathematical study of these random functions goes back at least 75 years, with pioneering works by Kac, Rice and Wiener, who were motivated both by applications in engineering and
by analytic questions about ``typical'' behavior in certain classes of functions.

Abstract: We will prove a sharp average-case variant of a classical embedding theorem of John through the theory of nonlinear spectral gaps. We will use this theorem to provide a new answer to questions of Johnson and Lindenstrauss (1983) and Bourgain (1985) on metric dimension reduction, and explain how it leads to algorithms for approximate nearest neighbor search.

In 1925, Tarski asked whether a disk in R^2 can be partitioned into finitely many pieces which can be rearranged by isometries to form a square of the same area. The restriction of having a disk and a square with the same area is necessary. In 1990, Laczkovich gave a positive answer to the problem using the axiom of choice. We give a completely explicit (Borel) way to break the circle and the square into congruent pieces. This answers a question of Wagon. Our proof has three main components. The first is work of Laczkovich in Diophantine approximation.

Abstract:
The Ising model, and its generalisation, the Potts model, are two classical graph-colouring models for magnetism and antiferromagnetism. Albeit their simple formulation, these models were instrumental in explaining many real-world magnetic phenomena and have found various applications in physics, biology and computer science. While our understanding of these models as modeling magnets has been constantly improving since the early twentieth century, little progress was made in treatment of Potts antiferromagnets.

Abstract:
A central problem in complex analysis is how to describe zero sets of power series in terms of their coefficients. In general, it is difficult to obtain precise results for a given function. However, when the function is defined by a power series, whose coefficients are independent random variables, such results can be obtained. Moreover, if the coefficients are complex Gaussians, the results are especially elegant. In particular, in this talk I will discuss some different notions of "rigidity" of the zero sets.

Given a convex polytope P, what is the number of integer points in P? This problem is of great interest in combinatorics and discrete geometry, with many important applications ranging from integer programming to statistics. From a computational point of view it is hopeless in any dimensions, as the knapsack problem is a special case. Perhaps surprisingly, in bounded dimension the problem becomes tractable. How far can one go? Can one count points in projections of P, finite intersections of such projections, etc?

From Betti cohomology to crystalline cohomology

Read more about Landau Lecture 1: From Betti cohomology to crystalline cohomology (colloquium)

Representation theory of non-compact real groups, such as SL(2,R), is a fundamental discipline with uses in harmonic analysis, number theory, physics, and more. This theory is analytical in nature, but in the course of the 20th century it was algebraized and geometrized (the key contributions are by Harish-Chandra for the former and by Beilinson-Bernstein for the latter). Roughly and generally speaking, algebraization strips layers from the objects of study until we are left with a bare skeleton, amenable to symbolic manipulation.