Activities This Week

It is a well-known fact that 1D systems and non-selfadjoint operators are closely related via the notion of operator colligation. The study of the characteristic function of a colligation is related to the study of de Branges spaces of analytic functions on an open set in the Riemann sphere. It allows us, for instance, to give an alternative proof for the Beurling’s Theorem using Livšic Colligations and de Branges spaces.

In this talk, I will characterize de Branges spaces, i.e. reproducing kernel Hilbert spaces of analytic sections defined on a real compact Riemann surface, rather than on the Riemann sphere. This is done through the vessel theory, a generalization of the colligation theory to the case of $n$-tuple commuting non-selfadjoint operators. The characteristic function of a vessel is then a bundle mapping defined on a compact Riemann surfaces and which also carries the input-output relation of a 2D system. As a consequence, I will introduce a Beurling-type Theorem on finite bordered Riemann surfaces.

Problems pertaining to approximation and their applications have been extensively studied in the theory of convex bodies. In this talk we discuss several such problems, and focus on their extension to the realm of measures. In particular, we discuss variations of problems concerning the approximation of convex bodies by polytopes with a given number of vertices. This is done by introducing a natural construction of convex sets from Borel measures. We provide several estimates concerning these problems, and describe an application to bounding certain average norms.

Based on joint work with Han Huang

The functoriality conjecture is a key ingredient in the theory of automorphic forms and the Langlands program. Given two reductive groups G and H, the principle of functoriality asserts that a map r:G^->H^ between their dual complex groups should naturally give rise to a map r*:Rep(G)->Rep(H) between their automorphic representations. In this talk, I will describe the idea of functoriality, its connection to L-functions and recent work on weak functorial lifts to the exceptional group of type G_2.

Given two permutations A and B which “almost” commute, are they “close” to permutations A’ and B’ which really commute? Arzhantseva and Paunescu (2015) formalized this question and answered affirmatively. This can be viewed as a property of the equation $XY=YX$, and turns out to be equivalent to the following property of the group $Z^2 = \langle X,Y \vert XY=YX \rangle$ Every “almost-everywhere locally-defined” action of $X^2$ is close to a genuine action of $Z^2$. This leads to the notion of locally testable groups (aka “stable groups”).

We will take the opportunity to say something about “local testability” in general, which is an important subject in computer science. We will then describe some results and methods to decide whether various groups are locally testable. This will bring in some important notions in group theory, such as amenability, Kazhdan’s Property (T) and sofic groups.

Based on joint work with Alex Lubotzky.