פעילויות השבוע

In this talk, we present several dynamical and fractal-dimensional ways of characterizing the spectral measures of Schrödinger operators, such as Rajchman behavior and Hausdorff/packing dimensions, and discuss the extent to which these properties are stable under rank-one perturbations.

We begin with the concrete setting of half-line Schrödinger operators, where a theorem of Gordon shows that generic rank-one perturbations eliminate pure point spectrum, ruling out the most extreme dynamical and dimensional behavior. I will then describe constructions demonstrating that properties only slightly weaker than pure point spectrum can, in fact, be entirely stable: for certain sparse half-line models, both packing-dimension-zero and non-Rajchman behavior persist for every rank-one perturbation.

In the second part, we examine how spectral dimensions behave when passing from the whole line to the half-line. I will present an operator whose spectral measure on the line has Hausdorff dimension one, whereas every half-line restriction - under any boundary condition - has dimension zero, even though the two settings differ only by a finite-rank perturbation.

I shall give simplicial reformulations of several equivalent characterisations of stability of a first order theory. In particular, I shall reformulate without any word of logic the statement of the Sela‘s theorem that the theory of a free group is stable.

I will make an effort to be self-contained and remind the definitions of my previous talk. In particular, I shall define a semantics of first order formulas with respect to any simplicial set, where a predicate is assigned a subset of simplicies rather than n-tuples. From this point of view, the type space functor defined last time, is the canonical model of a first order theory.

Delsarte‘s method provides a framework for analysis of sphere packing problems in terms of positive definite functions. Constrained packing, where only a prespecified set of angles between points may occur, is, in turn, naturally governed by the interpolation theory for positive definite functions. In this talk, we discuss the general theory of positive definite functions on the sphere, Delsarte‘s method and its geometric kernel embedding interpretation, related interpolation problems, and other topics.

Based on joining work with Sujit Sakharam Damase.

In this talk we will be concerned with actions of finite groups, and more specifically with elements fixing “few” points in such actions. One of the goals will be to explain why this is interesting, highlighting applications to polynomials, random generation and, time permitting, monodromy groups of curves.

We will discuss questions of invertibility of large random matrices, and estimating the smallest singular value of random matrices in the non-asymptotic setting, via methods of asymptotic geometric analysis. We will discuss a construction of an efficient net on the high-dimensional sphere, and its applications to the questions of invertibility of inhomogeneous and log-concave random matrices.