Colloquium Ical Atom Mailing List

We introduce a new class of locally compact groups, namely the strongly compactly covered groups, which are the Hausdorff topological groups G such that every element of G is contained in a compact open normal subgroup of G. For continuous endomorphisms ϕ:G→G of these groups we compute the algebraic entropy and study its properties. Also an Addition Theorem is available under suitable conditions.

This is joint work with Anna Giordano Bruno and Daniele Toller.

Bounded harmonic functions on groups are closely related to random walks on groups. It has long been known that all abelian groups, and more generally, virtually nilpotent groups are “Choquet-Deny groups”: these groups cannot support non-trivial bounded harmonic functions. Equivalently, their Furstenberg-Poisson boundary is trivial, for any random walk. I will present a recent result where we complete the classification of discrete countable Choquet-Deny groups, proving a conjuncture of Kaimanovich-Vershik. We show that any finitely generated group which is not virtually nilpotent, is not Choquet-Deny. Surprisingly, the key is not the growth rate of the group, but rather the algebraic infinite conjugacy class property (ICC).

This is joint work with Joshua Frisch, Omer Tamuz and Pooya Vahidi Ferdowsi.

We will explain the Geometrization Theorem proved by Perelman in 2003 and we will talk about the Virtual Fibering Theorem proved several years ago by Ian Agol and Dani Wise. I will not assume any previous knowledge of 3-manifold topology.

A subset S of a group G invariably generates G if for every choice of $g(s)\in G$ ,$s\in S$ the set ${s^g(s):s\in S}$ is a generating set of G. We say that a group G is invariably generated if such S exists, or equivalently if S=G invariably generates G. In this talk, we study invariable generation of Thompson groups. We show that Thompson group F is invariably generated by a finite set, whereas Thompson groups T and V are not invariable generated. This is joint work with Tsachik Gelander and Kate Juschenko.

Let M be a Riemannian manifold with a given volume form and hyperbolic fundamental group. We will explain how to construct coclasses in the cohomology of the group of volume preserving diffeomorphisms (or homeomorphisms) of M. As an application, we show that the 3rd bounded cohomology of those groups is highly non-trivial.

A long-standing open problem, known as Hadwiger’s covering problem, asks what is the smallest natural number N(n) such that every convex body in $R^n$ can be covered by a union of the interiors of at most N(n) of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of $\binom{2n}{n}n\ln n$.

In this talk, I will discuss some history of this problem and present a new result in which we improve this bound by a sub-exponential factor. Our approach combines ideas from previous work, with tools from Asymptotic Geometric Analysis. Namely, we make use of measure concentration in the form of thin-shell estimates for isotropic log-concave measures.

If time permits we shall discuss some other methods and results concerning this problem and its relatives.

Joint work with H. Huang, B. Vritsiou, and T. Tkocz

The Hardy space $H^2(\mathbb{D})$ is the Hilbert space of analytic functions on the unit disc with square summable Taylor coefficients is a fundamental object both in function theory and in operator algebras. The operator of multiplication by the coordinate function turns $H^2(\mathbb{D})$ into a module over the polynomial ring $\mathbb{C}[z]$. Moreover, this space is universal, in the sense that whenever we have a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z]$, such that $z$ acts by a pure row contraction, we have that $\cH$ is a quotient of several copies of $H^2(\mathbb{D})$ by a submodule.

There are two multivariable generalizations of this property, one commutative and one free. I will show why the free generalization is in several ways the correct one. We will then discuss quotients of the noncommutative Hardy space and their associated universal operator algebras. Each such quotient naturally gives rise to a noncommutative analytic variety and it is a natural question to what extent does the geometric data determine the operator algebraic one. I will provide several answers to this question.

Only basic familiarity with operators on Hilbert spaces and complex analysis is assumed.

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. No prior knowledge on quantum mechanics or representation theory will be assumed.

We will review some results about superanalogue of Borel-Weil-Bott theorem, explain the role of Weyl groupoid and prove a weak version of BGG reciprocity. Then we illustrate how BGG reciprocity can be used for computing the Cartan matrix of the category of finite dimensional representations of the nontivial central extension of the periplectic supergroup P(4).

The problem of computational super-resolution asks to recover fine features of a signal from inaccurate and bandlimited data, using an a-priori model as a regularization. I will describe several situations for which sharp bounds for stable reconstruction are known, depending on signal complexity, noise/uncertainty level, and available data bandwidth. I will also discuss optimal recovery algorithms, and some open questions.