Title: Topologies, idempotents and ideals

by Nico Spronk (University of Waterloo) as part of Topological Groups

Lecture held in Elysium.

Abstract

Let $G$ be a topological group. I wish to exhibit a bijection between (i) a certain class of weakly almost periodic topologies, (ii) idempotents in the weakly almost periodic compactification of $G$, and (iii) certain ideals of the algebra of weakly almost periodic functions. This has applications to decomposing weakly almost periodic representations on Banach spaces, generalizing results which go back to many authors.

Moving to unitary representations, I will develop the Fourier-Stieltjes algebra $B(G)$ of $G$, and give the analogous result there. As an application, I show that for a locally compact connected group, operator amenability of $B(G)$ implies that $G$ is compact, partially resolving a problem of interest for 25 years.

Title: Potential Theory on Stratified Lie Groups

by Mukund Madhav Mishra (Hansraj College) as part of Topological Groups

Lecture held in Elysium.

Abstract

Stratified Lie groups form a special subclass of the class of nilpotent Lie groups. The Lie algebra of a stratified Lie group possesses a specific stratification (and hence the name), and an interesting class of anisotropic dilations. Among the linear differential operators of degree two, there exists a family that is well behaved with the automorphisms of the stratified Lie group, especially with the anisotropic dilations. We shall see that one such family of operators mimics the classical Laplacian in many aspects, except for the regularity. More specifically, these Laplace-like operators are sub-elliptic, and hence referred to as the sub-Laplacians. We will review certain interesting properties of functions harmonic with respect to the sub-Laplacian on a stratified Lie group, and have a closer look at a particular class of stratified Lie groups known as the class of Heisenberg type groups.

Title: Compact Quantum Groups and their Semidirect Products

by Sutanu Roy (National Institute of Science Education and Research) as part of Topological Groups

Lecture held in Elysium.

Abstract

Compact quantum groups are noncommutative analogs of compact groups in the realm of noncommutative geometry introduced by S. L. Woronowicz back in the 80s. Roughly, they are unital C*-bialgebras in the monoidal category (given by the minimal tensor product) of unital C*-algebras with some additional properties. For real 0<|q|<1, q-deformations of SU(2) group are the first and well-studied examples of compact quantum groups. These examples were constructed independently by Vaksman-Soibelman and Woronowicz also back in the 80s. In fact, they are examples of a particular class of compact quantum groups namely, compact matrix pseudogroups. The primary goal of this talk is to motivate and discuss some of the interesting aspects of this theory from the perspective of the compact groups. In the second part, I shall briefly discuss the semidirect product construction for compact quantum groups via an explicit example. The second part of this will be based on a joint work with Paweł Kasprzak, Ralf Meyer and Stanislaw Lech Woronowicz.

Title: Towards a unifying approach to algebraic and coarse entropy

by Nicolò Zava (University of Udine) as part of Topological Groups

Lecture held in Elysium.

Abstract

In each situation, entropy associates to a self-morphism a value that estimates the chaos created by the map application. In particular, the algebraic entropy $h_{alg}$ can be computed for (continuous) endomorphisms of (topological) groups, while the coarse entropy $h_c$ is associated to bornologous self-maps of locally finite coarse spaces. Those two entropy notions can be compared because of the following observation. If $f$ is a (continuous) homomorphism of a (topological) group $G$, then $f$ becomes automatically bornologous provided that $G$ is equipped with the compact-group coarse structure. For an endomorphism $f$ of a discrete group, $h_{alg}(f)=h_c(f)$ if $f$ is surjective, while, in general, $h_{alg}(f)

eq h_c(f)$. That difference occurs because in many cases, if $f$ is not surjective, then $h_c(f)=0$.

In the first part of the talk, after briefly recalling the large-scale geometry of topological groups, we define the coarse entropy and discuss its relationship with the algebraic entropy. The second part is dedicated to the introduction of the algebraic entropy of endomorphisms of $G$-sets (i.e., sets endowed with group actions). We show that it extends the usual algebraic entropy of group endomorphisms and we provide evidence that it can represent a useful modification and generalisation of the coarse entropy that overcome the non-surjectivity issue.

Title: Topological Groups Seminar Two-Week Hiatus

by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract: TBA

Title: Topological Groups Seminar Two-Week Hiatus

by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract: TBA

Title: Abelian Varieties as Algebraic Protori?

by Wayne Lewis (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract

An outcome of the structure theory of protori (compact connected abelian groups) is their representability as quotients of $mathbb{A}^n$ for the ring of adeles $mathbb{A}$. $mathbb{A}$ does not contain zeros of rational polynomials, but rather representations of zeros. Investigating the relations between algebraicity of complex tori and algebraicity of protori leads one to the problem of computing the Pontryagin dual of $mathbb{A}/mathbb{Z}$. Applying an approach by Lenstra in the setting of profinite integers to the more general $mathbb{A}$ leads to a definition of the closed maximal $Lenstra$ $ideal$ $E$ of $mathbb{A}$, whence the locally compact field of $adelic$ $numbers$ $mathbb{F}=mathbb{A}/E$, providing a long-sought connection to $mathbb{C}$ enabling one to define a functor from the category of complex tori to the category of protori – is it possible to do so in a way that preserves algebraicity? While $mathbb{F}$ marks tentative progress, much work remains…

Title: Accounting with $mathbb{QP}^infty$

by Wayne Lewis (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract

Rational projective space provides a useful accounting tool in engineering decompositions of $mathbb{Q}[x]$ for desired effect. The device is useful for defining a correspondence between summands of such a decomposition and elements of a partition of $mathbb{A}$. This mechanism is applied to a decomposition of $mathbb{Q}[x]$ relative to which the correspondence gives the $Lenstra$ $ideal$ $E$, a closed maximal ideal yielding the $adelic$ $numbers$ $mathbb{F}=frac{mathbb{A}}{E}$.