Title: Dynamics of Distal Actions on Locally Compact Groups

by Riddhi Shah (Jawaharlal Nehru University, New Delhi, India) as part of Topological Groups

Lecture held in Elysium.

Abstract

Distal maps were introduced by David Hilbert on compact spaces to study non-ergodic maps. A homeomorphism T on a topological space X is said to be distal if the closure of every double T-orbit of (x, y) does not intersect the diagonal in X x X unless x=y. Similarly, a semigroup S of homeomorphisms of X is said to act distally on X if the closure of every S-orbit of (x,y) does not intersect the diagonal unless x=y. We discuss some properties of distal actions of automorphisms on locally compact groups and on homogeneous spaces given by quotients modulo closed invariant subgroups which are either compact or normal. We relate distality to the behaviour of orbits. We also characterise the behaviour of convolution powers of probability measures on the group in terms of the distality of inner automorphisms.

Title: The coding power of products of partitions

by Lu Liu (Central South University) as part of Computability theory and applications

Abstract

Given two combinatorial notions P0 and P1, can we encode P0 via P1. In this talk we address the question where P0 is a 3-partition of integers and P1 is a product of finitely many 2-partitions of integers.

We firstly reduce the question to a lemma which asserts that certain Pi01 class of partitions admit two members violating a particular combinatorial constraint. Then we took a digression to see how complex does the class has to be so as to maintain the cross constraint.

On the other hand, reducing the complexity of the two members in the lemma in certain ways will answer an open question concerning a sort of Weihrauch degree of stable Ramsey’s theorem for pairs. It turns out the resulted strengthen of the lemma is a basis theorem for Pi01 class with additional constraint. We look at several such variants of basis theorem, among them some are unknown.

We end up by introducing some results and questions concerning product of infinitely many partitions.

Title: The group algebra of a compact group and Tannaka duality for compact groups

by Karl Hofmann (Technische Universität Darmstadt) as part of Topological Groups

Lecture held in Elysium.

Abstract

In the 4th edition of the text- and handbook “The Structure of Compact Groups”,

de Gruyter, Berlin-Boston, having appeared June 8, 2020, Sidney A. Morris and

I decided to include, among material not contained in earlier editions, the Tannaka-Hochschild Duality Theorem which says that $the$ $category$ $of$ $compact$ $groups$ $is,dual$

$to$ $the$ $category,of,real,reductive$ $Hopf$ $algebras$. In the lecture I hope to explain

why this theorem was not featured in the preceding 3 editions and why we decided

to present it now. Our somewhat novel access led us into a new theory of real

and complex group algebras for compact groups which I shall discuss. Some Hopf

algebra theory appears inevitable. Recent source: K.H.Hofmann and L.Kramer,

$On$ $Weakly,Complete,Group,Algebras$ $of$ $Compact$ $Groups$, J. of Lie Theory $bold{30}$ (2020), 407-424.

Karl H. Hofmann

Title: Reduction games, provability, and compactness

by Sarah Reitzes (University of Chicago) as part of Computability theory and applications

Abstract

In this talk, I will discuss joint work with Damir D. Dzhafarov and Denis R. Hirschfeldt. Our work centers on the characterization of problems P and Q such that P is omega-reducible to Q, as well as problems

P and Q such that RCA_0 proves Q implies P, in terms of winning strategies in certain games. These characterizations were originally introduced by Hirschfeldt and Jockusch. I will discuss extensions and generalizations of these characterizations, including

a certain notion of compactness that allows us, for strategies satisfying particular conditions, to bound the number of moves it takes to win. This bound is independent of the instance of the problem P being considered.

Visualizations of Schottky GroupsOften times, to truly understand a mathematical object it must be viewed from several different perspectives, involving several different foundations. In this thesis, I will present eye-catching visualization techniques for something called Schottky groups, which are similar to general linear groups of degree n over the complex numbers. This is built on an understanding of complex numbers to explore the structure of specific kinds of linear groups. My discussions will include the mathematics needed to imagine such an object, as well as the numerics required to compute such an object. Parts of this will include discussion of programming these objects.This thesis is an exploration of a particular kind of projective linear group, Schottky groups, and their variations by understanding them geometrically. The general idea here is how we can study fractal-like sets by looking at images of circles representing group elements. This exploration will lead to the discovery and understanding of Schottky groups.Furthermore, I do this exploration using first and foremost, a background in complex analysis, and in particular, a deep understanding of Mobius maps. This leads to discovery and understanding of anti-Mobius maps, which will be our main tool in understanding reflections. Additionally, I will use Python3 to program examples and experiments of these ideas. The Python code will provide specific examples. as well as mathematical challenges of its own. These challenges will include creating Mobius and anti-Mobius classes, and using these classes to perform all of these operations.

Title: A Survey on Analog Models of Computation

by Amaury Pouly (CNRS) as part of Computability theory and applications

Abstract: TBA