Title: A local approach towards uniform Martin’s conjecture

by Vittorio Bard (Università degli Studi di Torino) as part of Computability theory and applications

Abstract

In 1967 Sacks asked whether there is degree invariant r.e. operator that maps x to a solution to Post’s problem relativized for x. In 1975, Lachlan proved that the answer is no if we require the operator to be degree invariant in a uniform way.

Sack’s question can be considered the forefather of Martin’s conjecture, a foundamental open problem that hypothizes that degree invariant functions under AD have limited possibilities of behavior. Following Lachlan’s example, in the late 80s Slaman and Steel proved Martin’s conjecture for unifromly degree invariant functions.

We will show that half of this result is actually the consequence of phenomena that unifromly degree invariant functions already manisfest on single Turing degrees. We also present a joint result with Patrick Lutz, in which we show that Lachlan’s result arises locally, too.

Title: Topological Groups Seminar One-Week Hiatus

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

Lecture held in Elysium.

Abstract: TBA

Title: Rogers semilattices in the analytical hierarchy

by Nikolai Bazhenov (Sobolev Institute of Mathematics) as part of Computability theory and applications

Abstract

For a countable set S, a numbering of S is a surjective map from ω onto S. A numbering ν is reducible to a numbering μ if there is a computable function f such that ν(x) = μ f(x) for all indices x. The notion of reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. We discuss recent results on Rogers semilattices induced by numberings in the analytical hierarchy. Special attention is given to the first-order properties of Rogers semilattices. The talk is based on joint works with Manat Mustafa, Sergei Ospichev, and Mars Yamaleev.

Title: Variants of Invariant Means of Amenability

by Ajit Iqbal Singh (Indian National Science Academy) as part of Topological Groups

Lecture held in Elysium.

Abstract

It all started, like many other amazing theories, in nineteen twenty-nine,

With John von Neumann, the greatest of the great.

The question of existence of a finitely additive measure on a group, a mean of a kind,

That is invariant, under any translation, neither gaining nor losing any weight.

Mahlon M. Day, in his zest and jest, giving double importance to semigroups, too,

Took up the study of conditions and properties, and named it amenability.

Erling Folner followed it up, more like a combinatorial maze to go through,

Whether or not translated set meets the original in a sizeable proportionality.

How could functional analysts sit quiet, who measure anything by their own norms,

Lo and behold, it kept coming back to the same concept over and over again.

Group algebras were just as good or bad, approximate conditions did no harms,

With the second duals of lofty Richard Arens, it became deeper, but still a fun-game.

Ever since, with the whole alphabet names, reputed experts or budding and slick,

Considering several set-ups and numerous variants of the invariance.

Actions on Manifolds or operators, dynamical systems nimble or quick,

We will have a look at some old and some new, closely or just from the fence.

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.