Title: The Semigroup $beta S$

by Dona Strauss (University of Leeds) as part of Topological Groups

Lecture held in Elysium.

Abstract

If $S$ is a discrete semigroup, the semigroup operation on $S$ can be extended to a semigroup operation on its Stone–Čech compactification $beta S$. The properties of the semigroup $beta S$ have been a powerful tool in topological dynamics and combinatorics.

I shall give an introductory description of the semigroup $beta S$, and show how its properties can be used to prove some of the classical theorems of Ramsey Theory.

Title: Part 1 of Martin’s Conjecture for Order Preserving Functions

by Patrick Lutz (UC Berkeley) as part of Computability theory and applications

Abstract

Martin’s conjecture is an attempt to make precise the idea that the only natural functions on the Turing degrees are the constant functions, the identity, and transfinite iterates of the Turing jump. The conjecture is typically divided into two parts. Very roughly, the first part states that every natural function on the Turing degrees is either eventually constant or eventually increasing and the second part states that the natural functions which are increasing form a well-order under eventual domination, where the successor operation in this well-order is the Turing jump.

In the 1980′s, Slaman and Steel proved that the second part of Martin’s conjecture holds for order-preserving Borel functions. In joint work with Benny Siskind, we prove the complementary result that (assuming analytic determinacy) the first part of the conjecture also holds for order-preserving Borel functions (and under AD, for all order-preserving functions). Our methods also yield several other new results, including an equivalence between the first part of Martin’s conjecture and a statement about the Rudin-Keisler order on ultrafilters on the Turing degrees.

In my talk, I will give an overview of Martin’s conjecture and then describe our new results.

Title: The characterization of Weihrauch reducibility in systems containing $E$-$PA^omega$ + $QF$-$AC^{0,0}$

by Patrick Uftring (TU Darmstadt) as part of Computability theory and applications

Abstract

We characterize Weihrauch reducibility in E-PAω + QF-AC0,0 and all systems containing it by the provability in a linear variant of the same calculus using modifications of Gödel’s Dialectica interpretation that incorporate ideas from linear logic, nonstandard arithmetic, higher-order computability, and phase semantics. A full preprint is available here: https://arxiv.org/abs/2003.13331

Title: Closed Lie Ideals and Center of Generalized Group Algebras

by Bharat Talwar (University of Dehli) as part of Topological Groups

Lecture held in Elysium.

Abstract

The closed Lie ideals of the generalized group algebra $L^1(G,A)$ are characterized in terms of elements of the group $G$, elements of the algebra $A$, and the modular function $Delta$ of the group $G$. Conditions under which for a given closed Lie ideal $Lsubseteq A$ the subspace $L^1(G,L)$ is a Lie ideal, and vice versa, are discussed. The center of $L^1(G,A)$ is characterized, followed by a discussion regarding a very special projection in $L^1(G,A)$. Finally, a few restrictions are imposed on $G$ and $A$ under which $mathcal{Z}(L^1(G,A))congmathcal{Z}(L^1(G))otimes^gammamathcal{Z}(A)$.

The presentation is based on joint work with Ved Prakash Gupta and Ranjana Jain.

Title: The higher levels of the Weihrauch lattice

by Alberto Marcone (Università di Udine) as part of Computability theory and applications

Abstract

The classification of mathematical problems in the Weihrauch lattice is a line of research that blossomed in the last few years. Initially this approach dealt mainly with statements which are provable in ACA_0 and showed that usually Weihrauch reducibility is more fine-grained than reverse mathematics.

In the last few years the study of multi-valued functions arising from statements laying at higher levels (such as ATR_0 and Pi^1_1-CA_0) of the reverse mathematics spectrum started as well. The multi-valued functions studied so far include those arising from the perfect tree theorem, comparability of well-orders, determinacy of open and clopen games, König’s duality theorem, various forms of choice, the open and clopen Ramsey theorem and the Cantor-Bendixson theorem.

At this level often a single theorem naturally leads to several multi-valued functions of different Weihrauch degree, depending on how the theorem is “read” from a computability viewpoint. A case in point is the perfect tree theorem: it can be read as the request to produce a perfect subtree of a tree with uncountably many paths, or as the request to list all paths of a tree which does not contain a perfect subtree. Similarly, the clopen Ramsey theorem leads to the multi-valued function that associates to every clopen subset of [N]^N an infinite homogeneous set on either side, and to the multi-valued function producing for each clopen subset which has an infinite homogeneous sets on one side a homogeneous set on that side. Similar functions can be defined similarly starting from the open Ramsey theorem.

In this talk I discuss some of these results, emphasizing recent joint work with my students Vittorio Cipriani and Manlio Valenti.

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: Noncomputable Coding, Density, and Stochasticity

by Justine Miller (University of Notre Dame) as part of Computability theory and applications

Abstract

We introduce the into and within set operations in order to construct sets of arbitrary intrinsic density from any Martin-Löf random. We then show that these operations are useful more generally for working with other notions of density as well, in particular for viewing Church and MWC stochasticity as a form of density.