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.

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: Which Lebesgue spaces are computably presentable?

by Timothy McNicholl (Iowa State University) as part of Computability theory and applications

Abstract

We consider the following question: “If there is a computably presentable $L^p$ space, does it follow that $p$ is computable?” The answer is of course `no’ since the 1-dimensional $L^p$ space is just the field of scalars. So, we turn to non-trivial cases. Namely, assume there is a computably presentable $L^p$ space whose dimension is at least $2$. We prove $p$ is computable if the space is finite-dimensional or if $p geq 2$. We then show that if $1 leq p < 2$, and if $L^p[0,1]$ is computably presentable, then $p$ is right-c.e.. Finally, we show there is no uniform solution of this problem even when given upper and lower bounds on the exponent. The proof of this result leads to some basic results on the effective theory of stable random variables. Finally, we conjecture that the answer to this question is `no’ and that right-c.e.-ness of the exponent is the best result possible.