Title: The higher levels of the Weihrauch lattice
by Alberto Marcone (Università di Udine) as part of Computability theory and applications
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.
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
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.
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
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.
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.
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: Effective Dimension and the Intersection of Random Closed Sets
by Christopher Porter (Drake University) as part of Computability theory and applications
The connection between the effective dimension of sequences and membership in algorithmically random closed subsets of Cantor space was first identified by Diamondstone and Kjos-Hanssen. In this talk, I highlight joint work with Adam Case in which we extend Diamondstone and Kjos-Hanssen’s result by identifying a relationship between the effective dimension of a sequence and what we refer to as the degree of intersectability of certain families of random closed sets (also drawing on work by Cenzer and Weber on the intersections of random closed sets). As we show, (1) the number of relatively random closed sets that can have a non-empty intersection varies depending on the choice of underlying probability measure on the space of closed subsets of Cantor space—this number being the degree of intersectability of a given family of random closed sets—and (2) the effective dimension of a sequence X is inversely proportional to the minimum degree of intersectability of a family of random closed sets, at least one of which contains X as a member. Put more simply, a sequence of lower dimension can only be in random closed sets with more branching, which are thus more intersectable, whereas higher dimension sequences can be in random closed sets with less branching, which are thus less intersectable, and the relationship between these two quantities (that is, effective dimension and degree of intersectability) can be given explicitly.
Title: The computable strength of Milliken’s Tree Theorem and applications
by Paul-Elliot Angles d’Auriac (University of Lyon) as part of Computability theory and applications
Devlin’s theorem and the Rado graph theorem are both variants of Ramsey’s theorem, where a structure is added but more colors are allowed: Devlin’s theorem (respectively the Rado graph theorem) states if S is ℚ (respectively G, the Rado graph), then for any size of tuple n, there exists a number of colors l such that for any coloring of [S]^n into finitely many colors, there exists a subcopy of S on which the coloring takes at most l colors. Moreover, given n, the optimal l is specified.
The key combinatorial theorem used in both the proof of Devlin’s theorem and the Rado graph theorem is Milliken’s tree theorem. Milliken’s tree theorem is also a variant of Ramsey’s theorem, but this time for trees and strong subtrees: it states that given a coloring of the strong subtrees of height n of a tree T, there exists a strong subtree of height ω of T on which the coloring is constant.
In this talk, we review the links between those theorems, and present the recent results on the computable strength of Milliken’s tree theorem and its applications Devlin and the Rado graph theorem, obtained with Cholak, Dzhafarov, Monin and Patey.