Title: Computing descending sequences in linear orderings

by Jun Le Goh (University of Wisconsin) as part of Computability theory and applications

Abstract

Let DS be the problem of computing a descending sequence in a given ill-founded linear ordering. We investigate the uniform computational content of DS from the point of view of Weihrauch reducibility, in particular its relationship with the analogous problem of computing a path in a given ill-founded tree (known as choice on Baire space).

First, we show that DS is strictly Weihrauch reducible to choice on Baire space. Our techniques characterize the problems which have codomain N and are Weihrauch reducible to DS, thereby identifying the so-called first-order part of DS.

Second, we use the technique of inseparable $Pi^1_1$ sets (first used by Angles d’Auriac, Kihara in this context) to study the strengthening of DS whose inputs are $Sigma^1_1$-codes for ill-founded linear orderings. We prove that this strengthening is still strictly Weihrauch reducible to choice on Baire space.

This is joint work with Arno Pauly and Manlio Valenti.

Title: Locally Compact Contraction Groups

by Helge Glöckner (Universität Paderborn) as part of Topological Groups

Lecture held in Elysium.

Abstract

Consider a locally compact group $G$, together with an automorphism $alpha$ which is $contractive$ in the sense that $alpha^nrightarrow{rm id}_G$ pointwise as $ntoinfty$. Siebert showed that $G$ is the direct product of its connected component $G_e$ and an $alpha$-stable, totally disconnected closed subgroup;

moreover, $G_e$ is a simply connected, nilpotent real Lie group.

I’ll report on research concerning the totally disconnected part, obtained jointly with G. A. Willis.

For each totally disconnected contraction group $(G,alpha)$, the set ${rm tor} G$ of torsion elements is a closed subgroup of $G$. Moreover, $G$ is a direct product

$$G=G_{p_1}times cdotstimes G_{p_n}times {rm tor} G$$ of $alpha$-stable $p$-adic Lie groups $G_p$ for certain primes $p_1,ldots, p_n$ and the torsion subgroup. The structure of $p$-adic contraction groups is known from the work of J. S. P. Wang; notably, they are nilpotent. As shown with Willis, ${rm tor} G$ admits a composition series and there are countably many possible composition factors, parametrized by the finite simple groups. More recent research showed that there are uncountably many non-isomorphic torsion contraction groups, but only countably many abelian ones. If a torsion contraction group $G$ has a compact open subgroup which is a pro-$p$-group, then $G$ is nilpotent. Likewise if $G$ is locally pro-nilpotent.

Title: Group dualities: G-barrelled groups

by Elena Martín-Peinador (University of Madrid) as part of Topological Groups

Lecture held in Elysium.

Abstract

A natural notion in the framework of abelian groups are the group dualities. The most efficient definition of a group duality is simply a pair $(G, H)$, where $G$ denotes an abstract abelian group and $H$ a subgroup of characters of $G$, that is $H leq {rm Hom}(G, mathbb T)$. Two group topologies for $G$ and $H$ appear from scratch in a group duality $(G, H)$: the weak topologies $sigma(G, H)$ and $sigma (H, G)$ respectively. Are there more group topologies either in $G$ or $H$ that can be strictly related with the duality $(G, H)$? In this sense we shall define the term “compatible topology” and loosely speaking we consider the compatible topologies as members of the duality.

The locally quasi-convex topologies defined by Vilenkin in the 50′s form a significant class for the construction of a duality theory for groups. The fact that a locally convex topological vector space is in particular a locally quasi-convex group serves as a nexus to emulate well-known results of Functional Analysis for the class of topological groups.

In this talk we shall

deal with questions of the sort:

Under which conditions is there a locally compact topology in a fixed duality?

The same question for a metrizable, or a $k$-group topology.

We shall also introduce the $g$-barrelled groups, a class for which the Mackey-Arens Theorem admits an optimal counterpart. We study also the existence of $g$-barrelled topologies in a group duality $(G, H)$.

Title: Redundancy of information: lowering effective dimension

by Joe Miller (University of Wisconsin) as part of Computability theory and applications

Abstract

A natural way to measure the similarity between two infinite

binary sequences X and Y is to take the (upper) density of their

symmetric difference. This is the Besicovitch distance on Cantor

space. If d(X,Y) = 0, then we say that X and Y are coarsely

equivalent. Greenberg, M., Shen, and Westrick (2018) proved that a

binary sequence has effective (Hausdorff) dimension 1 if and only if

it is coarsely equivalent to a Martin-Löf random sequence. They went

on to determine the best and worst cases for the distance from a

dimension t sequence to the nearest dimension s>t sequence. Thus, the

difficulty of increasing dimension is understood.

Greenberg, et al. also determined the distance from a dimension 1

sequence to the nearest dimension t sequence. But they left open the

general problem of reducing dimension, which is made difficult by the

fact that the information in a dimension s sequence can be coded (at

least somewhat) redundantly. Goh, M., Soskova, and Westrick recently

gave a complete solution.

I will talk about both the results in the 2018 paper and the more

recent work. In particular, I will discuss some of the coding theory

behind these results. No previous knowledge of coding theory is

assumed.

Title: Classification of Periodic LCA Groups of Finite Non-Archimedean Dimension

by Wayne Lewis (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract

A periodic LCA group such that the $p$-components all have $p$-rank bounded above by a common positive integer are classified via a complete set of topological isomorphism invariants realized by an equivalence relation on pairs of extended supernatural vectors.

Remaining time will be devoted to a facilitated discussion on how things are going this fall/winter academic semester in your part of the world as you see it.

Title: Discovering structure within the class of K-trivial sets

by Andre Nies (University of Auckland) as part of Computability theory and applications

Abstract

Joint work with Noam Greenberg, Joseph Miller, and Dan Turetsky

The K-trivial sets are antirandom in the sense that the initial segment complexity in terms of prefix-free Kolmogorov complexity K grows as slowly as possible. Chaitin introduced this notion in about 1975, and showed that each K-trivial is Turing below the halting set. Shortly after, Solovay proved that a K-trivial set can be noncomputable.

In the past two decades, many alternative characterisations of this class have been found: properties such as being low for K, low for Martin-Löf (ML) randomness, and a basis for ML randomness, which state in one way or the other that the set is close to computable.

Initially, the class of noncomputable K-trivials appeared to be amorphous. More recently, evidence of an internal structure has been found. Most of these results can be phrased in the language of a mysterious reducibility on the K-trivials which is weaker than Turing’s: A is ML-below B if each ML-random computing B also computes A.

Bienvenu, Greenberg, Kucera, Nies and Turetsky (JEMS 2016) showed that there an ML complete K-trivial set. Greenberg, Miller and Nies (JML, 2019) established a dense hierarchy of subclasses of the K-trivials based on fragments of Omega computing the set, and each such subclass is an initial segment for ML. More recent results generalise these approaches using cost functions. They also show that each K-trivial set is ML-equivalent to a c.e. K-trivial.

The extreme lowness of K-trivials, far from being an obstacle, allows for methods which don’t work in a wider setting. The talk provides an overview and discusses open questions. For instance, is ML-completeness an arithmetical property of K-trivials?