Title: On the Mackey Topology of an Abelian Topological Group

by Lydia Außenhofer (Universität Passau) as part of Topological Groups

Lecture held in Elysium.

Abstract

For a locally convex vector space $(V,tau)$ there exists a finest locally convex vector space topology $mu$ such that the topological dual spaces $(V,tau)’$ and $(V,mu)’$ coincide algebraically. This topology is called the $Mackey$ $topology$. If $(V,tau)$ is a metrizable locally convex vector space, then $tau$ is the Mackey topology.

In 1995 Chasco, Martín Peinador, and Tarieladze asked, “Given a locally quasi-convex group $(G,tau),$ does there exist a finest locally quasi-convex group topology $mu$ on $G$ such that the character groups $(G,tau)^wedge$ and $(G,mu)^wedge$ coincide?”

In this talk we give examples of topological groups which

1. have a Mackey topology,

2. do not have a Mackey topology,

and we characterize those abelian groups which have the property that every metrizable locally quasi-convex group topology is Mackey (i.e., the finest compatible locally quasi-convex group topology).

Title: Minimal pairs in the generic degrees

by Denis Hirschfeldt (University of Chicago) as part of Computability theory and applications

Abstract

Generic computability is a notion of “almost everywhere computability” that has been studied from a computability-theoretic perspective by several authors following work of Jockusch and Schupp. It leads naturally to a notion of reducibility, and hence to a degree structure. I will discuss the construction of a minimal pair in the generic degrees, which contrasts with Igusa’s result that there are

no minimal pairs for the similar notion of relative generic computability. I will then focus on several related questions that remain open.

Title: Simply Given Compact Abelian Groups

by Peter Loth (Sacred Heart University) as part of Topological Groups

Lecture held in Elysium.

Abstract

A compact abelian group is called simply given if its Pontrjagin dual is simply presented. Warfield groups are defined to be direct summands of simply presented abelian groups. They were classified up to isomorphism in terms of cardinal invariants by Warfield in the local case, and by Stanton and Hunter–Richman in the global case. In this talk, we classify up to topological isomorphism the duals of Warfield groups, dualizing Stanton’s invariants. We exhibit an example of a simply given group with nonsplitting identity component.

Title: Coding in the automorphism group of a structure

by Dan Turetsky (Victoria University of Wellington, New Zealand) as part of Computability theory and applications

Abstract

In this talk I will discuss a new technique for coding a closed set into the automorphism group of a structure. This technique has applications to problems in Scott rank, effective dimension, and degrees of categoricity. For instance, I will explain how it can be used to construct a computably categorical structure with noncomputable Scott rank.

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.