Nicolò Zava (University of Udine) @ Lecture held in Elysium
May 12 @ 6:00 am – 8:00 am

Title: The Large-Scale Geometry of Locally Compact Abelian Groups
by Nicolò Zava (University of Udine) as part of Topological Groups

Lecture held in Elysium.

Large-scale geometry, also known as coarse geometry, is the branch of mathematics that studies the global, large-scale properties of spaces. This theory is distinguished by its applications which include the Novikov and coarse Baum-Connes conjectures. Since the breakthrough work of Gromov, large-scale geometry has played a prominent role in geometric group theory, in particular, in the study of finitely generated groups and their word metrics. This large-scale approach was successfully extended to all countable groups by Dranishnikov and Smith. A further generalisation introduced by Cornulier and de la Harpe dealt with locally compact σ-compact groups endowed with particular pseudo-metrics.
To study the large-scale geometry of more general groups and topological groups, coarse structures are required. These structures, introduced by Roe, encode global properties of spaces. We also mention the equivalent approach provided by Protasov and Banakh using balleans. Coarse structures compatible with a group structure can be characterised by special ideals of subsets, called group ideals. While the coarse structure induced by the family of all finite subsets is well-suited for abstract groups, the situation is less clear for groups endowed with group topologies, as exemplified by the left coarse structure, introduced by Rosendal, and the compact-group coarse structure, induced by the group ideal of all relatively compact subsets, each suitable in disparate settings.
We present the large-scale geometry of groups via the historically iterative sequence of generalisations, enlisting illustrative examples specific to distinct classes of groups and topological groups. We focus on locally compact abelian groups endowed with compact-group coarse structures. In particular, we discuss the role of Pontryagin duality as a bridge between topological properties and their large-scale counterparts. An overriding theme is an evidence-based tenet that the compact-group coarse structure is the right choice for the category of locally compact abelian groups.

Matthew Harrison-Trainor (Victoria University of Wellington, New Zealand)
May 12 @ 3:00 pm – 4:00 pm

Title: The tree of tuples of a structure
by Matthew Harrison-Trainor (Victoria University of Wellington, New Zealand) as part of Computability theory and applications

Given a countable structure, one can associate a tree of finite tuples from that structure, with each tuple labeled by its atomic type. This tree encodes the back-and-forth information of the structure, and hence determines the isomorphism type, but it is still missing something: with Montalban I proved that there are structures which cannot be computably (or even hyperarithmetically) recovered from their tree of tuples. I’ll explain the meaning of this result by exploring two separate threads in computable structure theory: universality and coding.

Lydia Außenhofer (Universität Passau) @ Lecture held in Elysium
May 19 @ 6:00 am – 8:00 am

Title: On the Mackey Topology of an Abelian Topological Group
by Lydia Außenhofer (Universität Passau) as part of Topological Groups

Lecture held in Elysium.

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).

Denis Hirschfeldt (University of Chicago)
May 19 @ 10:00 am – 11:00 am

Title: Minimal pairs in the generic degrees
by Denis Hirschfeldt (University of Chicago) as part of Computability theory and applications

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.

Peter Loth (Sacred Heart University) @ Lecture held in Elysium
May 26 @ 6:00 am – 8:00 am

Title: Simply Given Compact Abelian Groups
by Peter Loth (Sacred Heart University) as part of Topological Groups

Lecture held in Elysium.

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.

Dan Turetsky (Victoria University of Wellington, New Zealand)
May 26 @ 3:00 pm – 4:00 pm

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

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.

Vittorio Bard (Università degli Studi di Torino)
Jun 2 @ 4:00 am – 5:00 am

Title: A local approach towards uniform Martin’s conjecture
by Vittorio Bard (Università degli Studi di Torino) as part of Computability theory and applications

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.

Break (University of Hawaiʻi) @ Lecture held in Elysium
Jun 2 @ 6:00 am – 8:00 am

Title: Topological Groups Seminar One-Week Hiatus
by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.
Abstract: TBA