Title: Topological Entropy and Algebraic Entropy on Locally Compact Abelian Groups

by Anna Giordano Bruno (University of Udine) as part of Topological Groups

Lecture held in Elysium.

Abstract

Since its origin, the algebraic entropy $h_{alg}$ was introduced in connection with the topological entropy $h_{top}$ by means of Pontryagin duality. For a continuous endomorphism $phicolon Gto G$ of a locally compact abelian group $G$, denoting by $widehat G$ the Pontryagin dual of $G$ and by $widehat phicolon Gto G$ the dual endomorphism of $phi$, we prove that $$h_{top}(phi)=h_{alg}(widehatphi)$$ under the assumption that $G$ is compact or that $G$ is totally disconnected. It is known that this equality holds also when $phi$ is a topological automorphism.

Title: A theorem from Rival and Sands and reverse mathematics

by Marta Fiori Carones (LMU Munich) as part of Computability theory and applications

Abstract

In 1980 Ivan Rival and Bill Sands proved that for each infinite poset P with finite width (i.e. such that there is a fixed finite bound on the size of antichains in P) there is an infinite chain C ⊆ P such that each element of P is comparable to none or to infinitely many elements of C. Moreover, if P is countable, C can be found such that each element of P is comparable to none or to cofinitely many elements of C.

We prove that some versions of the previous theorem are equivalent to the Ascending/descending sequence principle or to related known principles of the reverse mathematics zoo.

(Joint work with Alberto Marcone, Paul Shafer and Giovanni Soldà)

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.

Abstract

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.

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

Abstract

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.

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.