Title: Limiting Density and Free Structures

by Julia Knight (Notre Dame) as part of Computability theory and applications

Abstract

Gromov had asked what a random group looks like – in a limiting density sense. I conjectured that the elementary frst order theory of the random group on n >= 2 generators, and with a single relator matches

that of the non-Abelian free groups. Coulon, Ho, and Logan have proved that the theories match on universal sentences. We may ask Gromovs question for other varieties. Franklin and I looked for varieties for which

calculating the limiting densities is easier. We have examples for which the elementary frst order theory of the random structure matches that of the free structure, and other examples for which the theories differ.

(joint work with Johanna Franklin)

Title: The Connection between the von Neumann Kernel and the Zariski Topology

by Dikran Dikranjan (University of Udine) as part of Topological Groups

Lecture held in Elysium.

Abstract

Every group G carries a natural topology Z_G defined by taking as a pre-base of the family of all closed sets the solution sets of all one-variable equations in the group of the form (a_1)x^{ε_1}(a2)x^{ε_2}…(a_n)x^{ε_n} = 1, where a_i ∈ G, ε_i = ±1 for i = 1,2,…,n, n ∈ N. The topology was explicitly introduced by Roger Bryant in 1978, who named it the verbal topology, but the name Zariski topology was universally applied subsequently. As a matter of fact, this topology implicitly appeared in a series of papers by Markov in the 1940’s in connection to his celebrated problem concerning unconditionally closed sets: sets which are closed in any Hausdorff group topology on G. These are the closed sets in the topology M_G obtained as the intersection of all Hausdorff group topologies on G, which we call the Markov topology, although this topology did not explicitly appear in Markov’s papers. Both Z_G and M_G are T1 topologies and M_G ≥ Z_G, but they need not be group topologies. One can use these topologies to formulate Markov’s problem: does the equality M_G = Z_G hold? Markov proved that M_G = Z_G if the group is countable and mentioned that the equality holds also for arbitrary abelian groups (so one can speak about the Markov-Zariski topology of an abelian group). The aim of the presentation is to expose this history, to describe some problems of Markov related to these topologies, and to apply the theory to give a solution to the Comfort-Protasov-Remus problem on minimally almost periodic topologies of abelian groups. This problem is associated to a more general problem of Gabriyelyan concerning the realisation of the von Neumann kernel n(G) of a topological group; that is, the intersection of the kernels of the continuous homomorphisms G → T into the circle group. More precisely, given a pair consisting of an abelian group G and a subgroup H, one asks whether there is a Hausdorff group topology τ on G such that n(G,τ) = H. Since (G,τ) is minimally almost periodic precisely when n(G) = G, the solution of this more general problem also gives a solution to the Comfort-Protasov-Remus problem.

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.