Title: Adelic Theory of Protori

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

Lecture held in Elysium.

Abstract: TBA

Title: Classification of Finite-Dimensional Periodic LCA Groups

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

Lecture held in Elysium.

Abstract

Generalized resolutions of protori have non-Archimedean component a periodic LCA group with finite non-Archimedean dimension. The previous session introduced the notion of non-Archimedean dimension of LCA groups. Applying published results by Dikranjan, Herfort, Hofmann, Lewis, Loth, Mader, Morris, Prodanov, Ross, and Stoyanov, we introduce new minimalist notation and accompanying definitions to clarify the structure of these groups and their Pontryagin duals, enabling a parametrization of the spectrum of resolutions of finite-dimensional protori (the Grothendieck group is a moduli space).

Title: Recursion Theory and Diophantine Approximation

by Theodore Slaman (UC Berkeley) as part of Computability theory and applications

Abstract

We will give a survey of some connections between Recursion Theory, especially Algorithmic Randomness, and Diophantine Approximation, especially normality and exponents of irrationality. We will emphasize what we view as the contribution of a recursion theoretic perspective.

Title: Pontryagin Duals of Type Subgroups of Finite Rank Torsion-Free Abelian Groups

by Adolf Mader (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract

Pontryagin duals of type subgroups of finite rank torsion-free abelian groups are presented. The interplay between the intrinsic study of compact abelian groups, respectively torsion-free abelian groups, is discussed (how can researchers better leverage the published results in each setting so there is a dual impact?). A result definitively qualifying, in the torsion-free category, the uniqueness of decompositions involving maximal rank completely decomposable summands is given; the formulation of the result in the setting of protori is shown to optimally generalize a well-known result regarding the splitting of maximal tori from finite-dimensional protori.

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)