Logic Event

There will be a 2-day logic event in Cambridge on 9-10 October 2015. The talks will take place in the Isaac Newton Institute. On Friday, we will be in seminar room 2 (in the top of the gate house); the talks on Saturday take place in seminar room 1.

Timetable

Friday
12.30-13.25Hugo Nobrega Computable analysis and games in descriptive set theory
13.30-14.25Johannes Carmesin Min-Max theorems in infinite combinatroics
14.40-15.35Francesco Parente Saturated Boolean Ultrapowers
15.50-16.45Lovkush Agarwal Uncountably many maximal-closed subgroups of Sym(N) via reducts of Henson digraphs
Saturday
11.00-11.55Lorenzo Galeotti Weihrauch degrees for generalized Baire space
12.00-13.00lunch
13.00-13.55Jacob Hilton Topological Ramsey theory of countable ordinals
14.00-14.55Philipp Kleppmann Free groups and the Axiom of Choice

Abstracts

Hugo Nobrega
Computable analysis and games in descriptive set theory

We report on ongoing work with Arno Pauly, showing how concepts from computable analysis can be used to shed light and uniformize certain games for classes of functions which have been studied in descriptive set theory, such as Wadge's game for continuous functions, Duparc's eraser game for Baire class 1 functions, and Semmes' tree game for Borel functions.

As an application, for each finite n we obtain a game characterizing the Baire class n of functions.

Johannes Carmesin
Min-Max theorems in infinite combinatroics

The start of my talk is the extension of the marriage theorem to infinite bipartite graphs due to Aharoni, Nash-Williams and Shelah. This is implied by the Infinite Menger Conjecture, which was proved recently by Aharoni and Berger. Next I will talk about related packing and covering conjectures in infinite graphs.

Then I will give a short introduction to infinite matroids. The matroidal point of view allows us to understand the above statements as different perspectives or special cases of the same central problem of Infinite Matroid Theory, which can be traced back to Nash-Williams.

At the end, I will mention a link between Determinacy of infinite games and that conjecture of Nash-Williams. More precisely, there is a special case of the conjecture which is equivalent to the statement that a certain family of infinite games is determined if and only if a second family of infinite games is.

This talk is self contained and I will not assume any special knowledge of the audience.

Francesco Parente
Saturated Boolean Ultrapowers

In this talk I will survey the general theory of Boolean ultrapowers, starting from the beginnings and including many applications and some possible future developments. Also, the set-theoretic approach to Boolean ultrapowers, due to recent work of Hamkins and Seabold, will be discussed.

First developed by Mansfield as a purely algebraic construction, Boolean ultrapowers are a natural generalization of usual power-set ultrapowers. More specifically, I will focus on how some combinatorial properties of a ultrafilter U are related to the realization of types in the resulting Boolean ultrapower. Many results on κ-regular and κ-good ultrafilters, mostly due to Keisler, can be generalized to this context. In particular, I will sketch the construction of a κ-good ultrafilter on the Levy collapsing algebra Coll(ω, <κ). In addition to that, I will describe a possible application to Keisler's order on complete theories.

Lovkush Agarwal
Uncountably many maximal-closed subgroups of Sym(N) via reducts of Henson digraphs

This work contributes to the two closely related areas of countable homogeneous structures and infinite permutation groups. In the permutation group side, we answered a question of Macpherson that asked to show that there are uncountably many pairwise non-conjugate maximal-closed subgroups of Sym(\mathbb{N}). This was achieved by taking the automorphism groups of uncountably many pairwise non-isomorphic Henson digraphs. The fact these groups are maximal-closed follows from the classification of the reducts of Henson digraphs. In itself, this classification contributes to the building list of structures whose reducts are known and also provides further evidence that Thomas' conjecture is true.

In this talk, my main aim will be to describe the construction of these continuum many maximal-closed subgroups, which will include Henson's famous construction of continuum many countable homogeneous digraphs. Any remaining time will be spent giving some of the ideas behind how we prove these groups are maximal closed.

Lorenzo Galeotti
Weihrauch degrees for generalized Baire space

The theory of Weihrauch degrees is about representing classical theorems of analysis in Baire space and comparing their strength (measured as the Weihrauch degree). In this talk, we are exploring a version of this theory for generalized Baire space. The first step in this generalization is that of finding a generalization of R on which we can prove a version of theorems from classical analysis. The first part of the talk will be devoted to the presentation of the construction of an extension of R on which we can prove a version of the Intermediate Value Theorem. In the second part of the talk we will be focusing on generalizing notions from computable analysis. Finally we will show how this new framework can be used to characterize the strength of the generalized version of the version of the Intermediate Value Theorem we presented in the first half of the talk.

Jacob Hilton
Topological Ramsey theory of countable ordinals

Recall that the Ramsey number R(n, m) is the least k such that, whenever the edges of the complete graph on k vertices are coloured red and blue, then there is either a complete red subgraph on n vertices or a complete blue subgraph on m vertices - for example, R(4, 3) = 9. This generalises to ordinals: given ordinals α and β, let R(α, β) be the least ordinal γ such that, whenever the edges of the complete graph with vertex set γ are coloured red and blue, then there is either a complete red subgraph with vertex set of order type α or a complete blue subgraph with vertex set of order type β - for example, R(ω + 1, 3) = ω.2 + 1. We will prove the result of Erdos and Milner that R(α, k) is countable whenever α is countable and k is finite, and look at a topological version of this result. This is joint work with Andres Caicedo.

Philipp Kleppmann
Free groups and the Axiom of Choice

The role of the Axiom of Choice in Mathematics has been studied extensively. Given a theorem of ZFC, one may ask how strong it is compared to the Axiom of Choice. Although a large collection of results has been analysed in this way, there are still simple and elegant theorems that offer resistance. One such result is the Nielsen-Schreier theorem, which states that subgroups of free groups are free.

I will introduce recent results that help to establish the strength of Nielsen-Schreier, focussing on the method of representative functions. Then I discuss potential applications of this technique to other algebraic structures admitting a basis, such as free abelian groups and vector spaces.