Location: MALL
Title: Zarankiewicz’s Problem and Model Theory
NOTE: this is a 2-hour seminar for both model and set theorists.
A shower thought that anyone interested in graph theory must have had at some point in their lives is the following: `How “sparse" must a given graph be, if I know that it has no “dense” subgraphs?’. This curiosity definitely crossed the mind of Polish mathematician K. Zarankiewicz, who asked a version of this question formally in 1951. In the years that followed, many central figures in the development of extremal combinatorics contemplated this problem, giving various kinds of answers. Some of these will be surveyed in the first part of my talk.
So far so good, but this is a model (and set) theory seminar and the title does include the words “Model Theory"… In the second part of my talk, I will discuss how the celebrated Szemerédi-Trotter theorem gave a starting point to the study of Zarankiewicz’s problem in “geometric” contexts, and how the language of model theory has been able to capture exactly what these contexts are. I will then ramble about improvements to the classical answers to Zarankiewicz’s problem when we restrict our attention to one of: (a) semilinear/semibounded o-minimal structures; (b) Presburger arithmetic, and (c) various kinds of Hrushovski constructions. The second hour of the talk will essentially be devoted to proofs. Which of (a),(b), or (c) will occupy the second hour will depend on input from the audience.
The new results appearing in the talk were obtained jointly with Pantelis Eleftheriou.
Can topological compactness be expressed as a first-order property within tame topology? Let's find out. In this talk I will present various attempts in the literature to capture this notion. We will go over the model theory behind them and present open questions.
Iterated forcing is a powerful tool for ouroboric arguments in set theory that rely on repeatedly creating or destroying some property until your construction eats its own tail and gives you your final result (in fact a similar argument may be applied to many ideas in set theory, especially when ordinals are involved. A simple example would be the \omega_1th stage of the Borel/projective hierarchies being no more than the union of their prior stages). To this end, it is often a helpful feature of an iterated notion of forcing that in the final model one has not introduced any new reals (subsets of \kappa, functions \lambda\to\lambda, etc) that are not already present in some intermediate stage. This behaviour is precisely captured by finality, which we shall define and give an exact characterisation of.
Location: MALL
Title: Definable groups in valued fields
I will discuss joint work with Gismatullin and Halupczok, giving the structure of definably (almost) simple groups definable in Henselian valued fields, possibly equipped with extra structure. I will also describe some other work on definable groups in valued fields.
Location: MALL
Title: Omega-categorical pseudofinite groups
I will discuss recent joint work with Katrin Tent on omega-categorical groups which are pseudofinite, i.e. satisfy every first order sentence which is true of all finite groups. It is fairly easy to show that they are nilpotent-by-finite, and we conjecture that they are finite-by-abelian-by-finite, and can reduce this to the nilpotent class 2 case. We show that certain class 2 groups constructed by amalgamation are NOT pseudofinite – in particular there is an example with supersimple theory which is not pseudofinite.
Location: MALL
Title: Eventually different, refining and dominating families at the uncountable
NOTE: the speaker will be joining us online.
We will discuss some recent results, including ZFC inequalities, concerning the higher Baire spaces analogues of some of the classical combinatorial cardinal characteristics of the continuum.
Of special interest for the talk will be the generalized bounding number, relatives of the generalized almost disjointness number, as well as the generalized refining and dominating numbers.
Location: MALL
Title: Weak Threads for Ladder Systems at Inaccessible $\kappa$
"Every club sequence has a weak thread" is a compactness property that implies simutaneously stationary reflection. In this talk, we will first explore weak threads for various ladder systems. Then we show it is consistent that every club sequence has a weak thread and there exists an almost disjoint ladder system given by vanishing branches of a $\kappa$-Suslin tree. This is joint work with Assaf Rinot and Zhixing You.
Location: MALL
Title: Weakly immediate types and T-convexity
For $T$ an o-minimal theory expanding RCF, a $T$-convex valuation ring on an o-minimal expansion of a RCF is a convex subring closed under continuous $T$-definable functions. This was first defined by Van Den Dries and Leweneberg who proved that the common theory $T_{\mathrm{convex}}$ of the expansions of models of $T$ by a non-trivial $T$-convex valuation ring is complete and weakly o-minimal. One of the key properties of the valuation theory of $T_{\mathrm{convex}}$ for power bounded $T$ is the so called residue-valuation property which can be restated as saying that every model of $T_{\mathrm{convex}}$ has a spherically complete maximal immediate extension. This is known to be false if $T$ defines an exponential. The goal of the talk will be to discuss potential analogues of the residue-valuation property in the exponential context.
A collection of unbounded subsets of $\omega_1$ is strongly almost disjoint if each pairwise intersection is finite. I will present Baumgartner's thinning out technique and use it to show that under Martin's Axiom + "failure of weak CH" every $\omega_1$-mad family has an $\omega_1$-mad strongly almost disjoint refinement. My presentation will be a mix between lecture and research talk. It has come to my ears that students of set theory in Leeds have not learned about $\Delta$-systems. For this reason I will present this powerful combinatorial result which is the essence of most ccc proofs. All subject to my bad management of time.
