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Location: MALL
Title: Dividing lines for permutation models
Permutation models are models of ZFA (ZF with atoms) that are built from sets that are 'symmetric enough' with respect to a group action on the set of atoms. In the case that this group action is a closed permutation group---that is, the automorphism group of a first-order structure---one can analyse the preservation of choice principles through the tameness of the structure. I shall exhibit examples of this behaviour.
Location: MALL
Title: On the binary linear ordering
NOTE: this seminar will take place in the MAGIC Room (10.03).
Let us call an order-type "untranscendable" if it cannot be embedded into a product of two smaller ones(!). Ordinals are untranscendable if and only if they are multiplicatively indecomposable. Moreover untranscendability almost implies additive indecomposability, that is to say, there is but one linear order type which is additively decomposable yet untranscendable. However, using the Axiom of Choice one can prove that there is a different untranscendable order type which at least fails to be strongly indecomposable, the order type of the real number continuum. Moreover, we can show that there is nothing more among the sigma-scattered linear order types and consistently neither among the Aronszajn lines.
Towards the end of the talk I am going to sketch some open problems, both in the presence and the absence of the Axiom of Choice.
This is joint ongoing work with Garrett Ervin and Alberto Marcone and builds on previous work by Barbosa, Galvin, Hausdorff, Laver, Ranero, and others.
Location: MALL
Title: How large must a small measurable cardinal must be?
Every measurable cardinal is inaccessible. At least assuming the Axiom of Choice. Without the Axiom of Choice, however, we can have measurable cardinals which are successor cardinals. Discounting those, how "small" on the large cardinal hierarchy can the least measurable cardinal get? It turns out that the answer depends on available large cardinals in inner models.
Specifically, in a recent work with Gitik and Hayut, we show that if the least measurable is the least inaccessible cardinal, then in the core model there is a measurable of a relatively high Mitchell order. This is in contrast to an older work with Hayut where we show that for the least measurable to be the least Mahlo cardinal we only need a single measurable cardinal in an inner model.
Location: MALL
Title: Baumgartner's Axiom and its Higher Dimensional Versions
Note: the speaker will be joining us online.
A set of reals $A \subseteq \mathbb{R}$ is called $\aleph_1$-dense if its intersection with every nonempty open interval has size $\aleph_1$. Baumgartner's axiom (BA) is the statement that every pair of $\aleph_1$-dense set of reals are isomorphic as linear orders. BA is the most straightforward generalization of Cantor's theorem about countable dense linear orders to the uncountable. This axiom, proved consistent by Baumgartner in 1973, while seemingly innocuous is actually both very finnicky and also seems to induce a lot of structure on the reals. For instance (on the finnicky side) it is implied by PFA, but not MA, even in the presence of strong reflection principles. On the '"induces a lot of structure" side, it implies the bounding number is greater than $\aleph_1$ (Todorčević). BA also has a natural generalization to higher dimensions i.e. $\mathbb{R}^n$ for $n > 1$ and these versions do follow from MA and in fact weaker cardinal characteristic assumptions (Steprāns-Watson). In this talk we will discuss these issues and show that the higher dimensional versions however also induce a lot of structure on the reals, in particular for every natural number $n$ BA for $\mathbb{R}^n$ implies the bounding number is bigger than $\aleph_1$.
This is joint work with Juris Steprāns.
Location: Roger Stevens LT 04 (8.04)
Title: "Categorifying" Borel reducibility
NOTE: this is a 2-hour seminar for both model and set theorists.
Please make note of the unusual venue!
Borel reducibility is a framework for comparing the complexities of different equivalence relations, and it has been used to great effect showing that various old classification programmes were impossible tasks. However, these days classification maps are generally expected to be functorial, which the classical Borel reducibility framework takes no account of. After going through preliminaries of the classical set-up, I will present a natural framework of Borel categories and functorial Borel reducibility that remedies this oversight. Notably, many examples of classes of structures that were known to be universal in a Borel reducibility sense - Borel complete - are not universal for our functorial version. I'll give many examples, including new ones for the old hands who've seen me talk about this stuff before. This is joint work with Filippo Calderoni.
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.
Location: MALL
Title: Finality of forcing
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: 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: Strong almost disjointness
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.
