Location: MALL
Title: Convolution semigroup on Keisler measures and revised Newelski's conjecture

We study the structure of the convolution semigroups of measures over definable groups. We isolate the property of generic transitivity and demonstrate that it is sufficient (and necessary) to develop stable group theory localizing on a generically stable type. We establish generic transitivity of generically stable idempotent types in important new cases, including abelian groups in arbitrary theories and arbitrary groups in rosy theories, and characterize them as generics of connected type-definable subgroups. Using tools from Keisler's randomization theory, we generalize some of these results from types to generically stable Keisler measures, and classify idempotent generically stable measures in abelian groups as (unique) translation-invariant measures on type-definable fsg subgroups. This provides a partial definable counterpart to the classical work of Rudin, Cohen and Pym for locally compact topological groups. Working over a countable NIP structure, we provide an explicit construction of a minimal left ideal in the convolution semigroup of measures from a minimal left ideal of types and the unique Haar measure on the ideal group. As a key ingredient, we prove the revised Ellis group conjecture of Newelski saying that under NIP, the so-called tau-topology on the ideal group is Hausdorff.

Joint work with Kyle Gannon and Krzysztof Krupiński.

Location: MALL
Title: Well-founded models of fragments of Collection

Let $\mathsf{M}$ be the weak set theory (with powersets) axiomatised by: $\textsf{Extensionality}$, $\textsf{Pair}$, $\textsf{Union}$, $\textsf{Infinity}$, $\textsf{Powerset}$, transitive containment ($\textsf{TCo}$), $\Delta_0\textsf{-Separation}$ and $\textsf{Set-Foundation}$. In this talk I will discuss the relationship between two alternative versions of the set-theoretic collection scheme: $\textsf{Collection}$ and $\textsf{Strong Collection}$. Both of these schemes yield $\mathsf{ZF}$ when added to $\mathsf{M}$, but when restricted the $\Pi_n$-formulae (denoted $\Pi_n\textsf{-Collection}$ and $\textsf{Strong } \Pi_n\textsf{-Collection}$) these alternative versions of set-theoretic collection differ. In particular, over the theory $\mathsf{M}$, $\textsf{Strong }\Pi_n\textsf{-Collecton}$ is equivalent to $\Pi_n\textsf{-Collection}+\Sigma_{n+1}\textsf{-Separation}$. And, $\mathsf{M}+\textsf{Strong }\Pi_n\textsf{-Collection}$ proves the consistency of $\mathsf{M}+\Pi_n\textsf{-Collection}$. In this talk I will show that, despite this difference in consistency strength, every countable well-founded model of $\mathsf{M}+\Pi_n\textsf{-Collection}$ satisfies $\textsf{Strong } \Pi_n\textsf{-Collection}$. If time permits I will outline how this argument can be refined to show that $\mathsf{M}+\Pi_n\textsf{-Collection}+\Pi_{n+1}\textsf{-Foundation}$ proves $\Sigma_{n+1}\textsf{-Separation}$.

To celebrate Dugald's 30 years in Leeds we are holding a mini-conference with a focus on the various research themes that have occupied Dugald's succesful achievements over the last 30 years in the University of Leeds.

Location: MALL 1
Title: Continuous Actions of Monoids, a New Perspective on Large Cardinals

The standard "symmetric models" approach to building models of the failure of the axiom of choice relies on the action of non-discrete topological groups on a universe of sets (either sets-with-atoms or a boolean valued model of names). A new approach interprets large cardinal axioms as positing the action of non-discrete topological monoids on the universe of sets (a model of names, a model with atoms, or even $V$ itself). By understanding such actions, one can reinterpret (and reprove) the Kunen inconsistency theorem as a symmetric model theorem, as well as interpreting a version of the HOD conjecture in a natural way. The talk will attempt to place such techniques in their natural context and highlight some applications.

In 1904 Ernst Zermelo published his paper, Beweis, daß jede Menge wohlgeordnet werden kann, proving that every set can be well-ordered. This prompted the efforts to axiomatise and clarify the foundations of set theory and mathematics. This year, we are celebrating 120 years of choice and order.

There will be a tutorial session, 16 lectures, and 2 poster sessions.

Registration to the conference is open until 20 June 2024. Please consider registering early so we can better estimate the number of participants!

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.

I will present ongoing work joint with Müller, Ramsey and Siniora. A classical result of Macintyre and Saracino states that the theory of Lie algebras over a fixed field and of bounded nilpotency class does not admit a model-companion. We prove that by letting the field grow (i.e. with a separated sort for the field) the theory of Lie algebras of bounded nilpotency class admits a model-companion and that this theory relates asymptotically to the omega-categorical existentially closed c-nilpotent Lie algebra over a finite field F_p for c<p. We also prove that if the theory of the field is NSOP1 then the theory of the corresponding Lie algebra is NSOP4. We will explain how to get this result via a criterion for NSOP4 which does not use stationary independence relations.

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: 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.