Events

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 $\Sigma_{n+1}\textsf{-Separation}$.

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.