May 5

**Münster Set Theory Seminar****Time:** Tuesday, May 5, 4:15pm CEST**Speaker:** Andreas Lietz, University of Muenster**Title:** How to force (*) from less than a supercompact**Abstract:** Asperò-Schindler have shown that Woodin’s axiom (*) is a consequence of MM^{++} and the latter is known to be forceable from a supercompact cardinal. (*) however has consistency strength of \omega-many Woodin cardinals, so it should be possible to force it from a much weaker assumption. We present a construction that does so from strictly less than a \kappa^{+++}-supercompact cardinal \kappa (+GCH). The strategy will be to iterate the forcing from the proof of MM^{++}\Rightarrow(\ast). Two main difficulties arise: Whenever we want to use that forcing we will have to make sure that it is semiproper and that NS_{\omega_1} is saturated. We hope that the large cardinal assumption can be lowered to around the region of an inaccessible limit of Woodin cardinals. This is joint work with Ralf Schindler.**Information:** The seminar will be held remotely via zoom. Please contact rds@wwu.de ahead of time in order to participate.

**Cornell Logic Seminar****Time:** Tuesday, May 5, 2:55pm New York time (20:55pm CEST)**Speaker:** Andrew Zucker, University of Lyon**Title:** Topological dynamics beyond Polish groups**Abstract:** When *G* is a Polish group, one way of knowing that *G* has “nice” dynamics is to show that *M*(*G*), the universal minimal flow of *G*, is metrizable. However, works of Bartosova, Gheysens, and Krupinski–Pillay investigate groups beyond the Polish realm, such as *Sym*(*κ*), *Homeo*(*ω*1), and automorphism groups of uncountable, *ω*-homogeneous structures. For example, Bartosova shows that the universal minimal flow of *Sym*(*κ*) is the space of linear orders on *κ*–not a metrizable space, but still “nice.” In this talk, we seek to put these results into a general framework which encompasses all topological groups.

This is joint work with Gianluca Basso

May 6

**Paris-Lyon Séminaire de LogiqueTime:** Wednesday, May 6, 16:00-17:15 CEST

**Speaker:**Ilijas Farah – York University (Toronto)

**Title:**Between ultrapowers and reduced products.

**Abstract:**Ultrapowers and reduced powers are two popular tools for studying countable (and separable metric) structures. Once an ultrafilter on N is fixed, these constructions are functors into the category of countably saturated structures of the language of the original structure. The question of the exact relation between these two functors has been raised only recently by Schafhauser and Tikuisis, in the context of Elliott’s classification programme. Is there an ultrafilter on N such that the quotient map from the reduced product associated with the Fréchet filter onto the ultrapower has the right inverse? The answer to this question involves both model theory and set theory. Although these results were motivated by the study of C*-algebras, all of the results and proofs will be given in the context of classical (discrete) model theory.

**Information:**Join via the link on the seminar webpage 10 minutes before the talk.

**Oxford Set Theory SeminarTime:** Wednesday, May 6, 16:00-17:30 UK time (17:00-18:30 CEST)

**Speaker:**Victoria Gitman, City University of New York

**Title:**Elementary embeddings and smaller large cardinals

**Abstract:**A common theme in the definitions of larger large cardinals is the existence of elementary embeddings from the universe into an inner model. In contrast, smaller large cardinals, such as weakly compact and Ramsey cardinals, are usually characterized by their combinatorial properties such as existence of large homogeneous sets for colorings. It turns out that many familiar smaller large cardinals have elegant elementary embedding characterizations. The embeddings here are correspondingly ‘small’; they are between transitive set models of set theory, usually the size of the large cardinal in question. The study of these elementary embeddings has led us to isolate certain important properties via which we have defined robust hierarchies of large cardinals below a measurable cardinal. In this talk, I will introduce these types of elementary embeddings and discuss the large cardinal hierarchies that have come out of the analysis of their properties. The more recent results in this area are a joint work with Philipp Schlicht.

**Information:**For the Zoom access code, contact Samuel Adam-Day: me@samadamday.com.

May 7

**Kurt Gödel Research Center Seminar** (organised by Ben Miller)**Time:** Thursday, May 7, 16:00 CEST

**Speaker:**Stefan Hoffelner, University of Muenster

**Title:**TBA

**Abstract:**TBA

**Information:**Talk via zoom.

May 8

**Toronto Set Theory Seminar****Time:** Friday, May 8, 1:30-3:00pm EDT (19:30-21:00 CEST)**Speaker:** Dima Sinapova, University of Chicago**Title:** Iteration, reflection, and Prikry forcing**Abstract:** There is an inherent tension between stationary reflection and the failure of SCH. The former is a compactness type principle that follows from large cardinals. The latter is an instance of incompactness, and usually obtained using Prikry forcing. We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we discuss the situation at smaller cardinals. This is joint work with Alejandro Poveda and Assaf Rinot.**Information:** The talk will take place via zoom: https://yorku.zoom.us/j/96087161597.

**CUNY Set Theory Seminar****Time:** Friday, May 8, 2pm New York time (8pm CEST)**Speaker:** Sandra Mueller, KGRC Vienna**Title:** How to obtain lower bounds in set theory**Abstract:** Computing the large cardinal strength of a given statement is one of the key research directions in set theory. Fruitful tools to tackle such questions are given by inner model theory. The study of inner models was initiated by Gödel’s analysis of the constructible universe L. Later, it was extended to canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others.

We will outline two recent applications where inner model theory is used to obtain lower bounds in large cardinal strength for statements that do not involve inner models. The first result, in part joint with J. Aguilera, is an analysis of the strength of determinacy for certain infinite two player games of fixed countable length, and the second result, joint with Y. Hayut, involves combinatorics of infinite trees and the perfect subtree property for weakly compact cardinals κ. Finally, we will comment on obstacles, questions, and conjectures for lifting these results higher up in the large cardinal hierarchy.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.