The talks are listed in reverse chronological order, so please scroll down to find today’s talks!

We announce talks from the following set theory related seminars and colloquia:

– **European Set Theory Society Colloquia**, 17:00-19:00 CEST/CET, several times each year

– Kobe Set Theory Seminar, Monday 16:30 local time (09:30 CEST until 29 October 2023, 08:30 CET from 30 October 2023)

– Vienna Set Theory Seminar, Thursdays 11:30-13:00 OR Tuesdays 15:00-16:30 local time (15:00-16:00 CEST/CET)

– Baltic Set Theory Seminar, Tuesdays 15:00-16:30 local time (15:00-16:00 CEST/CET)

– Carnegie-Mellon University Pittsburgh Core Model Seminar, Tuesdays 1:30pm local time (19:30 CEST/CET, except in the week beginning with 30 October, then 18:30 CET)

– Carnegie-Mellon University Pittsburgh Logic Seminar, Tuesdays 3:30pm local time (21:30 CEST/CET, except in the week beginning with 30 October, then 20:30 CET)

– Leeds Models and Sets Seminar, Wednesdays, 13:45-15:00 local time (14:45-16:00 CEST/CET)

– Barcelona Set Theory Seminar, Wednesdays 4pm local time (15:00-16:30 CEST/CET)

– Caltech: Logic seminar, Wednesdays 12am-1pm local time (21:00-22:00 CEST/CET except in the week beginning with 30 October, then 20:00 CET)

– Renyi Institute Set Theory Seminar, Thursdays 10:30 – 12:00 CEST/CET

– Vienna Logic Colloquium, Thursdays 15:00 – 16:30 local time (15:00 – 16:30 CEST/CET)

– Bristol Logic and Set Theory Seminar, Thursdays 15:00 local time (16:00 CEST/CET)

– Cross-Alps Logic Seminar, Fridays 16:30 local time (16:30 CEST/CET)

– CUNY Set Theory Seminar, Fridays 12:15pm-1:45pm local time (18:15-19:45 CEST/CET except in the week beginning with 30 October, then 17:15-18:45 CET)

– University of Toronto Set Theory seminar, Fridays 1:30pm local time (19:30 CEST/CET except in the week beginning with 30 October, then 18:30 CET)

– CUNY Logic Workshop, Fridays 2:00pm-3:30pm local time (20:00-21:30 CEST/CET except in the week beginning with 30 October, then 19:00-20:30 CET)

**12 – 18 June**

**Kobe Set Theory SeminarTime:** Monday, 12 June, 16:30 local time (09:30 CEST)

**Speaker:**

**Title:**

**Abstract:**

**Information:**Please see the seminar webpage. This talk will be given in hybrid format. Please email Hiroshi Sakai in advance for the zoom information.

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 13 June, 15:00-16:30 CEST**Speaker:** **Title:** **Abstract:** **Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 13 June, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.

2. Paul Larson, A course on AD^+**Information:** Please see the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 13 June, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** **Title:** **Abstract: ****Information:** See the seminar webpage.

**Leeds Models and Sets Seminar****Time:** Wednesday, 14 June, 13:45-15:00 local time (14:45-16:00 CEST)**Speaker:** **Title:** **Abstract:** **Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 14 June, 16:00-17:30 CEST**Speaker:** **Title:** **Abstract:** **Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Caltech Logic Seminar****Time:** Wednesday, 14 June, 12:00am-13:00pm Pacific time (21:00-22:00 CEST)**Speaker:****Title:****Abstract:****Information:** Please see the seminar webpage.

**Renyi Institute Set Theory SeminarTime:** Thursday, 15 June, 10:30 – 12:00 CEST

**Speaker:**

**Title:**

**Abstract:**

**Information:**Please see the seminar webpage. This talk will be given in hybrid format.

**Vienna Logic ColloquiumTime:** Thursday, 15 June, 15:00 – 15:45 CEST

**Speaker:**

**Title:**

**Abstract:**

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Cross-Alps Logic Seminar****Time:** Friday, 16 June, 16.00-17.00 CEST**Speaker:** **Title:** **Abstract:** **Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**CUNY Set Theory Seminar****Time:** Friday, 16 June, 12:15pm New York time (18:15 CEST)**Speaker: ****Title:** **Abstract:** **Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 16 June, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** **Title:** **Abstract:** **Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 16 June, 2:00 – 3:30 New York time (20:00-21:30 CEST)**Speaker: ****Title: ****Abstract:** **Information:** The talk will take place in person. For more information, please see the seminar webpage or email Victoria Gitman.

**5 – 11 June**

**Kobe Set Theory SeminarTime:** Monday, 5 June, 16:30 local time (09:30 CEST)

**Speaker:**Sakaé Fuchino

**Title:**Resurrection and Maximality under the tightly Laver-generically ultrahuge cardinal (2/2)

**Abstract:**A (definable) class P of posets is said to be

**iterable**if ① P is closed with respect to forcing equivalence (i.e. if ℙ∈P and ℙ∼ℙ′ then ℙ′∈P ), ② closed wrt restriction (i.e. if ℙ∈P then ℙ↾𝕡∈P for any 𝕡∈ℙ ), and, ③ for any ℙ∈P and ℙ-name ℚ˙, ⫦ℙ“ℚ˙∈P” implies ℙ∗ℚ˙∈P.

For an iterable class P of posets, a cardinal

*κ*is said to be

**P-Laver-generically supercompact**if, for any

*λ*≥

*κ*and ℙ∈P, there is a ℙ-name ℚ˙ with ⫦ℙ“ℚ˙∈P” such that, for (𝖵,ℙ∗ℚ˙)-generic ℍ, there are

*j*,

*M*⊆𝖵[ℍ] with

(a)

*j*:𝖵≺_

*κ M*, (b)

*j*(

*κ*)>

*λ*, and (c) ℙ, ℙ∗ℚ˙, ℍ,

*j*′′

*λ*∈

*M*.

*κ*is said to be

**tightly P-Laver-generically supercompact**if additionally (d) ∣∣ℙ∗ℚ˙∣∣≤

*j*(

*κ*) holds.

Similarly, we can also define (tightly) P-Laver-generic versions of super almost-huge, superhuge, and ultrahuge cardinals.

In [ II ] it is shown that the existence of P-Laver-gen. supercompact cardinal (tightly P-Laver gen. superhuge in the case P = ccc posets) for a reasonable P highlights the situations with the continuum being ① ℵ1, ② ℵ2 or ③ very large.

In particular with P being the class of all ① σ-closed posets, ② semi-proper posets, or ③ ccc-posets, the existence of P-Laver-gen. supercompact cardinal (or tightly P-Laver gen. superhuge in the case P = ccc posets) implies a double-plused version of forcing axiom for the respective P and strong reflection properties down to less than

*κ_refl*:=max{ℵ2,2ℵ0} compatible with the forcing axiom.

In this talk we shall prove that the existence of tightly P-Laver-generically superhuge cardinal implies the boldface version of Resurrection Axiom ([Hamkins-Johnstone 1], [Hamkins-Johnstone 2] ) for P over H(

*κ_refl*).

We further show that the existence of tightly P-Laver-generically ultrahuge cardinal implies the Unbounded Resurrection Axiom of Tsaprounis ([Tsaprounis]) for P and strong version of local maximality principle ((slightly?) stronger than the one mentioned in [Minden]).

**References.**

[ I ] S.F., A. Ottenbreit Maschio Rodrigues, and H. Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, Archive for Mathematical Logic, Vol.60, 1-2, (2021), 17–47. https://fuchino.ddo.jp/papers/SDLS-x.pdf

[ II ] —–, Strong downward Löwenheim-Skolem theorems for stationary logics, II — reflection down to the continuum, Archive for Mathematical Logic, Vol.60, 3-4, (2021), 495–523. https://fuchino.ddo.jp/papers/SDLS-II-x.pdf

[Hamkins-Johnstone 1] Joel David Hamkins, and Thomas A.Johnstone, Resurrection axioms and uplifting cardinals, Archive for Mathematical Logic, Vol.53, Iss.3-4, (2014), 463-485.

[Hamkins-Johnstone 2] —–, Strongly uplifting cardinals and the boldface resurrection axioms, Archive for Mathematical Logic volume 56, (2017), 1115-1133.

[Minden] Kaethe Minden, Combining resurrection and maximality, The Journal of Symbolic Logic, Vol. 86, No. 1, (2021), 397–414.

[Tsaprounis 1] Konstantinos Tsaprounis, On resurrection axioms, The Journal of Symbolic Logic, Vol.80, No.2, (2015), 587–608.

[Tsaprounis 2] —–, Ultrahuge cardinals, Mathematical Logic Quarterly, Vol.62, No.1-2, (2016), 1–2.

**Information:**Please see the seminar webpage. This talk will be given in hybrid format. Please email Hiroshi Sakai in advance for the zoom information.

**Baltic Set Theory Seminar****Time:** Tuesday, 6 June, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.

2. Paul Larson, A course on AD^+**Information:** Please see the seminar webpage.

**Leeds Models and Sets Seminar****Time:** Wednesday, 7 June, 13:45-15:00 local time (14:45-16:00 CEST)**Speaker:** Richard Matthews, Université Paris-Est Créteil Val de Marne **Title:** Very large set axioms over Constructive Set Theories **Abstract:** One of the main areas of research in set theory is the study of large cardinal axioms and many of these can be characterised by the existence of elementary embeddings with certain properties. The guiding principle is then that the closer the domain and co-domain of the embedding is to the universe, the stronger the resulting large cardinal axiom. This leads naturally to the question of whether there is an elementary embedding of the universe into itself which is not the identity, and the least ordinal moved by such an embedding is known as a Reinhardt cardinal. While Kunen famously proved that no such embedding can exist if the universe satisfies ZFC, it is an open question in many subtheories of ZFC, most notably ZF (without Choice).

In this talk we will study elementary embeddings in the weaker context of intuitionistic set theories, that is set theories without the law of excluded middle. We shall observe that the ordinals can be very ill-behaved in this setting and therefore we will reformulate large cardinals by instead looking for large sets which capture the desired structural properties. We shall investigate the consistency strength of analogues to measurable cardinals, Reinhardt cardinals and many other similar ideas in terms of the standard ZFC large cardinal hierarchy.

This is joint work with Hanul Jeon.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 7 June, 12:00am-13:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Clinton Conley, CMU **Title:** Borel asymptotic dimension and hyperfiniteness **Abstract:** We introduce a “purely Borel” version of Gromov’s notion of asymptotic dimension, and show how to use it to establish hyperfiniteness of various equivalence relations. Time permitting, we discuss hyperfiniteness of orbit equivalence relations of free actions of lamplighter groups. This is joint work with Jackson, Marks, Seward, and Tucker-Drob. **Information:** Please see the seminar webpage.

**Cross-Alps Logic Seminar****Time:** Friday, 9 June, 16.00-17.00 CEST**Speaker:** Ulrich Kohlenbach, Technische Universität Darmstadt**Title:** Proof mining: Recent developments **Abstract:** In this talk we survey some recent developments in the project of applying proof-theoretic transformations to obtain new quantitative and qualitative information from given proofs in areas of core mathematics such as nonsmooth optimization, geodesic geometry and ergodic theory. We will discuss some of the following items:

(1) Proof mining in the context of set-valued monotone and accretive operators with applications in nonsmooth optimization such as inconsistent feasibility theorems (partly joint work with Nicholas Pischke).

(2) Recent linear rates of asymptotic regularity as well as rates of metastability for Tikhonov-regularization methods (joint work with Horaţiu Cheval and Laurenţiu Leuştean).

(3) The extraction of uniform rates of convergence for the ε-capture in the Lion-Man game in a general geodesic setting from a proof that made iterated use of sequential compactness arguments (i.e. arithmetical comprehension). The extraction also qualitatively generalizes previously known results (joint work with Genaro López-Acedo and Adriana Nicolae).

(4) Recent applications to ergodic theory (joint work with Anton Freund).**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**29 May – 4 June**

**Kobe Set Theory SeminarTime:** Monday, 29 May, 16:30 local time (09:30 CEST)

**Speaker:**Sakaé Fuchino, Kobe University

**Title:**Resurrection and Maximality under the tightly Laver-generically ultrahuge cardinal (1/2)

**Abstract:**A (definable) class P of posets is said to be

**iterable**if ① P is closed with respect to forcing equivalence (i.e. if ℙ∈P and ℙ∼ℙ′ then ℙ′∈P ), ② closed wrt restriction (i.e. if ℙ∈P then ℙ↾𝕡∈P for any 𝕡∈ℙ ), and, ③ for any ℙ∈P and ℙ-name ℚ˙, ⫦ℙ“ℚ˙∈P” implies ℙ∗ℚ˙∈P.

For an iterable class P of posets, a cardinal

*κ*is said to be

**P-Laver-generically supercompact**if, for any

*λ*≥

*κ*and ℙ∈P, there is a ℙ-name ℚ˙ with ⫦ℙ“ℚ˙∈P” such that, for (𝖵,ℙ∗ℚ˙)-generic ℍ, there are

*j*,

*M*⊆𝖵[ℍ] with

(a)

*j*:𝖵≺_

*κ M*, (b)

*j*(

*κ*)>

*λ*, and (c) ℙ, ℙ∗ℚ˙, ℍ,

*j*′′

*λ*∈

*M*.

*κ*is said to be

**tightly P-Laver-generically supercompact**if additionally (d) ∣∣ℙ∗ℚ˙∣∣≤

*j*(

*κ*) holds.

Similarly, we can also define (tightly) P-Laver-generic versions of super almost-huge, superhuge, and ultrahuge cardinals.

In [ II ] it is shown that the existence of P-Laver-gen. supercompact cardinal (tightly P-Laver gen. superhuge in the case P = ccc posets) for a reasonable P highlights the situations with the continuum being ① ℵ1, ② ℵ2 or ③ very large.

In particular with P being the class of all ① σ-closed posets, ② semi-proper posets, or ③ ccc-posets, the existence of P-Laver-gen. supercompact cardinal (or tightly P-Laver gen. superhuge in the case P = ccc posets) implies a double-plused version of forcing axiom for the respective P and strong reflection properties down to less than

*κ_refl*:=max{ℵ2,2ℵ0} compatible with the forcing axiom.

In this talk we shall prove that the existence of tightly P-Laver-generically superhuge cardinal implies the boldface version of Resurrection Axiom ([Hamkins-Johnstone 1], [Hamkins-Johnstone 2] ) for P over H(

*κ_refl*).

We further show that the existence of tightly P-Laver-generically ultrahuge cardinal implies the Unbounded Resurrection Axiom of Tsaprounis ([Tsaprounis]) for P and strong version of local maximality principle ((slightly?) stronger than the one mentioned in [Minden]).

**References.**

[ I ] S.F., A. Ottenbreit Maschio Rodrigues, and H. Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, Archive for Mathematical Logic, Vol.60, 1-2, (2021), 17–47. https://fuchino.ddo.jp/papers/SDLS-x.pdf

[ II ] —–, Strong downward Löwenheim-Skolem theorems for stationary logics, II — reflection down to the continuum, Archive for Mathematical Logic, Vol.60, 3-4, (2021), 495–523. https://fuchino.ddo.jp/papers/SDLS-II-x.pdf

[Hamkins-Johnstone 1] Joel David Hamkins, and Thomas A.Johnstone, Resurrection axioms and uplifting cardinals, Archive for Mathematical Logic, Vol.53, Iss.3-4, (2014), 463-485.

[Hamkins-Johnstone 2] —–, Strongly uplifting cardinals and the boldface resurrection axioms, Archive for Mathematical Logic volume 56, (2017), 1115-1133.

[Minden] Kaethe Minden, Combining resurrection and maximality, The Journal of Symbolic Logic, Vol. 86, No. 1, (2021), 397–414.

[Tsaprounis 1] Konstantinos Tsaprounis, On resurrection axioms, The Journal of Symbolic Logic, Vol.80, No.2, (2015), 587–608.

[Tsaprounis 2] —–, Ultrahuge cardinals, Mathematical Logic Quarterly, Vol.62, No.1-2, (2016), 1–2.

**Information:**Please see the seminar webpage. This talk will be given in hybrid format. Please email Hiroshi Sakai in advance for the zoom information.

**Baltic Set Theory Seminar****Time:** Tuesday, 30 May, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.

2. Paul Larson, A course on AD^+**Information:** Please see the seminar webpage.

**Leeds Models and Sets Seminar****Time:** Wednesday, 31 May, 13:45-15:00 local time (14:45-16:00 CEST)**Speaker:** Liuzhen Wu, Chinese Academy of Sciences **Title:** A surjection from square onto power **Abstract:** Cantor proves that for any set A, there is no surjection from A into its power set P(A). In this talk, we describe a construction of a ZF model. In this model, there is a set A and a surjection from its square set A^2 onto its power set P(A). This indicates Cantor’s Theorem is in some sense optimal. This is joint work with Guozhen Shen and Yinhe Peng.**Information:** Please see the seminar webpage.

**22 – 28 May**

**Kobe Set Theory SeminarTime:** Monday, 22 May, 16:30 local time (09:30 CEST)

**Speaker:**Takehiko Gappo, TU Wien

**Title:**Chang models over derived models with supercompact measures (2/2)

**Abstract:**The relationship between the Axiom of Determinacy and supercompactness of

*ω*1 has been studied by many people. In 1990’s, Woodin showed that assuming the existence of a proper class of Woodin limits of Woodin cardinals, a generalized Chang model satisfies “ADℝ +

*ω*1 is supercompact.” Recently he also showed that the regularity of Θ in the model follows from determinacy of a long game of length

*ω*1, which is, however, still unknown to be consistent. Based on these results, we conjecture that the following two theories are equiconsistent:

(1) ZFC + there is a Woodin limit of Woodin cardinals.

(2) ZF + ADℝ + Θ is regular +

*ω*1 is supercompact.

Toward this conjecture, we construct a new model of the Axiom of Determinacy, called the Chang model over the derived model with supercompact measures.We then prove that it is consistent relative to a Woodin limit of Woodin cardinals that our model satisfies “ADℝ + Θ is regular +

*ω*1 is <

*δ*-supercompact for some regular cardinal

*δ*> Θ.” This is joint work with Sandra Müller and Grigor Sargsyan.

**Information:**Please see the seminar webpage. This talk will be given in hybrid format. Please email Hiroshi Sakai for the zoom information in advance.

**Bristol Logic and Set Theory Seminar****Time:** Tuesday, 23 May, 1.30-2.30pm UK time (14:30-15:30 CET)**Speaker:** Beatrice Pitton, University of Lausanne**Title:** Definable subsets of the generalized Cantor and Baire spaces**Abstract:** Generalized descriptive set theory (GDST) aims at developing a higher analogue of classical descriptive set theory in which ω is replaced with an uncountable cardinal κ in all definitions and relevant notions. In the literature on GDST it is often required that κ<κ = κ, a condition equivalent to κ regular and 2<κ = κ. In contrast, in this paper we use a more general approach and develop in a uniform way the basics of GDST for cardinals κ still satisfying 2<κ = κ but independently of whether they are regular or singular. This allows us to retrieve as a special case the known results for regular κ, but it also uncovers their analogues when κ is singular. We also discuss some new phenomena specifically arising in the singular context (such as the existence of two distinct yet related Borel hierarchies), and obtain some results which are new also in the setup of regular cardinals, such as the existence of unfair Borel∗ codes for all Borel∗ sets. This is joint work with Luca Motto Ros.**Information:** Please see the seminar webpage https://www.bristolmathsresearch.org/events/logic-and-set-theory. If you want to participate online, please contact Philipp Schlicht in advance.

**Baltic Set Theory Seminar****Time:** Tuesday, 23 May, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.

2. Paul Larson, A course on AD^+**Information:** Please see the seminar webpage.

**Bristol Logic and Set Theory Seminar****Time:** Wednesday, 23 May, 3.00-4.00pm UK time (16:00-17:00 CET)**Speaker:** Bokai Yao, University of Notre Dame**Title:** Forcing with Urelements**Abstract:** I will begin by isolating a hierarchy of axioms based on ZFCU_R, which is ZFC set theory (with Replacement) modified to allow a class of urelements. For example, the Collection Principle is equivalent to the Reflection Principle over ZFCU_R, while it is folklore that neither of them is provable in ZFCU_R.

I then turn to forcing over countable transitive models of ZFU_R. A forcing relation is full just in case whenever a forcing condition p forces an existential statement, p also forces some instance of that statement. According to the existing approach, forcing relations are almost never full when there are urelements. I introduce a new forcing machinery to address this problem. I show that over ZFCU_R, the principle that every new forcing relation is full is equivalent to the Collection Principle. Furthermore, I show how forcing is able to preserve, destroy and resurrect the axioms in the hierarchy I introduced. In particular, the Reflection Principle is “necessarily forceble” in certain models of ZFCU_R. In the end, I will consider how the ground model definability can fail when the ground model contains a proper class of urelements.**Information:** Please see the seminar webpage https://www.bristolmathsresearch.org/events/logic-and-set-theory. If you want to participate online, please contact Philipp Schlicht in advance.

**Leeds Models and Sets Seminar****Time:** Wednesday, 24 May, 13:45-15:00 local time (14:45-16:00 CEST)**Speaker:** Adele Padgett, McMaster University **Title:** Regular solutions of systems of transexponential-polynomial equations**Abstract:** It is unknown whether there are o-minimal fields that are transexponential, i.e., that define functions which eventually grow faster than any tower of exponential functions. In past work, I constructed a Hardy field closed under a transexponential function E which satisfies E(x+1) = exp E(x). Since the germs at infinity of unary functions definable in an o-minimal structure form a Hardy field, this can be seen as evidence that the real field expanded by E could be o-minimal. To prove o-minimality, a better understanding of definable functions in several variable is likely needed. I will discuss one approach using a criterion for o-minimality due to Lion. This ongoing work is joint with Vincent Bagayoko and Elliot Kaplan.**Information:** Please see the seminar webpage.

**Renyi Institute Set Theory SeminarTime:** Thursday, 25 May, 10:30 – 12:00 CEST

**Speaker:**Dorottya Sziraki

**Title:**Dichotomies for open dihypergraphs on definable subsets of generalized Baire spaces

**Abstract:**The open graph dichotomy for a subset $X$ of the Baire space $\omega^\omega$ states that any open graph on $X$ either contains a large complete subgraph or admits a countable coloring. It is a definable version of the open coloring axiom for $X$ and it generalizes the perfect set property. The focus of this talk is a recent generalization to infinite dimensional directed hypergraphs by Carroy, Miller and Soukup. It is motivated by applications to definable sets of reals, in particular to the second level of the Borel hierarchy.

We show that this infinite dimensional dichotomy holds for all subsets of the Baire space in Solovay’s model. Our main results are versions of this theorem for generalized Baire spaces $\kappa^\kappa$ for uncountable regular cardinals $\kappa$. If time permits, we will also look at conditions under which this dichotomy can be strengthened and mention several applications in the setting of generalized Baire spaces.

This is joint work with Philipp Schlicht.

**Information:**Please see the seminar webpage. This talk will be given in hybrid format.

**Vienna Research Seminar in Set Theory****Time:** Thursday, 25 May, 11:30-13:00 CEST**Speaker:** F. Kaak, Universität Kiel **Title:** Set theory of a Suslin line**Abstract:** A Suslin line is a linear ordering, which is in some way quite similar to the real line. We will discuss in what ways the set theory of the real line can be adapted to a Suslin line. We give a characterization of Borel sets of the Suslin line, look at a few cardinal characteristics and play games on a Suslin tree.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Vienna Logic ColloquiumTime:** Thursday, 25 May, 15:00 – 15:45 CEST

**Speaker:**S. Starchenko, University of Notre Dame

**Title:**On Hausdorff limits of images of o-minimal families in real tori

**Abstract:**Let {Xs:x∈S} be a family of subsets of Rn definable in some o-minimal expansion of the real field. Let Γ⊆Rn be a lattice and π:Rn/Γ→T be the quotient map. In a series of papers (published and unpublished) together with Y. Peterzil we considered Hausdorff limits of the family {π(Xs):s∈S} and provided their description. In this talk I describe model theoretic tools used in the description.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Toronto Set Theory Seminar****Time:** Friday, 26 May, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Christina Brech**Title:** PID and uncountable biorthogonal systems in Banach spaces**Abstract:** The relation between the possible sizes of a biorthogonal system in a Banach space and its density is not completely understood. In this talk, we discuss this problem and present a result stating that, under the P-ideal dichotomy, the existence of uncountable biorthogonal systems in all nonseparable Banach spaces of a certain class is equivalent to the cardinal assumption 𝔟>*ω*1. This is a joint work with Stevo Todorcevic.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**15 – 21 May**

**Kobe Set Theory SeminarTime:** Monday, 15 May, 16:30 local time (09:30 CEST)

**Speaker:**Takehiko Gappo, TU Wien

**Title:**Chang models over derived models with supercompact measures (1/2)

**Abstract:**The relationship between the Axiom of Determinacy and supercompactness of

*ω*1 has been studied by many people. In 1990’s, Woodin showed that assuming the existence of a proper class of Woodin limits of Woodin cardinals, a generalized Chang model satisfies “ADℝ +

*ω*1 is supercompact.” Recently he also showed that the regularity of Θ in the model follows from determinacy of a long game of length

*ω*1, which is, however, still unknown to be consistent. Based on these results, we conjecture that the following two theories are equiconsistent:

(1) ZFC + there is a Woodin limit of Woodin cardinals.

(2) ZF + ADℝ + Θ is regular +

*ω*1 is supercompact.

Toward this conjecture, we construct a new model of the Axiom of Determinacy, called the Chang model over the derived model with supercompact measures. We then prove that it is consistent relative to a Woodin limit of Woodin cardinals that our model satisfies “ADℝ + Θ is regular +

*ω*1 is <

*δ*-supercompact for some regular cardinal

*δ*> Θ.” This is joint work with Sandra Müller and Grigor Sargsyan.

**Information:**Please see the seminar webpage. This talk will be given in hybrid format. Please email Hiroshi Sakai for the zoom information in advance.

**Baltic Set Theory Seminar****Time:** Tuesday, 16 May, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.

2. Paul Larson, A course on AD^+**Information:** Please see the seminar webpage.

**Leeds Models and Sets Seminar****Time:** Wednesday, 17 May, 13:45-15:00 local time (14:45-16:00 CEST)**Speaker:** Will Boney, Texas State University **Title:** tba **Abstract:** tba **Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 17 May, 12:00am-13:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Forte Shinko, UCLA **Title:** The generic action of a free group on Cantor space is hyperfinite**Abstract:** Let Γ be a countable free group. The set of continuous actions of Γ on Cantor space 2N admits a natural Polish topology, and hence we can talk about properties of the generic action. It was shown by Frisch-Kechris-Shinko-Vidnyánszky that the generic action is measure-hyperfinite, meaning that for every Borel probability measure μ on 2N, the action is hyperfinite modulo some μ-null set. Kwiatkowska showed using the methods of projective Fraïssé theory that there is only one generic action up to isomorphism. We use her techniques to investigate the generic action further, and in particular we show that the generic action is hyperfinite. This is joint work with Sumun Iyer.**Information:** Please see the seminar webpage.

**Renyi Institute Set Theory SeminarTime:** Thursday, 18 May, 10:30 – 12:00 CEST

**Speaker:**Amitayu Banerjee

**Title:**On a variant of Erdős–Dushnik–Miller theorem without the Axiom of Choice (AC)

**Abstract:**Theories: ZFC (Zermelo–Fraenkel set theory with the Axiom of Choice (AC)), ZF (Zermelo–Fraenkel set theory without AC), ZFA (ZF with the Axiom of Extensionality weakened to allow the existence of atoms).

Known informations:

In 1941, Ben Dushnik and Miller established the proposition “Every infinite graph contains either a countably infinite independent set or a clique with the same cardinality as the whole graph” in ZFC, and gave credit to Paul Erdős for the proof of the result for the case in which the cardinality of the graph is a singular cardinal. The above result is uniformly known as Erdős–Dushnik–Miller theorem.

Consider the following variant (abbreviated as EDM): “Every uncountable graph contains either a countably infinite independent set or an uncountable clique”. It is well-known that in ZFC, EDM implies the proposition “Any partially ordered set such that all of its antichains are finite and all of its chains are countable is countable” (we abbreviate by K) as well as the infinite Ramsey’s theorem (“Every infinite graph has either an infinite independent set or an infinite clique”).

In 1977, Andreas Blass studied the exact placement of the infinite Ramsey’s theorem in the hierarchy of weak forms of AC. In particular, he proved that the Boolean Prime Ideal Theorem (a weak form of AC) is independent of the infinite Ramsey’s theorem in ZF (i.e., there exists a ZF model where the Boolean Prime Ideal Theorem holds, but the infinite Ramsey’s theorem fails, and there exists a ZF model where the infinite Ramsey’s theorem holds, but the Boolean Prime Ideal Theorem fails) (see: https://doi.org/10.2307/2272866).

In 2021, I studied some relations of K with weak forms of AC. (see: arXiv:2009.05368v2; to appear).

In 2022, Eleftherios Tachtsis investigated the deductive strength of K without AC in more detail. Among several results, Tachtsis proved that DC_{\aleph_{1}} (Dependent Choices for \aleph_{1}, a weak form of AC stronger than Dependent Choices (DC)) implies K in ZF (see: https://link.springer.com/article/10.1007/s00605-022-01751-9).

New Results: We study the exact placement of EDM in the hierarchy of weak forms of AC. In particular, we prove the following results (see arXiv:2211.05665v3):

1. The strength of EDM is strictly between DC_{\aleph_{1}} and K in ZFA.

2. EDM is strictly stronger than the infinite Ramsey’s theorem in ZF (i.e., the infinite Ramsey’s theorem does not imply EDM in ZF).

3. The Boolean Prime Ideal Theorem is independent of EDM in ZFA (specifically, neither the Boolean Prime Ideal Theorem implies EDM in ZF, nor EDM implies the Boolean Prime Ideal Theorem in ZFA).

Finally, the speaker will state some open questions in this track.

**Information:**Please see the seminar webpage. This talk will be given in hybrid format.

**Cross-Alps Logic Seminar****Time:** Friday, 19 May, 16.00-17.00 CEST**Speaker:** J. Duparc,** **Université de Lausanne **Title:** The Wadge order on the Cantor Space and on the Scott Domain**Abstract:** The Cantor space — 2N — and the Scott domain — P(N) — are two topological spaces whose points are sets of integers. But if the Cantor space is equipped with a topology of positive and negative information (conveyed through characteristic functions via the product topology of the discrete topology on {0,1}), the Scott domain drops that condition of negative information, and only keeps the one of positive information through the topology generated by the basis {OF∣F⊆N, F finite} where OF={A⊆N∣F⊆A}.

As a consequence, the Scott domain is not anymore Hausdorff (T2), not even Fréchet (T1) but only Kolmogorov (T0). So, on one hand, the Scott domain seems far away from the Cantor space: it is not even metrizable while the latter is Polish. But on the other hand, they share some similarities: the Cantor space is universal for 0-dimensional Polish spaces, the Scott domain is universal for quasi-Polish spaces (de Brecht).

More results by de Brecht suggest that a reasonable descriptive set theory still holds in the quasi-Polish setting. However, despite works by Becher, Grigorieff, Motto Ros, Schlicht, Selivanov, and others, much less is known about the Wadge order in this context, rather than in the Polish one.

We outline the main features of the Wadge order on Borel subsets of the Cantor space and on Borel subsets of the Scott domain and compare these two (a joint work with Louis Vuilleumier).**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**CUNY Set Theory Seminar****Time:** Friday, 19 May, 12:30pm New York time (18:15 CEST)**Speaker: **Miha Habič, Bard College at Simon’s Rock**Title:** Some old and new results on nonamalgamable forcing extensions**Abstract:** Fixing some countable transitive model M of set theory, we can consider its generic multiverse, the family of all models obtainable from M by taking any sequence of forcing extensions and ground models. There is an attractive similarity between the generic multiverse and the Turing degrees, but the multiverse has the drawback (or feature?) that it contains nonamalgamable models, that is, models with no common upper bound, as was observed by several people, going back to at least Mostowski. In joint work with Hamkins, Klausner, Verner, and Williams in 2019, we studied the order-theoretic properties of the generic multiverse and, among other results, gave a characterization of which partial orders embed nicely into the multiverse. I will present our results in the simplest case of Cohen forcing, as well as existing generalizations to wide forcing, and some new results on non-Cohen ccc forcings.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 19 May, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Vera Fischer, University of Vienna**Title:** Spectra and Definability**Abstract:** Maximal almost disjoint families, maximal cofinitary groups and maximal independent families are among the combinatorial sets of reals, which are central to the study of the set theoretic properties of the real line. In this talk, we will discuss recent developments regarding the possible cardinalities of such extremal sets of reals, as well as their definability properties.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**8 – 14 May**

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 9 May, 15:00-16:30 CEST**Speaker:** Thilo Weinert, Universität Wien**Title:** On Unsound Linear Orderings**Abstract:** In the Eighties Adrian Mathias introduced the notion of soundness of an ordinal. An ordinal is sound if for any countable partition P of it only countably many ordinals are order-types of unions of subpartitionts of P. Mathias showed that the least unsound ordinal ζ is ωω+21 if ℵ1 can be embedded into the continuum but if ℵ1 is regular yet cannot be embedded into the continuum, ζ⩾ωω2+11.

I am going to discuss his findings and consider the notion for the more general class of linear orderings building on work by him, MacPherson, and Schmerl. I am also going to mention some open problems. This is joint ongoing work with Garrett Ervin and Jonathan Schilhan.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 9 May, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.

2. Paul Larson, A course on AD^+**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 10 May, 13:00-15:00 Israel Time (12:00-14:00 CEST)**Speaker:** Jouko Vaananen**Title:** Descriptive Set Theory in Generalized Baire Spaces**Abstract:** I will review the motivation and basic notions of Generalized Baire Spaces. I will then talk about the role of trees, such as wide Aronszajn trees, in the Descriptive Set Theory of Generalized Baire Spaces. This part is motivated by recent joint work with Omer Ben-Neria and Menachem Magidor. I will also talk about universally Baire sets in Generalized Baire Spaces. This part is joint work with Menachem Magidor.**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 10 May, 13:45-15:00 local time (14:45-16:00 CEST)**Speaker:** Victoria Gould, University of York**Title:** Pseudo-finite semigroups and diameter**Abstract:** A semigroup S is said to be (right) pseudo-finite if the universal right congruence S x S can be generated by a finite set U of pairs of elements of S and there is a bound on the length of derivations for an arbitrary pair as a consequence of those in U . The diameter of a pseudo-finite semigroup is the smallest such bound taken over all finite generating sets.

The notion of being pseudo-finite was introduced by White in the language of ancestry, motivated by a conjecture of Dales and Zelazko for Banach algebras. The property also arises from several other sources.

Without assuming any prior knowledge, this talk investigates the somewhat unpredictable notion of pseudo-finiteness. Some well-known uncountable semigroups have diameter 1; on the other hand, a pseudo-finite group is forced to be finite. Actions, presentations, Rees matrix constructions and some good old-fashioned semigroup tools all play a part.

This research sits in the wider framework of a study of finitary conditions for semigroups.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 10 May, 12:00am-13:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Allison Wang, CMU**Title:** Every CBER is smooth below the Carlson-Simpson generic partition**Abstract:** One difficulty that arises in studying the class of countable Borel equivalence relations (CBERs) is that in many cases, the complexity of a CBER lies on a “small” set. For instance, a result of Hjorth and Kechris states that every CBER on a Polish space is hyperfinite when restricted to some comeager set. Another result, due to Mathias, shows that every CBER on the Ellentuck Ramsey space is hyperfinite when restricted to some pure Ellentuck cube. In this talk, we will show that every CBER on the space of all infinite partitions of the natural numbers coincides with equality below a Carlson-Simpson generic element. This is joint work with Aristotelis Panagiotopoulos.**Information:** Please see the seminar webpage.

**Third Colloquium of the European Set Theory Society****Time:** Thursday, 11 May, 17:00-19:00 CEST**Panelists:** Alan Dow, University of North Carolina at Charlotte

Heike Mildenberger, University of Freiburg

Slawek Solecki, Cornell University

Matteo Viale, University of Torino**Title:** European Set Theory Society Panel Discussions**Abstract:** Four experts will describe the general area they represent, explain where the area is heading and discuss how it relates to other areas of set theory and mathematics.**Information:** Online. Zoom link for 11 May: https://univienna.zoom.us/j/61733153940?pwd=Wm5sNjczVTlUWVA3Vy9pZlZreFlDQT09

**CUNY Set Theory Seminar****Time:** Friday, 12 May, 12:30pm New York time (18:30 CEST)**Speaker: **Miha Habič, Bard College at Simon’s Rock**Title:** Some old and new results on nonamalgamable forcing extensions**Abstract:** Fixing some countable transitive model M of set theory, we can consider its generic multiverse, the family of all models obtainable from M by taking any sequence of forcing extensions and ground models. There is an attractive similarity between the generic multiverse and the Turing degrees, but the multiverse has the drawback (or feature?) that it contains nonamalgamable models, that is, models with no common upper bound, as was observed by several people, going back to at least Mostowski. In joint work with Hamkins, Klausner, Verner, and Williams in 2019, we studied the order-theoretic properties of the generic multiverse and, among other results, gave a characterization of which partial orders embed nicely into the multiverse. I will present our results in the simplest case of Cohen forcing, as well as existing generalizations to wide forcing, and some new results on non-Cohen ccc forcings.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**CUNY Logic Workshop****Time:** Friday, 12 May, 2:00 – 3:30 New York time (20:00-21:30 CEST)**Speaker: **Brian Wynne, CUNY**Title: **Recent developments in the model theory of Abelian lattice-ordered groups**Abstract:** An Abelian lattice-ordered group (ℓ-group) is an Abelian group with a partial ordering, invariant under translations, that is a lattice ordering. A prototypical example of an ℓ-group is C(X), the continuous real-valued functions on the topological space X with pointwise operations and ordering. Let A be the class of ℓ-groups, viewed as structures for the first-order language L={+,−,0,∧,∨}. After giving more background on ℓ-groups, I will survey what is known about the ℓ-groups existentially closed (e.c.) in A, including some new examples I constructed using Fraïssé limits. Then I will discuss some recently published work of Scowcroft concerning the ℓ-groups e.c. in W+, the class of nonzero Archimedean ℓ-groups with distinguished strong order unit (viewed as structures for L1=L∪{1}). Building on Scowcroft’s results, I will present new axioms for the ℓ-groups e.c. in W+ and show how they allow one to characterize those spaces X for which (C(X),1X) is e.c. in W+.**Information:** The talk will take place in person. For more information, please see the seminar webpage or email Victoria Gitman.

**Baltic Set Theory Seminar****Time:** Tuesday, 2 May, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.

2. Paul Larson, A course on AD^+**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 3 May, 13:00-15:00 Israel Time (12:00-14:00 CEST)**Speaker:** tba **Title:** tba **Abstract:** tba **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 3 May, 13:45-15:00 local time (14:45-16:00 CEST)**Speaker:** Vahagn Aslanyan, University of Leeds**Title:** Combining Manin-Mumford and weak Zilber-Pink**Abstract:** I will introduce some classical notions and problems in Diophantine geometry, including the Manin-Mumford and Zilber-Pink conjectures, and explain how model-theoretic tools are used to approach them. I will then talk about one of my recent theorems establishing a new partial result towards Zilber-Pink by combining Manin-Mumford and a weak version of Zilber-Pink (both are theorems). I am going to start with very basic things, give quite a few examples and define/explain all concepts that I am going to use, so I hope that most of the talk will be accessible to a wide range of people including those who have not heard about Diophantine geometry before.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 3 May, 12:00am-13:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Jan Grebík, University of Warwick**Title:** Measurable Vizing’s theorem**Abstract:** Vizing’s theorem asserts that every graph of degree bounded by Δ<+∞ admits a proper edge coloring with (Δ+1) colors. I will discuss versions of this theorem in the context of measurable graph combinatorics. I will mainly focus on the case when the graph in question is defined on a standard probability space (X,μ). In this situation, a combination of an augmenting chain technique developed earlier with Oleg Pikhurko (that was applied for graphings) together with a new result about quasi-invariant measures allows to deduce a full analogue of Vizing’s theorem.**Information:** Please see the seminar webpage.

**Renyi Institute Set Theory SeminarTime:** Thursday, 4 May, 10:30 – 12:00 CEST

**Speaker:**Lajos Soukup, Renyi Institute

**Title:**Minimal Vertex Covers in Infinite Hypergraphs

**Abstract:**A “vertex cover” of a hypergraph is a set of vertices which intersects each hyperedge. A hypergraph possesses “property

*C*(

*k*,

*ρ*)” iff |⋂E′|<

*ρ*for each

*k*element set E′ of hyperedges. Komjáth proved that every uniform hypergraph possessing property

*C*(2,

*r*) for some

*r*∈

*ω*has a minimal vertex cover. We could relax the assumption of uniformity to an assumption that the set of cardinalities of the hyperedges is a “small” set of infinite cardinals, e.g. it is countable, or it does not contain uncountably many limit cardinals. Komjáth also proved that GCH does not decide the following statement: “If a hypergraph

*G*possessing property

*C*(2,

*ω*) is

*μ*-uniform for some

*μ*≥

*ω*1, then

*G*has a minimal vertex cover.

Using Shelah’s Revised GCH theorem, we could show that if we strengthen the assumption

*μ*≥

*ω*1 to

*μ*≥

*bethω*, then we can prove the statement in ZFC! We also show that if all the hyperedges of a hypergraph are countably infinite, then instead of

*C*(2,

*r*) the assumption

*C*(

*k*,

*r*) (for some

*k*∈

*ω*) is enough to guarantee the existence of a minimal vertex cover. If every hyperedge has cardinality

*ω*1, then we can only prove that

*C*(3,

*r*) is enough.

**Information:**Please see the seminar webpage. This talk will be given in hybrid format.

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 4 May, 11:30-13:00 CEST**Speaker:** L. Zdomskyy, TU Wien**Title:** Cardinalities of sets of reals satisfying combinatorial covering properties**Abstract:** We shall discuss which cardinalities sets of reals satisfying Menger and Hurewicz covering properties may have in some standard models of ZFC. Most of the results may be thought of as consistent instances of the Perfect Set Property, since they state that in some models, a set of reals satisfying certain covering properties either contains a copy of the Cantor set, or has small size. In particular, we plan to outline the proof of the fact that in the Sacks model every Menger totally imperfect set of reals has size at most ω1.

This is a joint work with V. Haberl and P. Szewczak.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Vienna Logic ColloquiumTime:** Thursday, 4 May, 15:00 – 15:45 CEST

**Speaker:**H. Mildenberger, University of Freiburg

**Title:**Destroying Guessing Principles

**Abstract:**An Ostaszewski club sequence is a weakening of Jensen’s diamond. In contrast to the diamond, the club does not imply the continuum hypothesis. Numerous questions about the club stay open, and we know only few models in which there is just a club sequence but no diamond sequence. In recent joint work with Shelah we found that a winning strategy for the completeness player in a bounding game on a forcing order does not suffice to establish the club in the extension.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Cross-Alps Logic Seminar****Time:** Friday, 5 May, 16.00-17.00 CEST**Speaker:** D. Sinapova,** **Rutgers University**Title:** Mutual stationarity and the failure of SCH**Abstract:** Mutual stationarity is a compactness type property for singular cardinals. Roughly, it asserts that given a singular cardinal κ, stationary subsets of regular cardinals with limit κ have a “simultaneous witness” for their stationarity. This principle was first defined by Foreman and Magidor in 2001, who showed that it holds for every sequence of stationary sets of cofinality ω. They also showed that their ZFC result does not generalize to higher cofinality. Whether the principle can consistently hold for higher cofinalities remained open, until a few years ago Ben Neria showed that from large cardinals mutual stationarity at ⟨ℵn∣n<ω⟩ can be forced for any fixed cofinality.

We show that we can obtain mutual stationarity at ⟨ℵn∣n<ω⟩ for any fixed cofinality together with the failure of SCH at ℵω. This is joint work with Will Adkisson.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**CUNY Set Theory Seminar****Time:** Friday, 5 May, 10:00am New York time (16:00 CEST)**Speaker:** Joel David Hamkins, Notre Dame University**Title:** Realizing Frege’s Basic Law V, provably in ZFC**Abstract:** The standard set-theoretic distinction between sets and classes instantiates in important respects the Fregean distinction between objects and concepts, for in set theory we commonly take the universe of sets as a realm of objects to be considered under the guise of diverse concepts, the definable classes, each serving as a predicate on that domain of individuals. Although it is commonly held that in a very general manner, there can be no association of classes with objects in a way that fulfills Frege’s Basic Law V, nevertheless, in the ZF framework, it turns out that we can provide a completely deflationary account of this and other Fregean abstraction principles. Namely, there is a mapping of classes to objects, definable in set theory in senses I shall explain (hence deflationary), associating every first-order parametrically definable class F with a set object εF, in such a way that Basic Law V is fulfilled:

εF=εG↔∀x(Fx↔Gx). Russell’s elementary refutation of the general comprehension axiom, therefore, is improperly described as a refutation of Basic Law V itself, but rather refutes Basic Law V only when augmented with powerful class comprehension principles going strictly beyond ZF. The main result leads also to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument. A central goal of the project is to highlight the issue of definability and deflationism for the extension assignment problem at the core of Fregean abstraction.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 5 May, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Jeffrey Bergfalk, University of Barcelona**Title:** Applications of nontrivial coherence**Abstract:** This talk will survey some applications of infinitary combinatorics to superficially unrelated questions arising in the study of condensed mathematics, strong homology, and the derived limit functors, showing in the process that they all, combinatorially speaking, drink from the same well. At least as interesting as our ZFC answers to each of these questions are the further questions they open onto; we aim to survey these as well. This work is joint with Chris Lambie-Hanson, and carries a third aim of connecting with some of the material of his seminar talk of the preceding week; no dependency between the two talks, however, is intended.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 5 May, 2:00-3:30pm New York time (20:00-21:30 CEST)**Speaker: **Karen Lange, Wellesley College**Title: **Classification via effective lists**Abstract:** ‘Classifying’ a natural collection of structures is a common goal in mathematics. Providing a classification can mean different things, e.g., identifying a set of invariants that settle the isomorphism problem or creating a list of all structures of a given kind without repetition of isomorphism type. Here we discuss recent work on classifications of the latter kind from the perspective of computable structure theory. We’ll consider natural classes of computable structures such as vector spaces, equivalence relations, algebraic fields, and trees to better understand the nuances of classification via effective lists and its relationship to other forms of classification in this setting.**Information:** The talk will take place in person. For more information, please see the seminar webpage or email Victoria Gitman.

**24 – 30 April**

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 25 April, 15:00-16:30 CEST**Speaker:** T. Żuchowski, Wrocław University**Title:** Nonseparable growth of ω supporting a strictly positive measure**Abstract:** During the talk I will present a construction in ZFC of a compactification of ω such that its remainder is not separable and carries a strictly positive measure, i.e. measure positive on nonempty open subsets. The measure is defined using the asymptotic density of subsets of ω. The remainder is a Stone space of some Boolean subalgebra of Borel subsets of the Cantor space containing all clopen sets, constructed with an aid of an uncountable almost disjoint family of subsets of ω. This is a joint work with Piotr Borodulin-Nadzieja.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 25 April, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 26 April, 13:00-15:00 Israel Time (12:00-14:00 CEST)**Speaker:** tba **Title:** tba **Abstract:** tba **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**Renyi Institute Set Theory SeminarTime:** Thursday, 26 April, 10:30 – 12:00 CEST

**Speaker:**Lajos Soukup, Renyi Institute

**Title:**Elusive graph properties

**Abstract:**A graph property is said to be elusive (or evasive) if every algorithm testing this property by asking questions of the form “

*is there an edge between vertices x and y*?” requires, in the worst case, to ask about all pairs of vertices. The unsettled Aanderaa-Karp-Rosenberg conjecture is that every non-trivial monotone graph property is elusive for any finite vertex set.

We show that the situation is completely different for infinite vertex sets: the monotone graph properties “every vertex has degree at least n” and “every connected component has size at least m”, where n ≥ 1 and m ≥ 2 are natural numbers, are not elusive for infinite vertex sets, but the monotone graph property ” the graph contains a cycle” is elusive for arbitrary vertex set.

On the other hand, we also prove that every algorithm testing some natural monotone graph properties should check “lots of edges”, e.g, all the edges of an infnite complete subgraph.

**Information:**This talk will be given in hybrid format. Please see the seminar website.

**CUNY Set Theory Seminar****Time:** Friday, 28 April, 12:15pm New York time (18:15 CEST)**Speaker: **Will Boney, Texas State University**Title:** Building generalized indiscernibles in AECs with set theory**Abstract:** Generalized indiscernibles can be built in first-order theories by generalizing the combinatorial Ramsey’s Theorem to classes with more structure, which is an active area of study. Trying to do the same for infinitely theories (in the guise of Abstract Elementary Classes) requires generalizing the Erdos-Rado Theorem instead. We discuss various results about generalizations of the Erdos-Rado Theorem and techniques (including large cardinals and forcing) to build generalized indiscernibles.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 28 April, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Chris Lambie-Hanson**Title:** Whitehead’s problem and condensed mathematics**Abstract:** Whitehead’s problem, which asks whether every Whitehead abelian group is a free abelian group, was a prominent open question in group theory in the mid-20th century. In 1974, Shelah proved that the problem is independent of ZFC, which was a surprising development and provided one of the first instances of a major problem coming from outside logic and set theory to be proven independent of ZFC. In recent years, Clausen and Scholze have introduced the category of condensed abelian groups, which can be seen as an enrichment of the category of topological abelian groups with nicer algebraic properties. Through some deep structural analysis of this category, they showed that, when appropriately interpreted, Whitehead’s problem is not independent in the category of condensed abelian groups: it is provable in ZFC that every abelian group that is Whitehead in the condensed category must be free. In this talk, we sketch a new, more concrete proof of Clausen and Scholze’s result, in the process highlighting some connections between condensed mathematics and the theory of forcing. This is joint work with Jeffrey Bergfalk and Jan Šaroch.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**Baltic Set Theory Seminar****Time:** Tuesday, 18 April, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 18 April, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CEST) **Speaker:** Martin Zeman, University of California, Irvine**Title:** Distributivity of iterated club shooting and fine structural models, part 3**Abstract: **Iterative adding closed unbounded sets through stationary sets found quite a few applications in set theory. One natural way to do this is adding club sets using posets consisting of initial segments of the desired clubs. In such situations, one important property of such iterations is sufficient distributivity. In fact, establishing distributivity is often the main part of arguments that involve iterated club shooting.

There are two possible situations where one iteratively adds clubs. First, for a fixed cardinal $\kappa$, one iteratively adds club subsets of $\kappa^+$. This kind of construction proved to have many applications. Second, one may start with a cardinal $\delta$ and iteratively add club subsets of cardinals $\kappa^+$ where $\kappa$ ranges over some set above $\delta$. Surprisingly, this kind of construction has not been much studied. In this talk we will focus on this situation.

In order to add a club subset of some stationary set $S$ the set $S$ must be large in a certain sense; such sets are called fat. It is known that, consistently, iterative adding club subsets of fat stationary sets of $\omega_n$ on a tail-end of $n\in\omega$ followed by forming an inverse limit at the end may collapse $\aleph_n$ to $\omega$. A strong form of fatness is the property of being the complement of a non-reflecting stationary set. One can prove, using a fairly standard argument, that if the iteration described above uses complements of non-reflecting stationary sets instead of just fat sets, then such an iteration is $(\omega_{n+1},\infty)$-distributive where $\omega_n$ is the first active step in the iteration. One can also prove in ZFC that the analogous amount of distributivity holds of longer iterations, where the first active step is at $\delta$ and inverse limits are used at singular steps, as long as the singular steps are of cofinality $<\delta$. Passing through singular steps of cofinality $\ge\delta$ seems to be difficult, and we only know how to do this over a fine structural model where the non-reflecting stationary sets are carefully chosen. Even in such a seemingly special case, the method does have applications.

This is a part of a joint work of Foreman-Magidor-Zeman on games with filters.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Caltech Logic Seminar****Time:** Tuesday, 18 April, 11:00am-12:00pm Pacific time (20:00-21:00 CEST)**Speaker:** Anton Bernshteyn, Georgia Institute of Technology**Title:** The Local Lemma in descriptive combinatorics: survey and recent developments**Abstract:** The Lovász Local Lemma is a classical tool in probabilistic combinatorics with numerous and diverse applications. In this talk, I will survey what is known about the behavior of the Local Lemma in the Borel and measurable context, including some very recent progress, and state several open problems. Part of this talk is based on joint work with Felix Weilacher.**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 19 April, 13:00-15:00 Israel Time (12:00-14:00 CEST)**Speaker:** Yair Hayut**Title:** Sealing Kurepa trees, continued**Abstract:** In this talk, I’m going to describe Itamar Giron’s master thesis. Most of the results in this talk are due to him.

The main question of the thesis was whether there is a forcing notion that makes an arbitrary Kurepa tree into a non-distributive one, and how far can one go in this direction (can we get sealed Kurepa trees?).

We will start with the classical construction of a Kurepa tree in L (by Solovay). We will show that this tree is distributive in L. We will review the known constructions due to Poor and Shelah (generalized by Muller and me), of sealed Kurepa trees in L (can be generalized to canonical inner models).

Then, we will also find a forcing extension in which for every Kurepa tree, one can add a branch without collapsing cardinals. This means that even though it is easy to find non-distributive Kurepa trees, it is far less trivial to get from combinatorial assertions (such as diamond*), a sealed Kurepa tree.

Finally, I will talk about the forcing notion that “specializes” a Kurepa tree over an arbitrary model of ZFC. This is Giron’s main result, which requires the most sophisticated tools. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**Vienna Research Seminar in Set Theory****Time:** Thursday, 20 April, 11:30-13:00 CEST**Speaker:** L. Zdomskyy, TU Wien**Title:** Menger spaces everywhere**Abstract:** Combinatorial covering properties, which arose from the study of classical special sets of reals, appear in many contexts in topology and set theory. In this talk we shall discuss some applications of the Menger property and certain stronger versions thereof. It is planned to be a gentle introduction to the next talk on April 27.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Vienna Logic ColloquiumTime:** Thursday, 20 April, 15:00 – 15:45 CEST

**Speaker:**Z. Vidnyánszky, Eötvös Loránd University

**Title:**Homomorphism problems in the infinite context

**Abstract:**The CSP dichotomy of Bulatov and Zhuk is a celebrated theorem of computer science: it states that given a finite structure H, deciding whether a structure G admits a homomorphism to G is either easy (in P) or hard (NP-complete). We will discuss two infinitary versions of this theorem. First, following Thornton, in the Borel con ext. Here a striking difference from the finite world emerges: we will show that solving linear equations over a finite field is already hard (Σ12-complete). Second, assuming only ZF, we will consider the relationship of the H-compactness properties, that is, the statement that for every G if every finite substructure of G admits a homomorphism to H then so is G. Here we show that there exists a model M of ZF, such that M⊨H-compactness iff the H-homomorphism problem is easy.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Cross-Alps Logic Seminar****Time:** Friday, 21 April, 16.00-17.00 CEST**Speaker:** M. Elekes, Rényi Institute and Eötvös Loránd University**Title:** tba**Abstract:** tba**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**CUNY Set Theory Seminar****Time:** Friday, 21 April, 12:15pm New York time (18:15 CEST)**Speaker: **Mohammad Golshani, Institute for Research in Fundamental Sciences**Title:** The proper forcing axiom for ℵ1-sized posets and the continuum**Abstract:** We discuss Shelah’s memory iteration technique and use it to show that the PFA for posets of size ℵ1 is consistent with large continuum. This is joint work with David Aspero. **Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 21 April, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Cesar Corral, Universidad Nacional Autónoma de México**Title:** tba **Abstract:** tba **Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 21 April, 2:00 – 3:30 New York time (20:00-21:30 CEST)**Speaker: **James Hanson, University of Maryland**Title: **How bad could it be? The semilattice of definable sets in continuous logic**Abstract:** Continuous first-order logic is a generalization of discrete first-order logic suited for studying structures with natural underlying metrics, such as operator algebras and R-trees. While many things from discrete model theory generalize directly to continuous model theory, there are also new subtleties, such as the correct notion of ‘definability’ for subsets of a structure. Definable sets are conventionally taken to be those that admit relative quantification in an appropriate sense. An easy argument then establishes that the union of definable sets is definable, but in general the intersection of definable sets may fail to be. This raises the question of which semilattices arise as the partial order of definable sets in a continuous theory.

After giving an overview of the basic properties of definable sets in continuous logic, we will give a largely visual proof that any finite semilattice (and therefore any finite lattice) is the partial order of definable sets in some superstable continuous first-order theory. We will then discuss a partial extension of this to certain infinite semilattices.**Information:** The talk will take place in person. For more information, please see the seminar webpage or email Victoria Gitman.

**10 – 16 April**

**Baltic Set Theory Seminar****Time:** Tuesday, 11 April, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 11 April, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CEST) **Speaker:** Martin Zeman, University of California, Irvine**Title:** Distributivity of iterated club shooting and fine structural models, part 2**Abstract: **Iterative adding closed unbounded sets through stationary sets found quite a few applications in set theory. One natural way to do this is adding club sets using posets consisting of initial segments of the desired clubs. In such situations, one important property of such iterations is sufficient amount of distributivity. In fact, establishing distributivity is often the main part of arguments that involve iterated club shooting.

There are two possible situations where one iteratively adds clubs. First, for a fixed cardinal $\kappa$, one iteratively adds club subsets of $\kappa^+$. This kind of construction proved to have many applications. Second, one may start with a cardinal $\delta$ and iteratively add club subsets of cardinals $\kappa^+$ where $\kappa$ ranges ofer some set above $\delta$. Surprisingly, this kind of construction has not been much studied. In this talk we will focus on this situation.

In order to add a club subset of some stationary set $S$ the set $S$ must be large in certain sense; such sets are called fat. It is known that, consistently, iterative adding club subsets of fat stationary sets of $\omega_n$ on a tail-end of $n\in\omega$ followed by forming an inverse limit at the end may collapse $\aleph_n$ to $\omega$. A strong form of fatness is the property of being the complement of a non-reflecting stationary set. One can prove, using a fairly standard argument, that if the iteration described above uses complements of non-reflecting stationary sets instead of just fat sets, then such an iteration is $(\omega_{n+1},\infty)$-distributive where $\omega_n$ is the first active step in the iteration. One can also prove in ZFC that the analogous amount of distributivity holds of longer iterations, where the first active step is at $\delta$ and inverse limits are used at singular steps, as long as the singular steps are of cofinality $<\delta$. Passing through singular steps of cofinality $\ge\delta$ seems to be difficult, and we only know how to do this over a fine structural model where the non-reflecting stationary sets are carefully chosen. Even in such a seemingly special case, the method does have applications.

This is a part of a joint work of Foreman-Magidor-Zeman on games with filters.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Caltech Logic Seminar****Time:** Tuesday, 11 April, 11:00am-12:00pm Pacific time (20:00-21:00 CEST)**Speaker:** William Chan, University of North Texas**Title:** Cardinalities Below the Power Set of the First Uncountable Cardinals**Abstract:** This talk will survey the known structure of cardinalities below the power set of the first uncountable cardinal under various determinacy assumptions. Regularity and cofinality of cardinalities will be formulated. Combinatorial aspects of cardinalities such as primeness and Jónssonness may also be discussed. Portions of this talk includes joint work with Jackson and Trang.**Information:** Please see the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 11 April, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Samuele Giraudo, Université du Québec à Montréal**Title:** Some enumerative and order-theoretic properties in combinatory logic**Abstract: **In the field of combinatory logic, every term specifies a multigraph consisting of the set of terms that can be reached from it. By adopting an algebraic and combinatorial approach, we investigate various properties of these multigraphs. Specifically, we address enumerative questions, such as whether the multigraph of a term is finite, and if so, how many elements it contains. We also explore order-theoretic questions, such as whether the reflexive and transitive closure of the rewrite relation constitutes a partially ordered set or even a lattice. In this work-in-progress, we present our initial findings in this area.**Information:** See the seminar webpage.

**CUNY Set Theory Seminar****Time:** Friday, 14 April, 12:15pm New York time (18:15 CEST)**Speaker: **Gabriel Goldberg, University of California, Berkeley**Title:** Cardinal preserving embeddings and strongly compact cardinals**Abstract:** Kunen’s theorem that there is no elementary embedding from V to V seems to set an upper bound on the hierarchy of large cardinal axioms. Challenging this, Caicedo asked what happens when V is replaced with an inner model M that is very close to V in the sense that M correctly computes the class of cardinals. Assuming the existence of strongly compact cardinals, we show that there is no elementary embedding from such an inner model M into V or from V into M. The former result (M into V) is joint work with Sebastiano Thei. Without strong compactness assumptions, both questions remain open, but we’ll discuss some partial results.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 14 April, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** David Fernández-Bretón, Instituto Politécnico Nacional**Title:** Hindman’s theorem in the hierarchy of choice principles**Abstract:** We will explain how to state Hindman’s theorem in a way that provides a statement provable from ZFC but not from ZF alone. This gives us a way of considering Hindman’s theorem as a weak version of the Axiom of Choice; we will discuss where this statement fits within the hierarchy of weak choice principles, and provide a few independence proofs, involving this principle, within ZF —this will require us to outline the Fraenkel-Mostowski method for providing independence proofs in ZFA.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**3 – 9 April**

**Baltic Set Theory Seminar****Time:** Tuesday, 4 April, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 4 April, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CEST) **Speaker:** Martin Zeman, University of California, Irvine**Title:** Distributivity of iterated club shooting and fine structural**Abstract: **Iterative adding closed unbounded sets through stationary sets found quite a few applications in set theory. One natural way to do this is adding club sets using posets consisting of initial segments of the desired clubs. In such situations, one important property of such iterations is sufficient amount of distributivity. In fact, establishing distributivity is often the main part of arguments that involve iterated club shooting. There are two possible situations where one iteratively adds clubs.

First, for a fixed cardinal $\kappa$, one iteratively adds club subsets of $\kappa^+$. This kind of construction proved to have many applications. Second, one may start with a cardinal $\delta$ and iteratively add club subsets of cardinals $\kappa^+$ where $\kappa$ ranges over some set above $\delta$. Surprisingly, this kind of construction has not been much studied. In this talk we will focus on this situation.

In order to add a club subset of some stationary set $S$ the set $S$ must be large in a certain sense; such sets are called fat. It is known that, consistently, iteratively adding club subsets of fat stationary sets of $\omega_n$ on a tail-end of $n\in\omega$ followed by forming an inverse limit at the end may collapse $\aleph_n$ to $\omega$. A strong form of fatness is the property of being the complement of a non-reflecting stationary set. One can prove, using a fairly standard argument, that if the iteration described above uses complements of non-reflecting stationary sets instead of just fat sets, then such an iteration is $(\omega_{n+1},\infty)$-distributive where $\omega_n$ is the first active step in the iteration. One can also prove in ZFC that the analogous amount of distributivity holds of longer iterations, where the first active step is at $\delta$ and inverse limits are used at singular steps, as long as the singular steps are of cofinality $<\delta$. Passing through singular steps of cofinality $\ge\delta$ seems to be difficult, and we only know how to do this over a fine structural model where the non-reflecting stationary sets are carefully chosen. Even in such a seemingly special case, the method does have applications.

This is a part of a joint work of Foreman-Magidor-Zeman on games with filters.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**27 March – 2 April**

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 28 March, 15:00-16:30 CEST**Speaker:** Miguel Moreno, Universität Wien**Title:** Generalised Descriptive Set Theory, part III**Abstract:** Following part I and part II in this three part series, during this talk we will discuss where in the generalized Borel-reducibility hierarchy are the isomorphism relation of first order complete theories. These theories are divided into two kinds: classifiable and non-classifiable. To study the classifiable theories case is needed the use of Ehrenfeucht-Fraïssé games. On the other hand the study of the non-classifiable theories is done by using colored ordered trees. The goal of the talk is to see the classifiable theories case and sketch the ideas of non-classifiable theories.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 28 March, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 28 March, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Matthieu Joseph, Université Paris Saclay**Title:** Several notions of freeness for actions of countable groups**Abstract: **In this talk we will study different notions of freeness for actions of countable groups by homeomorphisms on compact spaces. Genuine freeness, freeness in the sense of (some) invariant measure and freeness in the sense of the topology will be compared, via concrete examples. We will bring out and discuss the class of allosteric groups, which is the class of groups that admit actions on compact spaces with an invariant measure that are free in the sense of the topology but non-free in the sense of the measure.**Information:** See the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 29 March, 13:00-15:00 Israel Time (12:00-14:00 CEST)**Speaker:** Yair Hayut**Title:** Sealing Kurepa trees**Abstract:** In this talk, I’m going to describe Itamar Giron’s master thesis. Most of the results in this talk are due to him.

The main question of the thesis was whether there is a forcing notion that makes an arbitrary Kurepa tree into a non-distributive one, and how far can one go in this direction (can we get sealed Kurepa trees?).

We will start with the classical construction of a Kurepa tree in L (by Solovay). We will show that this tree is distributive in L. We will review the known constructions due to Poor and Shelah (generalized by Muller and me), of sealed Kurepa trees in L (can be generalized to canonical inner models).

Then, we will also find a forcing extension in which for every Kurepa tree, one can add a branch without collapsing cardinals. This means that even though it is easy to find non-distributive Kurepa trees, it is far less trivial to get from combinatorial assertions (such as diamond*), a sealed Kurepa tree.

Finally, I will talk about the forcing notion that “specializes” a Kurepa tree over an arbitrary model of ZFC. This is Giron’s main result, which requires the most sophisticated tools. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**Vienna Logic ColloquiumTime:** Thursday, 30 March, 15:00 – 15:45 CEST

**Speaker:**M. Pinsker, TU Wien

**Title:**Constraint Satisfaction Problems: algebraic and model-theoretic challenges to distinguish the easy from the hard

**Abstract:**I will give a gentle introduction to current algebraic and model-theoretic methods in the computational complexity of Constraint Satisfaction Problems (CSPs).

A CSP is a computational problem where we are given variables and constraints about them; the question is whether the variables can be assigned values such that all constraints are satisfied. Numerous natural computational problems, such as satisfiability of a given system of equations over a field, are CSPs; in fact, any computational problem is Turing-equivalent to a CSP.

Any CSP can be modeled by a relational structure, and conversely every relational structure naturally defines a CSP. In view of humanity’s continuing quest to distinguish easy from hard problems in general, and the class P (polynomial-time solvable problems, e.g. satisfiability of linear equations over a field) from the class NP (polynomial-time verifiable problems, e.g. satisfiability of a propositional formula) in particular, the question arises which mathematical properties of a relational structure make the corresponding CSP easy and which make it hard. It turns out that particular algebraic invariants of the structure often determine the borderline between different complexity classes. Hence algebraic methods, combined with concepts from model theory as well as from Ramsey theory in the case of infinite structures, yield appropriate tools to determine the computational complexity of CSPs.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Bourbaki Friday Seminar****Time:** Friday, 31 March, 14.00-17.30 CEST

14h **Romain Tessera — Limites d’actions de groupes sur des espaces hyperboliques**

Dans cet exposé introductif, nous tenterons d’expliquer dans les grandes lignes des notions introduites indépendamment par Paulin et Bestvina à la fin des années 80. L’idée est de considérer une topologie naturelle sur un ensemble d’espaces métriques munis d’actions par isométries d’un groupe fixé . Dans les exemples qui nous intéressent, ces espaces sont hyperboliques au sens de Gromov, et nous nous intéresserons tout particulièrement aux points d’accumulation, qui sont des arbres réels.

15h

**Mirna**

*Džamonja*— Les axiomes de forcingLe forcing est une méthode qui permet de passer d’un univers (modèle) de la théorie de ZFC des ensembles, à un autre , enrichi par un nouvel objet générique qui est construit à partir d’approximations organisées par un ordre partiel, nommé l’ordre de forcing. Le mot générique signifie que est choisi de la manière à assurer un certain nombre de conditions, qui sont exprimées en ayant une intersection non-vide avec des ensembles dits denses. À la différence des constructions inductives, où l’existence de l’objet construit par les approximations organisées sur un bon ordre est assuré par les principes de ZFC, les constructions organisées par un ordre partiel ne donnent pas lieu en général à un objet générique qui existe dans le même univers de ZFC. Pour cette existence, il faut payer le prix en enrichissant l’univers (cela ressemble à la théorie de Galois), d’où le passage à . Néanmoins, il est possible d’avoir des univers de ZFC qui sont saturés par rapport de l’existence de quelques-uns de ces génériques. Par exemple, un univers qui satisfait l’Axiome de Martin (MA), possédera un objet générique pour tout forcing avec la propriété ccc et pour toute famille d’ensembles denses qui est de cardinalité. La cohérence de ZFC implique la cohérence de ZFC+MA et, avec des hypothèses plus fortes que la cohérence de ZFC, on peut obtenir la cohérence de modèles qui vérifient des axiomes encore plus forts que MA. L’exposé expliquera quelques-uns de ces axiomes et leurs conséquences, notamment sur l’hypothèse du continu.

16h30

**Maxime**

*Bourrigan*— Homologie de Morse et simplicité de groupes de transformationsNous présenterons deux « miniatures » évoquant certains aspects du théorème énoncé dans le titre de l’exposé d’Étienne Ghys.

Nous aborderons d’abord le thème de la simplicité des groupes d’homéomorphismes et de difféomorphismes à travers le cas du cercle, déjà étonnamment subtil.

Dans un deuxième temps, nous introduirons l’homologie de Morse, une manière de calculer l’homologie d’une variété à partir d’une fonction définie dessus, et plus précisément à partir de l’étude dynamique de son gradient. Cette construction est une espèce de modèle réduit de l’outil crucial pour le théorème de Cristofaro-Gardiner, Humilière et Seyfaddini : l’homologie de Floer.

**Information:**For the live stream, click on [Live IHP] on the seminar wepage.

**Cross-Alps Logic Seminar****Time:** Friday, 31 March, 16.00-17.00 CEST**Speaker:** L. Patey, CNRS**Title:** Canonical notions of forcing in Reverse Mathematics**Abstract:** In Reverse Mathematics, a proof of non-implication from a statement P to a statement Q consists of creating a model of P which is not a model of Q. To this end, one usually create a complicated instance I of Q, and then, build iteratively a model of P containing I while avoiding every solution to I. The difficult part consists in building solutions to instances of P which will not compute any solution to I. This is usually done by forcing. Moreover, by some empirical observation, the notion of forcing used in a separation of P from Q usually does not depend on Q. For example, constructing models of WKL is usually done by forcing with Pi^0_1 classes. This tends to show that P admits a “canonical” notion of forcing. In this talk, we provide a formal framework to discuss this intuition, and study the canonical notions of forcing associated to some important statements in Reverse Mathematics. This is a joint work with Denis Hirschfeldt.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**CUNY Set Theory Seminar****Time:** Friday, 31 March, 12:15pm New York time (18:15 CEST)**Speaker: **Benjamin Goodman, CUNY**Title:** Σn-correct forcing axioms**Abstract:** The standard method of producing a model of a forcing axiom from a supercompact cardinal in fact gives a model of an even stronger principle: that for every small name a and every Σ2 formula arphi such that φ(a) is forceable by and preserved under further forcing in our forcing class, there is a filter F which meets a desired collection of dense sets and also interprets a such that φ(aF) already holds. I will show how to generalize this result to formulas of higher complexity by starting with slightly stronger large cardinal assumptions, then discuss the bounded versions of these enhanced forcing axioms, their relationships to other similar principles, and their consequences.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 31 March, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Keegan Dasilva-Barbosa**Title:** Box Ramsey and Canonical Partitions**Abstract:** The KPT correspondence gives a full characterization of the dynamics of automorphism groups of Fra\”iss\’e structures through finite combinatorics. There is still much open however on whether or not there is a full correspondence between big Ramsey degrees and topological dynamics. While a partial answer has been found by Zucker by considering structures that admit a big Ramsey structure, the question still remains open. Motivated by this problem, we aim to answer a related question. Namely, what are the necessary and sufficient conditions needed for a structure to admit a finite list of canonical relations? We do so by developing a natural productive analogue to big Ramsey we call the Box Ramsey degree, solving a question of Masulovic. Our techniques will be reminiscent of Rado’s proof of the Erd\”os-Rado theorem, or more recently, works on canonical equivalence relations done by Laflamme, Sauer, and Vuksanovic. We will conclude with an analysis of canonical equivalence relations.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 31 March, 2:00 – 3:30 New York time (20:00-21:30 CEST)**Speaker: **Corey Switzer, University of Vienna**Title: **Galois-Tukey reductions and canonical structure in the Cichoń diagram**Abstract:** Cardinal invariants of the continuum are cardinal numbers which, roughly, measure how ‘badly’ CH fails in various mathematical contexts such as analysis and topology. For instance the cardinal add(N) is the least κ for which there are κ many Lebesgue measure zero sets of reals whose union is not measure zero. Classical facts imply ℵ1≤add(N)≤2ℵ0 but the precise value is undetermined in ZFC and depends heavily on the axioms of set theory. Other numbers follow a similar pattern of ‘the least size of a set of reals (Borel sets, etc) lacking a classical smallness property’.

The Cichoń diagram displays cardinal invariants related to Lebesgue measure (the null ideal), Baire category (the meager ideal) as well as the bounding and dominating numbers which concern growth rates of functions. Many surprising ZFC-inequalities exist between these cardinals suggesting a rich world living on the reals in various models of set theory. At the combinatorial heart of every proof of a ZFC inequality derives from a Galois-Tukey reduction: the (ZFC-provable) existence of a pair of continuous maps with simple properties that make sense outside of the context of logic and indeed would be sensible to any analyst or topologist.

In this talk we will discuss some recent work in progress on the descriptive complexity of maps witnessing consistent but non-provable implications. We will show using largely computability theoretic methods that in Gödel’s constructible universe there are low level projective reductions between any two cardinal invariants – thus CH holds in a very ‘definable’ way, while in Solovay’s model of ‘all sets of reals are Lebesgue measurable’ (and therefore the axiom of choice fails) there are no non-ZFC provable implications thus these cardinals are all as different as possible.**Information:** The talk will take place in person. For more information, please see the seminar webpage or email Victoria Gitman.

**Bourbaki Seminar****Time:** Saturday, 1 April, 10.00-17.00 CEST

10h **Jonathan Hickman — Pointwise convergence for the Schrödinger equation**,

*after Xiumin Du and Ruixiang Zhang*

For an initial datum f ∈ L2(Rn), consider the linear Schrödinger equation iut − ∆x u = 0, u(x, 0) = f (x) (x, t) ∈ Rn × R. In 1980, Carleson asked which additional conditions on f guarantee (⋆) lim t→0 u(x, t) = f (x) for almost every x ∈ Rn. More precisely, what is the minimal Sobolev regularity index s such that (⋆) holds whenever f ∈ Hs(Rn) ? Whilst the n = 1 case was fully understood by the early 1980s, in higher dimensions the situation is much more nuanced. Nevertheless, a recent series of dramatic developments brought about an almost complete resolution of the problem. First Bourgain 2016 produced a subtle counterexample demonstrating that pointwise convergence can fail for certain f ∈ Hs(Rn) with s < n 2(n+1) . Complementing this, convergence was then shown to hold for s > n 2(n+1) when n = 2 in a landmark paper of Du, Guth and Li 2017 and later in all dimensions in equally important work of Du and Zhang 2019.

This seminar will explore the positive result of Du and Zhang 2019. The argument combines sophisticated modern machinery from harmonic analysis such as the multilinear Strichartz estimates of Bennett, Carbery and Tao 2006 and the ℓ2 decoupling theory of Bourgain and Demeter 2015. However, equally important are a variety of elementary guiding principles, rooted in Fourier analysis, which govern the behaviour of solutions to the Schrödinger equation. The talk will focus on these basic Fourier analytic principles, building intuition and presenting a powerful toolbox for tackling problems in modern PDE and harmonic analysis.

11h30

**Matteo**,

*Viale*— Strong forcing axioms and the continuum problem*following Aspéro’s and Schindler’s proof that*

**MM**^{ + +}implies Woodin’s Axiom (*)A topological approach to forcing axioms considers them as strong forms of the Baire category theorem; an algebraic approach describes certain properties of “algebraic closure” for the universe of sets that can be derived from them. Our goal is to show how the theorem of Aspéro and Schindler links the geometric and algebraic points of view. Drawing on Gödel’s program, we connect these mathematical results to the philosophical debate on what could constitute a viable solution of the continuum problem.

14h30

**Clara**,

*Löh*— Exponential growth rates in hyperbolic groups*after Koji Fujiwara and Zlil Sela*

A classical result of Jørgensen and Thurston shows that the set of volumes of finite volume complete hyperbolic 3-manifolds is a well-ordered subset of the real numbers of order type ωω ; moreover, they showed that each volume can only be attained by finitely many isometry types of hyperbolic 3-manifolds.

Fujiwara and Sela established a group-theoretic companion of this result : If Γ is a non-elementary hyperbolic group, then the set of exponential growth rates of Γ is well-ordered, the order type is at least ωω, and each growth rate can only be attained by finitely many finite generating sets (up to automorphisms).

In this talk, I will outline this work of Fujiwara and Sela and discuss related results.

16h

**Étienne**

homeomorphisms of the 2-dimensional sphere is not a simple

group

*Ghys*— The group of area-preserving and orientation-preservinghomeomorphisms of the 2-dimensional sphere is not a simple

group

*, after D. Cristofaro-Gardiner, V. Humilière and S. Seyfaddini*

Since the late 1970s, it has been known that the neutral component of the

group of compactly supported diffeomorphisms of a connected manifold is a

simple group. In the case of diffeomorphisms preserving a volume form or a

symplectic form, we have a similar result : there is then an “obvious” subgroup that is

simple. For volume-preserving homeomorphisms, the situation is understood when

the dimension is greater than or equal to 3. The case of surfaces, and especially of

the 2-dimensional sphere, has resisted many efforts over the last forty years. years.

The theorem of D. Cristofaro-Gardiner, V. Humilière and S. Seyfaddini is a surprise : the

group of homeomorphisms of the 2-dimensional sphere which respect the area and

orientation is not a simple group. The proof is a tour de force and makes extensive

use of periodic Floer homology. I will try to present the context as well as the main

lines of this beautiful result.

**Information:**For the live stream, click on [Live IHP] on the seminar wepage.

**20 – 26 March**

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 21 March, 15:00-16:30 CEST**Speaker:** Miguel Moreno, University of Vienna**Title:** Generalised Descriptive Set Theory, part II**Abstract:** We have introduced the notions of K-Borel class, K-analytic class, K-analytic-coanalytic class, K-Borel* class in the previous talk. In descriptive set theory the Borel class, the analytic-coanalytic class, and the Borel* class are the same class, we showed that this doesn’t hold in the generalized descriptive set theory.

In this talk, we will show the consistency of “K-Borel* class is equal to the K-analytic class”. This was initially proved by Hyttinen and Weinstein (former Kulikov), under the assumption V=L. We will show a different proof that shows that this holds in L but also can be forced by a cofinality-preserving GCH-preserving forcing from a model of GCH, but also by a <κ-closed κ+‑cc forcing.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 21 March, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 21 March, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Benjamin Siskind, Carnegie Mellon University**Title:** The short extender of the Martin measure**Abstract: **Under AD, using the uniform cofinalities analysis on function from finite products of omega_1 into omega_1, we show that the short extender of the the Martin measure ultrapower comes from the iterated ultrapower by the club filter on omega_1 and discuss some applications. We’ll also give an iterated ultrapowers proof of the uniform cofinality analysis. (We suspect that these observations have been noticed before but don’t know a reference.)**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 22 March, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Bea Adam-Day, University of Leeds**Title:** Indestructibility and $C^{(n)}$-supercompact cardinals**Abstract:** In the 70’s Laver showed that a supercompact cardinal $\kappa$ may be made indestructible by a suitable class of forcings—namely, after a preparatory forcing, the supercompactness of $\kappa$ will not be destroyed by any further $<\kappa$-directed closed forcing. Many indestructibility results have since been written, as well as those demonstrating the impossibility of indestructibility (or even preservation) of many large cardinals. In this talk we will consider the case of $C^{(n)}$-supercompact cardinals—a stronger and more slippery variant of supercompact cardinals—and how they can be made indestructible for $n\leq 2$. **Information:** Please see the seminar webpage.

**Bristol Logic and Set Theory Seminar****Time:** Wednesday, 22 March, 2-3.30pm UK time (15:00-16:30 CET)**Speaker:** Juan Aguilera, University of Ghent**Title:** A generalization of Borel determinacy**Abstract:** We present a theorem which is, in some sense, the provably optimal generalization of Martin’s Borel determinacy for infinite games on integers.**Information:** Please see the seminar webpage https://www.bristolmathsresearch.org/events/logic-and-set-theory for the zoom link.

**Toronto Set Theory Seminar****Time:** Friday, 24 March, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** tba **Title:** tba **Abstract:** tba **Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 24 March, 2:00 – 3:30 New York time (20:00-21:30 CET)**Speaker:** Victoria Gitman, CUNY**Title:** Parameter-free comprehension in second-order arithmetic**Abstract:** Second-order arithmetic has two types of objects: numbers and sets of numbers, which we think of as the reals. The second-order arithmetic framework has been used successfully to investigate what kinds of real numbers need to exist to prove various significant results in analysis. One of the strongest second-order arithmetic axiomatizations is the theory Z2 consisting of the axioms PA (for numbers), the set induction axiom, and comprehension for all second-order formulas with set parameters. How significant is the inclusion of set parameters in the comprehension scheme? Let Z−p2 be like Z2, but where set parameters are not allowed in the comprehension scheme. Harvey Friedman showed that Z2 and Z−p2 are equiconsistent because parameter-free comprehension suffices to build a model’s version of the constructible universe L inside the model and the ‘constructible’ reals satisfy Z2. Kanovei recently showed that models of Z−p2 can be very badly behaved, for example, their sets may not even be closed under complement. Kanovei also showed that there can be nicely behaved models of Z−p2 in which Σ12-comprehension (with set parameters) holds. He constructed his model in a forcing extension by a tree iteration of Sacks forcing. In Kanovei’s model, Σ14-comprehension (with set parameters) fails and he asked whether this can be improved to Σ13-comprehension. In this talk, I will show how to construct a model of Σ12-comprehension and Z−p2 in which Σ13-comprehension fails. The model will be constructed in a forcing extension by a tree iteration of Jensen’s forcing. Jensen’s forcing is a sub-poset of Sacks forcing constructed by Jensen to show that it is consistent to have a non-constructible Π12-definable singleton real (every Σ12-definable set of reals is constructible by Shoenfield’s Absoluteness).**Information:** The talk will take place in person. For more information, please see the seminar webpage or email Victoria Gitman.

**13 – 19 March**

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 14 March, 15:00-16:30 CEST**Speaker:** Miguel Moreno, University of Vienna**Title:** Generalised Descriptive Set Theory, part I**Abstract:** This is the first of three talks about Generalised Descriptive Set Theory. The aim of this talk is to introduce the notions of K‑Borel class, K‑analytic class, K‑analytic-coanalytic class, K‑Borel* class, and show the relation between these classes.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 14 March, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 14 March, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Dominik Adolf, University of North Texas**Title:** Rudin-Keisler Capturing and Mutual Stationarity at successors of singulars**Abstract: **Mutual Stationarity, first introduced by Foreman and Magidor, is a key concept for the study of stationary subsets on the powersets of singular cardinals. In this talk we will introduce a new way to add sequences with strong mutual stationarity properties by forcing. Towards this end we introduce a concept we dubbed Rudin-Keisler Capturing, a new large cardinal property living in the gaps of the Mitchell-Order. This is joint work with Omer Ben-Neria.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**CMU Logic Seminar****Time:** Tuesday, 14 March, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Alejandro Poveda Ruzafa, Harvard University**Title:** The Gluing Property**Abstract: **In this talk, I’ll report on a recent joint project with Y. Hayut where we introduce a new compactness principle called ‘’The Gluing Property’’. This concept was isolated from a former argument by Gitik saying that the existence of a \kappa-compact cardinal entails an inner model with a strong cardinal. During the talk, we shall introduce the gluing property and show that compact-like cardinals (such as strong compacts, \Pi^1_1-subcompacts, etc) do satisfy a certain amount of gluing. Contrarily, non-compact cardinals like strong cardinals are known to fail to have the gluing property. We shall conclude the exposition by discussing how to force the \omega gluing property from optimal assumptions. This latter being surprisingly mild – just “\exists \kappa (o(\kappa)=\omega_1)”.**Information:** See the seminar webpage.

**Vienna Logic ColloquiumTime:** Thursday, 16 March, 15:00 – 15:45 CET

**Speaker:**Silvain Rideau-Kikuchi, University of Paris

**Title:**Multi topological fields, approximations and NTP2

**Abstract:**(Joint work with S. Montenegro)

The striking resemblance between the behaviour of pseudo-algebraically closed, pseudo real closed and pseudo p-adically fields has lead to numerous attempts at describing their properties in a unified manner. In this talk I will present another of these attempts: the class of pseudo-

*T*-closed fields, where

*T*is an enriched theory of fields. These fields verify a «local-global» principle with respect to models of

*T*for the existence of points on varieties. Although it very much resembles previous such attempts, our approach is more model theoretic in flavour, both in its presentation and in the results we aim for.

The first result I would like to present is an approximation result, generalising a result of Kollar on PAC fields, respectively Johnson on Henselian fields. This result can be rephrased as the fact that existential closeness in certain topological enrichments come for free from existential closeness as a field. The second result is a (model theoretic) classification result for bounded pseudo-

*T*-closed fields, in the guise of the computation of their burden. One of the striking consequence of these two results is that a bounded perfect PAC field with

*n*independent valuations has burden

*n*and, in particular, is NTP

_{2}.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate

**Cross-Alps Logic Seminar****Time:** Friday, 17 March, 16.00-17.00 CEST**Speaker:** Victor Selivanov, Institute of Informatics Systems, Novosibirsk**Title:** Boole vs Wadge: comparing basic tools of descriptive set theory**Abstract:** We systematically compare ω-Boolean classes and Wadge classes, e.g. we complement the result of W. Wadge that the collection of non-self-dual levels of his hierarchy coincides with the collection of classes generated by Borel ω-ary Boolean operations from the open sets in the Baire space. Namely, we characterize the operations, which generate any given level in this way, in terms of the Wadge hierarchy in the Scott domain. As a corollary, we deduce the non-collapse of the latter hierarchy. Also, the effective version of this topic and its extension to k-partitions are developed.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**CUNY Set Theory Seminar****Time:** Friday, 17 March, 12:15pm New York time (18:15 CET)**Speaker: **Jonathan Osinski, University of Hamburg**Title:** Model-Theoretic Characterizations of Weak Vopěnka’s Principle**Abstract:** It has been known since the 1980s that Vopěnka’s Principle (VP) is equivalent to certain statements about logics, e.g. to the schema ‘Every logic has a compactness cardinal.’ On the other hand, it was only recently shown by Trevor Wilson that a related statement statement called Weak Vopěnka’s Principle (WVP) is strictly weaker than VP. In fact, Joan Bagaria and Wilson showed that WVP is equivalent to the existence of Πn-strong cardinals for all natural numbers n. We generalize logical characterizations of strong cardinals to achieve a characterization of Πn-strong cardinals and therefore of WVP in terms of properties of strong logics. This is partly joint work with Will Boney and partly with Trevor Wilson.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 17 March, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** George Domat**Title:** Big Mapping Class Groups in Dimensions 0,1, and 2**Abstract:** I’ll first give a broad introduction to big mapping class groups and why they may be of interest to set theorists. These groups are a rich class of non-archimedean Polish groups coming from the worlds of low-dimensional topology and geometric group theory. After this broad overview I will talk in more detail about some results (in parts joint with Mladen Bestvina, Ryan Dickmann, and Kasra Rafi) around automatic continuity properties of big mapping class groups of surfaces.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 17 March, 2:00 – 3:30 New York time (20:00-21:30 CET)**Speaker: **Filippo Calderoni, Rutgers University**Title: **Rotation equivalence and rigidity**Abstract:** The theory of countable Borel equivalence relations analyzes the actions of countable groups on Polish spaces. The main question studied is how much information is encoded by the corresponding orbit space. The amount of encoded information reflects the extent to which the action is rigid.

In this talk we will discuss rigidity results for the action of the group of rational rotations. In particular we will analyze the rotation equivalence on spheres in higher dimension. This is connected to superrigidity results of Margulis and to Zimmer’s program about the actions of discrete subgroups of Lie groups on manifolds. **Information:** The talk will take place in person. For more information, please see the seminar webpage or email Victoria Gitman.

**6 – 12 March**

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 7 March, 15:00-16:30 CEST**Speaker:** Juris Steprāns, York University**Title:** Selective and Milliken-Taylor ultrafilters**Abstract:** I will report on joint work with Dilip Raghavan solving a question of Blass about whether the existence of many selective ultrafilters implies the existence of Milliken-Taylor ultrafilters. The first part of the talk will provide the historical context of what was known in the mid 80s that prompted Blass to ask his question. The second part will discuss the key technical advance in our argument.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 7 March, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Tuesday, 7 March, 11:00am-12:00pm Pacific time (20:00-21:00 CET)**Speaker:** Martino Lupini, University of Bologna**Title:** Definable refinements of classical algebraic invariants**Abstract:** In this talk I will explain how methods from logic allow one to construct refinements of classical algebraic invariants that are endowed with additional topological and descriptive set-theoretic information. This approach brings to fruition initial insights due to Eilenberg, Mac Lane, and Moore (among others) with the additional ingredient of recent advanced tools from logic. I will then present applications of this viewpoint to invariants from a number of areas in mathematics, including operator algebras, group theory, algebraic topology, and homological algebra.**Information:** Please see the seminar webpage.

**Leeds Models and Sets Seminar****Time:** Wednesday, 8 March, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Soinbhe Nic Dhonncha, Manchester University**Title:** Purity in chains of modules**Abstract:** The model theory of modules extends naturally to certain functor categories. One such category is that of representations of the biinfinite quiver A_{\infty}^{\infty}, where each object can be thought of as a biinfinite chain of R-modules. This raises the question of how the objects and morphisms of (model theoretic) interest for this category relate to those of Mod R. In the simplest case, we take R to be von Neumann regular.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 8 March, 16:00-17:30 CET**Speaker:** Radek Honzik**Title:** Compactness principles at small cardinals and their preservation**Abstract:** It is known that if k is a (typical) large cardinal and P is a forcing notion of size < k, then k is preserved as a large cardinal in the forcing extension V[P]. If k is a successor cardinal, for instance w2, and satisfies some compactness principles such the tree property or stationary reflection, the preservation of these compactness properties by forcing notions is more complicated. We will survey recent results in this area and focus on the result that over models of PFA, Cohen forcing at w of any length preserves the tree property at w2 and the negation of the weak Kurepa hypothesis at w1. **Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Bristol Logic and Set Theory Seminar****Time:** Wednesday, 8 March, 16:30-17:30 UK time (17:30-18:30 CET) **Speaker:** Minh Tran, University of Oxford**Title:** Toward classifying the reducts of the complex fields**Abstract:** We will discuss some recent progress on the problem of classifying the reducts of the complex field (with named parameters and up to interdefinability). The tools we use include the recent solutions of the Restricted Trichotomy Conjecture in characteristic 0 and a generalized sumproduct result from additive combinatorics. (Joint with Benjamin Castle)**Information:** The talk can be streamed on zoom, please contact Philipp Schlicht in advance.

**CUNY Set Theory Seminar****Time:** Friday, 10 March, 12:15pm New York time (18:15 CET)**Speaker: **James Holland, Rutgers University**Title:** Forcing more choice over the Chang model**Abstract:** The ordinal Θ has lots of interesting results in the context of L(R). Here, we try to find an analogue of Θ for the Chang model, and see what assumptions about it are natural. These assumptions come out of the process of forcing more dependent choice over the Chang model.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 10 March, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Natasha Dobrinen, University of Notre Dame**Title:** tba **Abstract:** tba **Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**27 February – 5 March**

**Baltic Set Theory Seminar****Time:** Tuesday, 28 February, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 28 February, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Gabriel Goldberg, University of California, Berkeley**Title:** Inner models from stationary logic, part 3**Abstract: **We discuss the inner model C(aa) introduced by Kennedy-Magidor-Väänänen, and answer several questions posed in their paper “Inner models from strong logics”. Assuming large cardinal axioms, we’ll show that this model satisfies GCH (this is joint work with John Steel) and that C(aa) satisfies o(κ) = κ^{++} for all regular cardinals κ > ω_{1} (this is joint work with Otto Rajala).**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Caltech Logic Seminar****Time:** Tuesday, 28 February, 11:00am-12:00pm Pacific time (20:00-21:00 CET)**Speaker:** Julien Melleray, Université Lyon 1**Title:** Clopen type semigroups of actions on 0-dimensional compact spaces **Abstract:** Consider an action of a discrete group G on a compact, 0-dimensional space X. Its clopen type semigroup is an algebraic structure which encodes the equidecomposability relation between clopen subsets of X (two clopen subsets A,B of X are equidecomposable if there is a clopen partition A1,…,An of A and elements g1,…,gn of G such that g1A1,…,gnAn form a partition of B). I will discuss how some properties of the action can be studied via the clopen type semigroup; I will focus in particular on the dynamical comparison property (following Kerr and Ma), and the existence of a dense locally finite group in the topological full group associated to the action. I will also try to outline some consequences for generic properties of minimal actions of a given countable group on the Cantor space, and discuss some open problems.**Information:** Please see the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 28 February, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Marcin Sabok, McGill University**Title:** Perfect matchings in hyperfinite graphings**Abstract: **The talk will focus on recent results on measurable perfect matchings in hyperfintie graphings. In particular, we will discuss a result saying that every regular hyperfinite one-ended bipartite graphing admits a measurable perfect matching. We will also see some applications of these results, answering several questions in the field. For instance we will characterize the existence of factor of iid perfect matchings in bipartite Cayley graphs, extending a result of Lyons and Nazarov. We will also answer a question of Bencs, Hruskova and Toth arising in the study of balanced orientations in graphings. Finally, we see how the results imply the measurable circle squaring. This is joint work with Matt Bowen and Gabor Kun.**Information:** See the seminar webpage.

**Leeds Models and Sets Seminar****Time:** Wednesday, 1 March, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Jiachen Yuan, University of Leeds**Title:** How far is almost strong compactness from strong compactness **Abstract:** Almost strong compactness of $\kappa$ can be characterized as follows: for every $\delta < \kappa < \lambda$, there is an elementary embedding $j_{\delta,\lambda}: V \rightarrow M$ with critical point $\geq \delta$, so that $j_{\delta,\lambda}“ \lambda \subseteq D \in M$ and $M \vDash |D|< j_{\delta,\lambda}(\kappa)$. Boney and Brooke-Taylor were wondering whether almost strong compactness is essentially the same as strong compactness. Recently, Goldberg showed that if $\kappa$ is of uncountable cofinality and SCH holds from below then these two closely related concepts are the same. In this joint work with Zhixing You, we show that these two can be different in general cases. **Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 1 March, 16:00-17:30 CET**Speaker:** Boriša Kuzeljević **Title:** Lower bounds of sets of P-points **Abstract:** We will sketch the proof that MA(k) implies that each

collection of Pc-points of size at most k which has a Pc-point as an

RK upper bound also has a Pc-point as an RK lower bound. This is

joint work with Dilip Raghavan and Jonathan Verner. **Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Vienna Logic ColloquiumTime:** Thursday, 2 March, 15:00 – 15:45 CET

**Speaker:**Katrin Tent, University of Münster

**Title:**Simplicity of automorphism groups of homogeneous structures

**Abstract:**We discuss some general criteria that can be used to show that the automorphism group of a homogeneous structure (such as metric space, right-angled building, graph or hypergraphs) are simple groups or have simple quotients.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Cross-Alps Logic Seminar****Time:** Friday, 3 March 16.00-18.00 CET**Speaker:** Dugald Macpherson, University of Leeds**Title:** Uniform families of definable sets in finite structures **Abstract:** A 1992 theorem of Chatzidakis, van den Dries and Macintyre, stemming ultimately from the Lang-Weil estimates, asserts, roughly, that if ϕ(x,y) is a formula in the language of rings (where x,y are tuples) then the size of the solution set of ϕ(x,a) in any finite field F of size q (where a is a parameter tuple from F) takes one of finitely many dimension-measure pairs as F and a vary: for a finite set E of pairs (μ,d) (μ rational, d integer) dependent on ϕ, any set ϕ(F,a) has size roughly μqd for some (μ,d)∈E.

This led in work of Elwes, Steinhorn and myself to the notion of ‘asymptotic class’ of finite structures (a class satisfying essentially the conclusion of Chatzidakis-van den Dries-Macintyre). As an example, by a theorem of Ryten, any family of finite simple groups of fixed Lie type forms an asymptotic class. There is a corresponding notion for infinite structures of ‘measurable structure’ (e.g. a pseudofinite field, by the Chatzidakis-van den Dries-Macintyre theorem). Any ultraproduct of an asymptotic class is measurable, and in particular has supersimple theory (in the sense of stability theory).

I will discuss a body of work with Sylvy Anscombe, Charles Steinhorn and Daniel Wolf which generalises this, incorporating a richer range of examples with fewer model-theoretic constraints; for example, the corresponding infinite ‘generalised measurable’ structures, for which the definable sets are assigned values in some ordered semiring, need no longer have ‘simple’ theory. I will also discuss a variant in which sizes of definable sets in finite structures are given exactly rather than asymptotically.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Toronto Set Theory Seminar****Time:** Friday, 3 March, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Slawomir Solecki, Cornell University **Title:** Closed groups generated by generic measure preserving transformations**Abstract:** The talk will be about the Polish group of all measure preserving transformations. The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation $T$ has emerged. This picture included substantial evidence that pointed to these groups (for a generic $T$) being all topologically isomorphic to a single group, namely, $L^0$—the Polish group of all Lebesgue measurable functions from $[0,1]$ to the circle. In fact, Glasner and Weiss asked if this is the case. I will describe the background touched on above. I will indicate a proof of the following theorem that answers the Glasner–Weiss question in the negative: for a generic measure preserving transformation $T$, the closed group generated by $T$ is {\bf not} topologically isomorphic to $L^0$. The proof rests on an analysis of unitary representations of the non-locally compact group $L^0$ **Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**20-26 February**

**Baltic Set Theory Seminar****Time:** Tuesday, 21 February, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**Leeds Models and Sets Seminar****Time:** Wednesday, 22 February, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Gabriel Ng, Manchester University**Title:** Differentially Large Fields and Taylor Morphisms **Abstract:** Differential largeness is a generalisation of the notion of largeness for pure fields, introduced by Leon-Sanchez and Tressl. This class of differential fields contains many of the model-theoretically tame classes, such as differentially closed fields, closed ordered differential fields, etc. One of the tools that have been developed to study such fields is known as the `twisted Taylor morphism’, which essentially transforms ring homomorphisms into differential ring homomorphisms into the ring of power series in a uniform way. We generalise this notion, and show that differential largeness can also be characterised in terms of generalised Taylor morphisms. If time allows, we will talk about the structure of these generalised Taylor morphisms.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 22 February, 16:00-17:30 CET**Speaker:** Maxwell Levine **Title:** On Disjoint Stationary Sequences**Abstract:** Disjoint stationary sequences were introduced by Krueger to answer

questions about forcings that add clubs through stationary sets. We will discuss

a version of Mitchell forcing that adds a disjoint stationary sequence (given a

sufficient large cardinal). The benefit of this version is that it comes with an

Abraham-style projection analysis. This allows us to obtain disjoint stationary

sequences on successive cardinals, thus answering one of Krueger’s questions.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Toronto Set Theory Seminar****Time:** Friday, 24 February, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Assaf Rinot **Title:** Sums of triples in Abelian groups **Abstract:** Motivated by a problem in additive Ramsey theory, we extend Todorcevic’s partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for every Abelian group G of size w2, there exists a coloring c of G by the integers, such that for every uncountable subset X of G, and every integer k, there are three distinct elements x,y,z of X such that c(x+y+z)=k. This is joint work with Ido Feldman. **Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 24 February, 2:00 – 3:30 New York time (20:00-21:30 CET)**Speaker: **Johann Franklin,** **Hofstra University**Title: **Generalizing a question of Gromov**Abstract:** When Gromov asked ‘What is a typical group?’, he was thinking of finitely presented groups, and he proposed an approach involving limiting density. Here, we reframe this question in the context of universal algebra and discuss some examples illustrating the behaviors of some of these algebraic varieties and then general conditions that imply some of these behaviors. Our primary general result states that for a commutative generalized bijective variety and presentations with a single generator and single identity, the zero-one law holds and, furthermore, that the sentences with density 1 are those true in the free structure. The proof of this result requires a specialized version of Gaifman’s Locality Theorem that enables us to get a better bound on the complexity of the formulas of interest to us.

This work is joint with Meng-Che ‘Turbo’ Ho and Julia Knight.**Information:** The talk will take place in person. For more information, please see the seminar webpage or email Victoria Gitman.

**13-19 February**

**Baltic Set Theory Seminar****Time:** Tuesday, 14 February, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 14 February, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Gabriel Goldberg, University of California, Berkeley **Title:** Inner models from stationary logic, part 2 **Abstract: **We discuss the inner model C(aa) introduced by Kennedy-Magidor-Väänänen, and answer several questions posed in their paper “Inner models from strong logics”. Assuming large cardinal axioms, we’ll show that this model satisfies GCH (this is joint work with John Steel) and that C(aa) satisfies o(κ) = κ^{++} for all regular cardinals κ > ω_{1} (this is joint work with Otto Rajala). **Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 15 February, 13:00-15:00 Israel Time (12:00-14:00 CET)**Speaker:** Omer Ben-Neria **Title:** Rudin Keisler capturing and mutual stationarity at successors of singulars **Abstract:** The notion of mutually stationary sequences was introduced by Foreman and Magidor in the 90s as a certain notion of stationarity at singular cardinals. Not all sequences of stationary sets are mutually stationary however Foreman and Magidor proved that every sequence (S_n : n < omega) of stationary sets S_n consisting of countable cofinality ordinals is mutually stationary. In the first part of the talk we will introduce the notion of mutually stationary sequences and go over their theorem. In the second part of the talk we will connect the question about mutual stationarity of other sequences to a certain type of ultrafilters that show up in sufficiently strong extenders. The resulting construction will allow us to prove new consistency results regarding mutually stationary sequences. In particular, we will show it is consistent that every sequence of a fixed cofinality stationary subsets S_n of the first successors of singular cardinals Aleph_{omega * n + 1}, n < omega, is mutually stationary. This is a joint work with Dominik Adolf. **Abstract:** tba **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 15 February, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Matteo Viale, University of Turin **Title:** Absolute Model Companionship, the AMC-spectrum of set theory, and the continuum problem **Abstract:** We introduce a classification tool for mathematical theories based on Robinson’s notion of model companionship; roughly the idea is to attach to a mathematical theory T those signatures L such that T as axiomatized in L admits a model companion. We also introduce a slight strengthening of model companionship (absolute model companionship – AMC) which characterize those model companionable L-theories T whose model companion is axiomatized by the \Pi_2-sentences for L which are consistent with the universal and existential theory of any L-model of T.

We use the above to analyze set theory, and we show that the above classification tools can be used to extract (surprising?) information on the continuum problem.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 15 February, 16:00-17:30 CET**Speaker:** Andreas Lietz, University of Münster **Title:** Forcing “NS is omega_1-dense” from large cardinals**Abstract:** Two different sets of tools have been developed to tackle independence problems for H_omega2, namely Shelah’s theory of iterated semiproper forcing and Woodin’s Pmax technique. A number of results achieved with the latter method are known to be possible using the former method instead. One interesting exception to this rule has been finding models of ZFC in which the nonstationary ideal on omega_1 is omega_1-dense, i.e. there are omega_1-many stationary sets so that one of them is contained (on a club) in any given stationary set. Woodin has shown that this holds true in the extension of L(R) by a Pmax-variation (assuming AD in L(R)). We will prove that if there is an inaccessible limit of supercompact cardinals then there is a stationary set preserving forcing extension in which the nonstationary ideal on on omega_1 is omega_1-dense. This answers a question of Woodin positively.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Toronto Set Theory Seminar****Time:** Friday, 17 February, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Jing Zhang**Title:** Partition relations and ideal hypothesis**Abstract:** We will discuss and address some problems concerning a family of weakenings of the usual Ramsey theorem. Instead of asking for a monochromatic complete graph, very roughly speaking, these versions ask for variations of a monochromatic topological copy of the complete graph, i.e. each edge is replaced by a path. This allows the possibility of consistency at small uncountable cardinals or the continuum where the usual Ramsey theorem is inconsistent. We will also say something about motivation and some recent applications. Joint work with Hrusak and Shelah.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 17 February, 2:00 – 3:30 New York time (20:00-21:30 CET)**Speaker: **Russell Miller, CUNY**Title: **Computability and the Absolute Galois Group of Q**Abstract:** Fix a computable presentation ¯Q of the algebraic closure of the rational numbers. The absolute Galois group of the rational numbers, which is precisely the automorphism group of the field ¯Q, may then be viewed as a collection of paths through a finite-branching tree. Each individual automorphism has a Turing degree. We will use known results in computability to try to build natural countable elementary subgroups of the absolute Galois group. Several intriguing questions in number theory will appear as we measure the extent to which we succeed in doing so.**Information:** The talk will take place in person. For more information, please see the seminar webpage or email Victoria Gitman.

**6-12 February**

**Baltic Set Theory Seminar****Time:** Tuesday, 7 February, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 7 February, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Gabriel Goldberg, University of California, Berkeley **Title:** Inner models from stationary logic, part 1 **Abstract: **We discuss the inner model C(aa) introduced by Kennedy-Magidor-Väänänen, and answer several questions posed in their paper “Inner models from strong logics”. Assuming large cardinal axioms, we’ll show that this model satisfies GCH (this is joint work with John Steel) and that C(aa) satisfies o(κ) = κ^{++} for all regular cardinals κ > ω_{1} (this is joint work with Otto Rajala). **Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Caltech Logic Seminar****Time:** Tuesday, 7 February, 11:00am-12:00pm Pacific time (20:00-21:00 CET)**Speaker:** Patrick Lutz, UCLA**Title:** The Solecki dichotomy and the Posner Robinson theorem**Abstract:** The Solecki dichotomy in descriptive set theory, roughly stated, says that every Borel function on the real numbers is either a countable union of partial continuous functions or at least as complicated as the Turing jump. The Posner-Robinson theorem in computability theory, again roughly stated, says that every non-computable real looks like 0′ relative to some oracle. Superficially, these theorems look similar: both roughly say that some object is either simple or as complicated as the jump. However, it is not immediately apparent whether this similarity is more than superficial. If nothing else, the Solecki dichotomy is about third order objects—functions on the real numbers—while the Posner-Robinson theorem is about second order objects—individual real numbers. We will show that there is a genuine mathematical connection between the two theorems by showing that the Posner-Robinson theorem plus determinacy can be used to give a short proof of a slightly weakened version of the Solecki dichotomy. We will explain the idea of this proof and then discuss its connections to some questions about well-foundedness of various reducibility notions on functions.**Information:** Please see the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 17 February, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Tom Benhamou, University of Illinois at Chicago **Title:** Saturation properties of ultrafilters**Abstract: **In this talk, we will focus on certain saturation properties of filters and ultrafilters which generalizes the so-called Galvin property. In the first part of the talk, we will present a connection between such ultrafilters and the existence of Slim-Kurepa trees. We will then present several results regarding the existence of non-Galvin ultrafilters under several large cardinal assumptions. Finally, if time permits, we will present a recent application to canonical inner models and some open related questions.**Information:** See the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 8 February, 13:00-15:00 Israel Time (12:00-14:00 CET)**Speaker:** Omer Ben-Neria **Title:** Rudin Keisler capturing and mutual stationarity at successors of singulars **Abstract:** The notion of mutually stationary sequences was introduced by Foreman and Magidor in the 90s as a certain notion of stationarity at singular cardinals. Not all sequences of stationary sets are mutually stationary however Foreman and Magidor proved that every sequence (S_n : n < omega) of stationary sets S_n consisting of countable cofinality ordinals is mutually stationary. In the first part of the talk we will introduce the notion of mutually stationary sequences and go over their theorem. In the second part of the talk we will connect the question about mutual stationarity of other sequences to a certain type of ultrafilters that show up in sufficiently strong extenders. The resulting construction will allow us to prove new consistency results regarding mutually stationary sequences. In particular, we will show it is consistent that every sequence of a fixed cofinality stationary subsets S_n of the first successors of singular cardinals Aleph_{omega * n + 1}, n < omega, is mutually stationary. This is a joint work with Dominik Adolf. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 8 February, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Elliot Glazer, Harvard University **Title:** Foundationless geology and a Foundation conservativity result **Abstract:** It is well-understood that the Axiom of Foundation has no “mathematical consequences” over ZFC – Foundation, since every mathematical structure is isomorphic to one whose universe is an ordinal by the well-ordering theorem. Over ZF – Foundation, there are mathematical consequences to adding Foundation, e.g. the sentence “if all orderable sets are well-orderable, then every set is well-orderable.” In joint work with Asaf Karagila, we identify a precise sense in which there is no simpler consequence of adding Foundation. In particular, for any $\varphi$ a sentence in second-order logic, adding Foundation does not refute the existence of a set model of $\varphi.$ This talk will focus on applying techniques of set-theoretic geology in a context without Choice or Foundation, which is a key ingredient in the proof of this theorem. **Information:** Please see the seminar webpage.

**CUNY Set Theory Seminar****Time:** Friday, 10 February, 12:15pm New York time (18:15 CET)**Speaker: **Davide Leonessi, CUNY**Title:** Strategy and determinacy in infinite Hex**Abstract:** The popular game of Hex can be extended to the infinite hexagonal lattice, defining a winning condition which formalises the idea of a chain of colored stones stretching towards infinity. The descriptive-set-theoretic complexity of the set of winning positions is unknown, although it is at most Σ^1_1, and it is conjectured to be Borel; this has implications on whether games of infinite Hex are determined from all initial positions as either first-player wins or draws.

I will show that, unlike the finite game, infinite Hex with an initially empty board is a draw. But is the game still a draw when starting from a non-empty board? This open question can be partially answered in the positive by assuming the existence of certain local strategies, and in the negative by giving the advantage of placing two stones at each turn to one of the players. This is joint work with Joel David Hamkins.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 10 February, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Justin Moore, Cornell University **Title:** On minimal non-sigma-scattered linear orders**Abstract:** We present new constructions of linear orders which are minimal with respect to being non-sigma-scattered. Specifically, we will show that Jensen’s diamond principle implies that there is a minimal Countryman line, answering a question of Baumgartner. We will also construct the first consistent examples of minimal non-$\sigma$-scattered linear orders of cardinality greater than aleph1. In fact this can be achieved at any successor cardinal kappa+, both via forcing constructions and via axiomatic principles which hold in Gödel’s Constructible Universe. These linear orders of cardinality kappa+ have the property that their square is the union of kappa-many chains. This is joint work with James Cummings and Todd Eisworth. **Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 10 February, 2:00 – 3:30 New York time (20:00-21:30 CET)**Speaker: **Athar Abdul-Quader, Purchase College**Title: **Satisfaction and saturation**Abstract:** It is well known that a countable model of PA has a truth predicate if and only if it is recursively saturated. It is also well known that not all countable recursively saturated models of PA have *inductive* or even Δ0-inductive truth predicates: indeed, such models must satisfy Con(PA), for example. Recent work by Enayat-Pakhomov and Cieśliński-Łełyk-Wcisło explored the principle of ‘disjunctive correctness’, asserting that every disjunction is true if and only if it has a true disjunct. In particular, one can show that every countable model of PA has a ‘disjunctively trivial’ elementary extension: that is, an elementary extension with a truth predicate in which all nonstandard length disjunctions are evaluated as true. In this talk, we will see that such ‘disjunctively trivial’ models are necessarily arithmetically saturated; indeed, we will see that a countable model of PA is arithmetically saturated if and only if it has a disjunctively trivial truth predicate. We will explore related pathologies in truth predicates, and classify the sets which can be defined using such pathologies. We find other surprising connections between arithmetic saturation and these questions of definability. This is joint work with Mateusz Łełyk, based heavily on unpublished work by Jim Schmerl.**Information:** The talk will take place in person. For more information, please see the seminar webpage or email Victoria Gitman.

**30 January – 5 February**

**Baltic Set Theory Seminar****Time:** Tuesday, 31 January, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 31 January, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Takehiko Gappo, TU Wien **Title:** Determinacy in the Chang model from a hod pair **Abstract: **We will show that the Chang model satisfies determinacy from the existence of an excellent least branch hod pair with a Woodin limit of Woodin cardinals. The proof is based on Sargsyan’s result on Chang models over derived models. This is joint work with Sargsyan. **Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Caltech Logic Seminar****Time:** Tuesday, 31 January, 11:00am-12:00pm Pacific time (20:00-21:00 CET)**Speaker:** Shaun Allison, University of Toronto**Title:** Classification strength of Polish groups and involving S∞**Abstract:** We say that a Polish group GG has stronger classification strength than a Polish group HH iff every orbit equivalence relation induced by a continuous action of HH on a Polish space is Borel-reducible to an orbit equivalence relation induced by a continuous action of GG on a Polish space (or in other words, “classified” by GG. The notion of classification strength gives a way to measure the inherent complexity of a Polish group, and gives rise to interesting hierarchies.

We say that GG involves HH iff there is a closed subgroup G′G′ of GG and a continuous surjective homomorphism from G′G′ onto HH. If GG involves HH then it has greater classification strength, a result of Mackey and Hjorth. Also, a result of Hjorth implies that any Polish group which has greater classification strength than S∞S∞, the Polish group of permutations of a countably-infinite set, involves S∞S∞. In other words, the non-Archimedean Polish groups (i.e. the closed subgroups of S∞S∞) with maximal classification strength are exactly those which involve S∞S∞.

We will describe several new necessary and sufficient conditions for a non-Archimedean Polish group to involve S∞S∞, some of which may have independent interest in model theory. One result of particular significance is that if GG classifies =+=+, a natural equivalence relation very low in the Borel hierarchy, then GG must involve S∞S∞. Moreover, a natural rank function of model-theoretic flavor arises, measuring how close a non-Archimedean Polish group is from involving S∞S∞, which yields an interesting hierarchy of classification strength. I will also mention previous and ongoing work with Aristotelis Panagiotopoulos relating to another hierarchy of classification strength among the cli (complete left-invariant) Polish groups.**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 1 February, 13:00-15:00 Israel Time (12:00-14:00 CET)**Speaker:** Yair Hayut **Title:** The Gluing Property **Abstract:** One of the characterizations of strongly compact cardinals is the ability to extend every $\kappa$-complete filter to a $\kappa$-complete ultrafilter. By restricting this property to different families of filters we obtain a variety of compactness principles, of varying strengths. For example, by restricting our attention to $\kappa$-complete filters on $\kappa$, we obtain the notion of $\kappa$-compactness, which is still rather strong. Restricting the filters to be normal does not reduce the strength.

In this sequence of talks, I will discuss the problem of restricting the filter extension property to filters which are obtained by trying to “glue together” $\kappa$-complete ultrafilter. As those filters are not very far from being ultrafilters already, it is expected that this property will be weaker, and it is consistently much weaker than $\kappa$-compactness. We will start by drawing some connections between strong compactness and this property. Then, we will start analysing how to get the consistency of this property from large cardinals in the level of strong cardinals. The final goal is to obtain it from a measurable cardinal of high Mitchell order. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 1 February, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Pablo Andujar Guerrero, University of Leeds**Title:** O-minimal tame set-theoretic topology**Abstract:** We give a positive answer in the o-minimal setting to a conjecture in set-theoretic topology and explore similar open problems in topology from the point of view of o-minimality.**Information:** Please see the seminar webpage.

**CUNY Set Theory Seminar****Time:** Friday, 3 February, 12:30pm New York time (18:30 CET)**Speaker: **Jing Zhang, University of Toronto**Title:** Highly connected Ramsey theory **Abstract:** A typical Ramsey statement is the following: given a coloring of a complete graph, we aim to find a ‘large’ complete subgraph that is monochromatic. The weaker variation we are considering here (introduced by Bergfalk-Hrusak-Shelah) is to relax the ‘complete subgraph’ in the goal. More precisely, we aim to find a certain ‘large’ connected monochromatic subgraph. We will discuss the motivation and the connections with other combinatorial and algebraic problems. We demonstrate consistently, such partition relations can hold at small uncountable cardinals like aleph_2, and successors of singular cardinals like aleph_{omega+1}. Joint work with Hrusak and Shelah.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 3 February, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Jan Hubička, Charles University **Title:** Ramsey theorem for trees with successor operation and big Ramsey degrees of relational structures **Abstract:** We discuss a new Ramsey type theorem for trees with successor operation. On boundedly branching trees it can be seen as a common generalization of the Carlson-Simpson theorem and Milliken tree theorem.

For trees with unbounded branching it however leads to a different notion of subtrees which are useful to give upper bounds on big Ramsey degrees of relational structures. We outline recent progress on giving such bound to relational structures with relations of arity 3 and more.

This is a joint work with Natasha Dobrinen, David Chodnouský, Matěj Konečný, Jaroslav Nešetřil, Stevo Todorčevič and Andy Zucker.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 3 February, 2:00 – 3:30 New York time (20:00-21:30 CET)**Speaker: **Roman Kossak, CUNY**Title: **Absolute Undefinability**Abstract:** I call a subset of the domain of a countable model absolutely undefinable if the set of its images under automorphisms of the model is uncountable. By the Kueker-Reyes theorem, all sets that are not absolutely undefinable are parametrically definable in Lω1ω. I will survey classical results about first-order undefinability in the standard model of arithmetic, and I will contrast them with some old and some new results about absolute undefinability in nonstandard models of PA.**Information:** The talk will take place in person. For more information, please see the seminar webpage or email Victoria Gitman.

**23-29 January**

**Baltic Set Theory Seminar****Time:** Tuesday, 24 January, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 24 January, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Takehiko Gappo, TU Wien**Title:** Chang models over derived models**Abstract: ** We will give a proof outline of the main theorem in Sargsyan’s paper, “Covering with Chang models over derived models.” In this paper, he constructed a new model of determinacy extending the derived model, called the Chang model over the derived model, inside of a symmetric extension of a least branch hod mouse with Woodin cardinals. This result will be used in the next talk.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Caltech Logic Seminar****Time:** Tuesday, 24 January, 11:00am-12:00pm Pacific time (20:00-21:00 CET)**Speaker:** Takehiko Gappo, TU Wien**Title:** Chang models over derived models**Abstract:** We will give a proof outline of the main theorem in Sargsyan’s

paper, “Covering with Chang models over derived models.” In this paper, he

constructed a new model of determinacy extending the derived model, called

the Chang model over the derived model, inside of a symmetric extension of

a least branch hod mouse with Woodin cardinals. This result will be used

in the next talk.**Information:** Please see the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 24 January, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Paolo Marimon, Imperial College London**Title:** Invariant Keisler measures in simple omega-categorical structures**Abstract: **We study invariant Keisler measures in the context of omega-categorical structures. For a first-order structure M, these are finitely additive probability measures on the Boolean algebra of definable subsets Def_x(M) which are invariant under the action of the automorphism group Aut(M). A natural notion of smallness for a definable subset of M is that it is assigned measure 0 by any invariant Keisler measure. Another natural model-theoretic notion of smallness is forking over the empty set. These two notions coincide in various classes of structures, including stable and NIP omega-categorical ones. In a recent article, Chernikov, Hrushovski, Kruckman, Krupinski, Moconja, Pillay and Ramsey find the first examples of simple structures with formulas which do not fork over the empty set but are universally measure zero. I give the first known simple omega-categorical examples exhibiting this property. These are various omega-categorical Hrushovski constructions. In order to prove this, I use a probabilistic independence result by Jahel and Tsankov to show that a stronger version of the independence theorem holds for simple omega-categorical structures where a formula forks over the empty set if and only if it is universally measure zero.**Information:** See the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 25 January, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Jonathan Kirby**Title:** Around Zilber’s quasiminimality conjecture**Abstract:** About 25 years ago, Zilber conjectured that the complex field with the exponential function is quasiminimal: every definable subset is countable or co-countable. This conjecture has sparked a lot of activity over that time. For example, Zilber’s part of the Zilber-Pink conjecture and the related work on functional transcendence came out of his early work towards the quasiminimality conjecture. Recently there has been significant progress towards proving the conjecture itself.

I will survey some of the work around the conjecture, including the nature of quasiminimality and its relationship to infinitary and classical first-order logic, and the recent result of Gallinaro and myself that the complex field equipped with complex power functions is quasiminimal.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 25 January, 13:00-15:00 Israel Time (12:00-14:00 CET)**Speaker:** Yair Hayut **Title:** The Gluing Property, continued**Abstract:** One of the characterizations of strongly compact cardinals is the ability to extend every $\kappa$-complete filter to a $\kappa$-complete ultrafilter. By restricting this property to different families of filters we obtain a variety of compactness principles, of varying strengths. For example, by restricting our attention to $\kappa$-complete filters on $\kappa$, we obtain the notion of $\kappa$-compactness, which is still rather strong. Restricting the filters to be normal does not reduce the strength.

In this sequence of talks, I will discuss the problem of restricting the filter extension property to filters which are obtained by trying to “glue together” $\kappa$-complete ultrafilter. As those filters are not very far from being ultrafilters already, it is expected that this property will be weaker, and it is consistently much weaker than $\kappa$-compactness. We will start by drawing some connections between strong compactness and this property. Then, we will start analysing how to get the consistency of this property from large cardinals in the level of strong cardinals. The final goal is to obtain it from a measurable cardinal of high Mitchell order. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**KGRC Logic Colloquium, ViennaTime:** Friday, 27 January, 15:00 – 15:45 CET

**Speaker:**M. Hils, University of Münster

**Title:**Lang-Weil type bounds in finite difference fields

**Abstract:**We establish Lang-Weil type bounds for the number of rational points of difference varieties over finite difference fields, in terms of the transformal dimension of the variety and assuming the existence of a smooth rational point. It follows that, working in any non-principle ultraproduct K of finite difference fields, the normalized pseudofinite dimension of a quantifier free partial type p is equal to the transformal dimension of p, i.e., to the maximal transformal transcendence degree over K of a realization of p.

This is joint work with Ehud Hrushovski, Jinhe Ye and Tingxiang Zou.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Cross-Alps Logic Seminar****Time:** Friday, 27 January, 16.00-17.00 CEST**Speaker:** K. Kowalik, University of Warsaw**Title:** Reverse mathematics of some Ramsey-theoretic principles over a weak base theory**Abstract:** The logical strength of Ramsey-theoretic principles has been one of the main areas of research in reverse mathematics. We study some of these combinatorial statements over a weak base theory RCA*_0, which is obtained from the usual RCA_0 by replacing Sigma^0_1 induction with Delta^0_1 induction. The weaker base theory allows for a finer analysis of the principles considered, but at the same time makes the notion of an infinite set unstable. Namely, it is consistent with RCA*_0 that there is an unbounded subset of natural numbers which is not in bijective correspondence with N. Thus, there are different ways of formalizing in RCA*_0 Ramsey-theoretic statements since they often assert the existence of some infinite sets (homogeneous sets for colourings, chains or antichains in partial orders etc.). For this reason, the reverse-mathematical zoo gets bigger over RCA*_0. However, there are certain general patterns of behaviour among our principles: some of them are Pi^0_3 conservative over RCA*_0 whereas some others imply I Sigma^0_1. In this talk I will present our main results on the topic and explain what it is like to work without assuming I Sigma^0_1. This is joint work with Marta Fiori Carones, Leszek Kolodziejczyk and Keita Yokoyama.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Toronto Set Theory Seminar****Time:** Friday, 27 January, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Forte Shinko, The Fields Institute and University of California**Title:** Realizations of equivalence relations and subshifts**Abstract:** Every continuous action of a countable group on a Polish space induces a Borel equivalence relation. We are interested in the problem of realizing (i.e. finding a Borel isomorphic copy of) these equivalence relations as continuous actions on compact spaces. We provide a number of positive results on this problem, and we investigate the connection to subshifts. Joint with Joshua Frisch, Alexander Kechris and Zoltán Vidnyánszky.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**16-22 January**

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 17 January, 15:00-16:30 CEST**Speaker:** Peter Holy, TU Wien**Title:** A weak form of Global Choice under the GCH, part 2**Abstract:** In 2012, Joel Hamkins asked (on MathOverflow) whether it is possible for the universe of sets to have a linear ordering, but no wellordering (that is, global choice fails). This question, which I consider very interesting, appears to still be open. In my talk, I want to present a somewhat related result. After providing a gentle introduction to second order set theory and the principle of global choice (no knowledge on these matters is assumed), we consider a different weakening of global choice under the GCH: The minimal ordinal-connection axiom MOC due to Rodrigo Freire. It is equivalent to the statement that the universe of sets can be stratified by a subset-increasing hierarchy ⟨Kα|α∈Ord⟩ with each Kα of the same size as α, and such that Kκ=H(κ), the collection of sets of hereditary size less than κ, for every regular infinite cardinal κ.

In this form, it clearly implies the GCH, and is easily seen to be a weak form of global choice under the GCH. We will show, using class forcing products of adding Cohen subsets of regular cardinals (without assuming any particular knowledge regarding the technique of class forcing), that MOC can consistently fail in models of the GCH, and that MOC can consistently hold while global choice fails.

This is joint work with Rodrigo Freire (University of Brasilia).**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 17 January, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Tuesday, 17 January, 11:00am-12:00pm Pacific time (20:00-21:00 CET)**Speaker:**Asger Tornquist, University of Copenhagen**Title:** Almost disjoint families in dimension 2 and higher**Abstract:** A classical almost disjoint family is a family of subsets of the natural numbers such that any two non-identical elements of the family intersect finitely, that is, their intersection is in the ideal FIN. A “mad family” is, of course, a *maximal* almost disjoint family. Definability problems related to classical mad families have been studied intensively in the past few years. This talk is about extending and generalizing the classical notion of an almost disjoint family by replacing the ideal of finite sets FIN with other ideals, and in this talk, this specifically means replacing it with the iterated Frechet ideals FIN2,FIN3,⋯. We call mad families with respect to the iterated Frechet ideals “higher dimensional” mad families. In this talk, I will try to give a fairly detailed account of the argument in “dimension 2”, i.e., when we consider FIN2. This is joint work with David Schrittesser.**Information:** Please see the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 18 January, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Vadim Weinstein **Title:** Blurry filters and classification by countable structures (part II: proofs) (Joint work with Martina Lannella)**Abstract:** This is a continuation of the talk given on Nov 9^{th} 2022. In the first talk we gave an overview of the field, main ideas, and results. In this, second, talk we dive into the details of the central proofs. The same abstract as for the first talk follows:

The Stone duality gives a neat way to go back-and-forth between totally disconnected Polish spaces and countable Boolean algebras. The main ingredient is the Stone space of all ultrafilters on a Boolean algebra. In this talk we introduce a weaker concept which we call the “blurry filter”. Using blurry filters instead of ultrafilters enables one to extend the class of spaces under consideration from totally disconnected ones to a larger class. As an application of this method, we show that the following are completely classifiable by countable structures: the homeomorphism on 3-manifolds (also applicable to 2-manifolds; but this was known since 1971), and wild embeddings of Cantor sets in R³. By “classification” in this talk we mean classical Borel-reducibility.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 18 January, 13:00-15:00 Israel Time (12:00-14:00 CET)**Speaker:** Yair Hayut **Title:** The Gluing Property**Abstract:** One of the characterizations of strongly compact cardinals is the ability to extend every $\kappa$-complete filter to a $\kappa$-complete ultrafilter. By restricting this property to different families of filters we obtain a variety of compactness principles, of varying strengths. For example, by restricting our attention to $\kappa$-complete filters on $\kappa$, we obtain the notion of $\kappa$-compactness, which is still rather strong. Restricting the filters to be normal does not reduce the strength.

In this sequence of talks, I will discuss the problem of restricting the filter extension property to filters which are obtained by trying to “glue together” $\kappa$-complete ultrafilter. As those filters are not very far from being ultrafilters already, it is expected that this property will be weaker, and it is consistently much weaker than $\kappa$-compactness. We will start by drawing some connections between strong compactness and this property. Then, we will start analysing how to get the consistency of this property from large cardinals in the level of strong cardinals. The final goal is to obtain it from a measurable cardinal of high Mitchell order. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**Toronto Set Theory Seminar****Time:** Friday, 20 January, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba **Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**9-15 January**

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 10 January, 15:00-16:30 CEST**Speaker:** Peter Holy, TU Wien**Title:** A weak form of Global Choice under the GCH, part 1**Abstract:** In 2012, Joel Hamkins asked (on MathOverflow) whether it is possible for the universe of sets to have a linear ordering, but no wellordering (that is, global choice fails). This question, which I consider very interesting, appears to still be open. In my talk, I want to present a somewhat related result. After providing a gentle introduction to second order set theory and the principle of global choice (no knowledge on these matters is assumed), we consider a different weakening of global choice under the GCH: The minimal ordinal-connection axiom MOC due to Rodrigo Freire. It is equivalent to the statement that the universe of sets can be stratified by a subset-increasing hierarchy ⟨Kα|α∈Ord⟩ with each Kα of the same size as α, and such that Kκ=H(κ), the collection of sets of hereditary size less than κ, for every regular infinite cardinal κ.

In this form, it clearly implies the GCH, and is easily seen to be a weak form of global choice under the GCH. We will show, using class forcing products of adding Cohen subsets of regular cardinals (without assuming any particular knowledge regarding the technique of class forcing), that MOC can consistently fail in models of the GCH, and that MOC can consistently hold while global choice fails.

This is joint work with Rodrigo Freire (University of Brasilia).**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 10 January, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 11 January, 13:00-15:00 Israel Time (12:00-14:00 CET)**Speaker:** Yair Hayut**Title:** The Gluing Property**Abstract:** One of the characterizations of strongly compact cardinals is the ability to extend every $\kappa$-complete filter to a $\kappa$-complete ultrafilter. By restricting this property to different families of filters we obtain a variety of compactness principles, of varying strengths. For example, by restricting our attention to $\kappa$-complete filters on $\kappa$, we obtain the notion of $\kappa$-compactness, which is still rather strong. Restricting the filters to be normal does not reduce the strength.

In this sequence of talks, I will discuss the problem of restricting the filter extension property to filters which are obtained by trying to “glue together” $\kappa$-complete ultrafilter. As those filters are not very far from being ultrafilters already, it is expected that this property will be weaker, and it is consistently much weaker than $\kappa$-compactness. We will start by drawing some connections between strong compactness and this property. Then, we will start analysing how to get the consistency of this property from large cardinals in the level of strong cardinals. The final goal is to obtain it from a measurable cardinal of high Mitchell order. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**Vienna Logic Colloquium****Time:** Thursday, 12 January, 15:00 – 16:30 CET

**Speaker:**Benjamin Siskind, Carnegie Mellon University

**Title:**Order-preserving Martin’s Conjecture

**Abstract:**We’ll talk about the current status of Martin’s Conjecture, a conjecture positing that, up to equivalence almost-everywhere, the only natural functions on the Turing Degrees are the well-known ones: constant functions, the identity, and transfinite iterates of the Turing Jump. While the full conjecture is open even for low-level Borel functions, the order-preserving case seems much more tractable. We’ll discuss recent progress on this order-preserving version of Martin’s Conjecture.

This is joint work with Patrick Lutz.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Cross-Alps Logic Seminar****Time:** Friday, 13 January 16.00-18.00 CET**Speaker:** Vasco Brattka, Universität der Bundeswehr München**Title:** Some fascinating topics in logic around reducibilities — UNESCO World Logic Day Seminar**Abstract:** In mathematical logic and theoretical computer science a reducibility is a relation that allows one to describe the transformation of one problem A into another problem B. Such reducibilities can vary in terms of what kind of problems are eligible and they can also vary with respect to the way in which problem B can be used to solve problem A.

Such reducibilities were first considered in computability theory, but they eventually conquered other areas related to logic, such as computational complexity theory, descriptive set theory, computable analysis, and reverse mathematics, where they turned out to be very powerful tools.

Some particularly fascinating questions in mathematical logic are intrinsically tied to certain reducibilities. For instance, Post’s problem and Martin’s conjecture are related to Turing reducibility, the P-NP problem is based on polynomial-time reducibility.

We will also discuss more recent types of reducibilities, such as the Wadge and the Weihrauch reducibilities and show how they might be helpful in addressing questions from descriptive set theory, such as the decomposability conjecture.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Toronto Set Theory Seminar****Time:** Friday, 13 January, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Frank Tall, University of Toronto**Title:** A Pandemic Smorgasbord**Abstract:** 1. The Grothendieck property in C_p-theory

An important theorem of Grothendieck used in functional analysis essentially says that countably compact subspaces of certain function spaces are compact. J. Iovino has applied this to stability and definability in Model Theory. Arhangel’skii vastly generalized Grothendieck’s theorem but left open some fundamental problems. We show they are undecidable. Clovis Hamel and I applied the work of Iovino and Arhangel’skii to Gowers’ problem on the definability of pathological Banach spaces. He will speak on this later this semester.

2. K-sigma-projective sets – a new topological generalization of descriptive set theory

I became interested in sigma-projective sets of reals when they appeared in my study with C. Eagle, C. Hamel, and S. Muller of the number of non-isomorphic countable models of countable theories in second order logic. I had been studying with I. Ongay-Valverde various possibilities for generalizing descriptive set theory beyond Polish spaces, following the path of Frolik, Jayne & Rogers, etc., but assuming determinacy axioms consistent with ZFC. Our conclusion is that the K-sigma-projective spaces – a natural generalization of the K-analytic spaces – are a useful venue to work in.

3. Counting the number of equivalence classes of sigma-projective equivalence relations and applications to second order logic

This is ongoing work with Eagle, Hamel, Muller, and now J. Zhang. We consider a second order version of Morley’s theorem on the number of countable models. This will be the first of several talks on this subject, Connections to work of Foreman, Magidor, Shelah, and Woodin on determinacy and large cardinals will appear.

This is ongoing work with Eagle, Hamel, Muller, and now J. Zhang. We consider a second order version of Morley’s theorem on the number of countable models. This will be the first of several talks on this subject, Connections to work of Foreman, Magidor, Shelah, and Woodin on determinacy and large cardinals will appear.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**Second Colloquium of the European Set Theory Society****Time:** Saturday, 14 January, 17:00-19:00 CEST**Panelists:** Natasha Dobrinen, University of Notre Dame

Mirna Džamonja, IRIF, CNRS-Université de Paris

Ilijas Farah, York University, Toronto

Ralf Schindler, University of Münster**Title:** European Set Theory Society Panel Discussions**Abstract:** Four experts will describe the general area they represent, explain where the area is heading and discuss how it relates to other areas of set theory and mathematics.**Information:** Online. Zoom link for 14 January: https://univienna.zoom.us/j/69960203361?pwd=Skx1OUlSOEs3Q3d2OHhNdFRzbVNEZz09

**2-8 January**

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 4 January, 13:00-15:00 Israel Time (12:00-14:00 CET)**Speaker:** Yair Hayut**Title:** The Gluing Property, continued**Abstract:** One of the characterizations of strongly compact cardinals is the ability to extend every $\kappa$-complete filter to a $\kappa$-complete ultrafilter. By restricting this property to different families of filters we obtain a variety of compactness principles, of varying strengths. For example, by restricting our attention to $\kappa$-complete filters on $\kappa$, we obtain the notion of $\kappa$-compactness, which is still rather strong. Restricting the filters to be normal does not reduce the strength.

In this sequence of talks, I will discuss the problem of restricting the filter extension property to filters which are obtained by trying to “glue together” $\kappa$-complete ultrafilter. As those filters are not very far from being ultrafilters already, it is expected that this property will be weaker, and it is consistently much weaker than $\kappa$-compactness. We will start by drawing some connections between strong compactness and this property. Then, we will start analysing how to get the consistency of this property from large cardinals in the level of strong cardinals. The final goal is to obtain it from a measurable cardinal of high Mitchell order. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.