For a list of talks in the coming weeks, see https://ests.wordpress.com/online-activities-2021.

**Caltech Logic Seminar**

**Time:** Monday, 26 April, 12:00 – 1:00pm Pacific time (21:00 CET)

**Speaker:** Noé de Rancourt, University of Vienna

**Title:** A dichotomy for countable unions of smooth Borel equivalence relations

**Abstract:** I will present a dichotomy for equivalence relations on Polish spaces that can be expressed as countable unions of smooth Borel subequivalence relations. It can be seen as an extension of Kechris-Louveau’s dichotomy for hypersmooth Borel equivalence relations. A generalization of our dichotomy, for equivalence relations that can be expressed as countable unions of Borel equivalence relations belonging to certain fixed classes, will also be presented. This is a joint work with Benjamin Miller.

**Information:** Check on the seminar webpage if the seminar will take place.

**Hebrew University-Bar Ilan University Set Theory seminar**

**Time:** Wednesday, 28 April, 14:00-16:00 Israel Time (13:00-15:00 CET)

**Speaker:** Menachem Magidor

**Title:** When is a nice complicated equivalence relation Borel some where

**Abstract:** The original problem is due to Kanovei, Sabok and Zapletal: Given an analytic equivalence relation , which is not Borel. Can we find a non trivial Borel set , such that the restriction of the relation to it is Borel.

‘Non trivial” here means positive with respect to some sigma-complete ideal on the Borel algebra It turns out that in order to avoid simple counter example we have to make some assumptions about the equivalence relation and about the ideal.There are some results due (independently) to Chan and Drucker, about this problem assuming some large cardinals.

We shall survey some of these results and discuss the issue of trying to generalize these results to larger family of equivalence relations (e.g. Universally Baire)-These are joint results with W. Chan.

**Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Münster research seminar on set theory**

**Time:** Wednesday, 28 April, 15:15-16:45 CET

**Speaker:** Farmer Schlutzenberg

**Title:** Local mantles of L[x]

**Abstract:** Recall that for a cardinal κ, a <κ-ground is an inner model W of ZFC such that V is a set-generic extension of W, as witnessed by a forcing of size <κ, and the κ-mantle is the intersection of all <κ-grounds. We will start with a brief overview

of some known facts on the κ-mantle. Following this, assuming sufficient large cardinals, we will analyze the κ-mantle M of L[x], where x is a real of sufficiently high complexity, and κ is a limit cardinal of uncountable cofinality in L[x]. We will show in particular that M models ZFC + GCH + “There is a Woodin cardinal”. We will also discuss a variant, joint with John Steel, for the κ-cc mantle, where κ is regular uncountable in L[x] and κ≤ the least Mahlo of L[x]. The proof relies on Woodin’s analysis of HODL[x,G] and Schindler’s generation of grounds, and is motivated by work of Fuchs, Sargsyan, Schindler and the author on Varsovian models and the mantle.

**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Barcelona Set Theory Seminar**

**Time:** Wednesday, 28 April, 16:00-17:30 CET

**Speaker:** Yair Hayut

**Title:** omega-strongly measurable cardinals

**Abstract:** In his profound work towards the identification of the ultimate-L (the ultimate canonical inner model), Woodin isolated a key ingredient: the w-

strongly measurable cardinals. Those cardinals are regular in V and measurable in HOD for a very simple reason – the intersection of their club filter

with HOD splits into a small collection of isolated normal measures. Woodin’s

HOD-dichotomy implies that if one can prove that there are class many regular

cardinals which are not strongly measurable, and there exists an extendible

cardinal, then some covering theorem holds between HOD and V.

In this talk I will present a recent joint result with Omer Ben-Neria, proving the

consistency of the existence of class many strongly measurable cardinals

(indeed, all the successors of regular cardinals), from a rather mild large

cardinal hypothesis, in the realm of o(κ) = κ.

I will focus on the details of the proof for the first two cardinals À1 and À2.

**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**KGRC Research Seminar, Vienna**

Time: Thursday, 29 April, 15:00-16:30 CET

**Speaker:** Moreno Pierobon, Università di Pisa, Italy

**Title:** Fullness and mixing property for boolean valued models

**Abstract:** Besides being one of the classical approaches to forcing, boolean valued models provide a flexible tool to produce a variety of structures.

In this talk, we will investigate in details the fullness property and the mixing property for boolean valued models. The former is necessary to control the semantics when quotienting a boolean valued model by an ultrafilter. The latter implies the former and it is easier to check.

We will show that not every model is full, and the mixing property in not equivalent to fullness. Moreover, we will improve the classical Łoś Theorem for boolean valued models.

In the end, we will give a simple characterization of the mixing property using étalé spaces. This last result is an easy corollary of a more general study we made on the categorical equivalence between boolean valued models and presheaves.

This is a joint work with Matteo Viale.

**Information:** Talk via zoom.

**Turin-Udine logic seminar**

**Time:** Friday, 30 April, 16:30-18:30 CET

**Speaker:** S. Barbina, Open University

**Title:** The theory of the universal-homogeneous Steiner triple system

**Abstract:** A Steiner triple system is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite Steiner triple systems has a Fraïssé limit, the countable homogeneous universal Steiner triple system M. In joint work with Enrique Casanovas, we have proved that the theory T of M has quantifier elimination, is not small, has TP2, NSOP1, eliminates hyperimaginaries and weakly eliminates imaginaries. In this talk I will review the construction of M, give an axiomatisation of T and prove some of its properties.

**Information:** Please check on the semianr webpage if the seminar will take place. Online on WebEx. Please see the seminar webpage.

**Toronto Set Theory Seminar**

**Time:** Friday, 20 April, 1.30-3pm Toronto time (19:30-21:00 CET)

**Speaker:** tba

**Title:** tba

**Abstract:** tba

**Information:** No webpage available. Email Ingay Valverde to receive the seminar announcements and for the zoom link.

**CUNY Set Theory Seminar**

**Time:** Friday, 30 April, 2pm New York time (20:00 CET)

**Speaker:** Elliot Glazer, Harvard University

**Title:** Paradoxes of perfectly small sets

**Abstract:** We define a set of real numbers to be perfectly small if it has perfectly many disjoint translates. Such sets have a strong intuitive claim to being probabilistically negligible, yet no non-trivial measure assigns them all a value of 0. We will prove from a moderate amount of choice that any total extension of Lebesgue measure concentrates on a perfectly small set, suggesting that for any such measure, translation-invariance fails ‘as badly as possible.’ From the ideas of this proof, we will also derive analogues of well-known paradoxes of randomness, specifically Freiling’s symmetry paradox and the infinite prisoner hat puzzle, in terms of perfectly small sets. Finally, we discuss how these results constrain what a paradox-free set theory can look like and some related open questions.

**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.