#### Talks during the fall term 2021

Wednesday 8.9.2021 12-14

Mark Kamsma: Independence Relations in Abstract Elementary Categories

**Abstract**: In Shelah's stability hierarchy we classify theories using combinatorial properties. Some important classes are: stable, simple and NSOP1 each being contained in the next. We can characterise these classes by the existence of a certain independence relation. For example, in vector spaces such an independence relation comes from linear independence. Part of this characterisation is canonicity of the independence relation: there can be at most one nice enough independence relation in a theory.

Lieberman, Rosický and Vasey proved canonicity of stable-like independence relations in accessible categories. Accessible categories are a very general framework. The category of models of some theory is an accessible category, every AEC (abstract elementary class) is an accessible category, but even then accessible categories are more general. Inspired by this we introduce the framework of AECats (abstract elementary categories) and prove canonicity for simple-like and NSOP1-like independence relations. This way we reconstruct part of the hierarchy that we have for first-order theories, but now in the very general category-theoretic setting.

Wednesday 15.9.2021 12-14

Jonathan Kirby: Independence in exponential fields

Abstract: In any field, an element can be algebraic or transcendental over a subfield, and we can build on this idea to give an independence relation: two subfields can be independent or not over a third subfield. An exponential field is a field equipped with a homomorphism from its additive group to its multiplicative group, like the usual real and complex exponential maps. There is a corresponding notion of exponential algebraicity / exponential transcendence, which can be used to give an independence relation. However it is very coarse. In this talk I will explain three finer independence relations which are useful in constructing exponential fields. If we restrict to categories of exponential fields with certain properties, such as having a kernel of the form 2iπ Z, we can show these independence relations can be used to prove that the category (or the relevant theory) falls into a particular place in the stability hierarchy: quasiminimal, stable, or NSOP1.

This is joint work with Robert Henderson, Mark Kamsma, and Vahagn Aslanyan

Wednesday 22.9.2021 16-18 **NOTE THE TIME CHANGE**

Gabriel Goldberg

**Title:** Large cardinals, maximality principles, and the multiverse

**Abstract:** We discuss some mathematical results indicating a tension between large cardinal axioms and maximality principles such as forcing axioms and the Axiom of Choice. Topics will include the failure of the Ground Axiom in natural models, the optimality of Usuba's theorem, the analogy between large cardinal axioms and determinacy principles, and the role of forcing in theory selection. Video.

Wednesday 29.9.2021 12-14

Adrien Deloro

Groups of finite Morley rank: from Algebraicity to Zilber

This survey talk combines model theory and group theory. Following Morley's celebrated `categoricity theorem', a certain notion of dimension (now called Morley Rank) was recognised as fundamental; it turns out to be a generalisation of the geometers' Zariski dimension. Now in well-behaved cases, resemblance with algebraic geometry is so striking that one may conjecture that: groups of finite Morley rank belong to geometry, are usual matrix groups. This was proposed in the 1970s by Cherlin and Zilber and is still open

despite gallant assaults through the connection with finite group theory. I shall try to survey the topic. No knowledge of geometry or group theory is needed, but I expect the audience to know what `definable in a first-order theory' means.

Wednesday 6.10.2021 12-14

Ulla Karhumäki

Title: Pseudofinite groups of finite centraliser dimension

Abstract: We first discuss important results by Wilson on (pseudo)finite groups. Then, I will present and prove a structural theorem for pseudofinite groups of finite centraliser dimension. Our proof is essentially made by combining results and techniques from the studies of locally finite groups of finite centraliser dimension (Borovik and Karhumäki/Buturlakin) and from the studies of pseudofinite groups with some model-theoretic tameness assumptions (Macpherson and Tent/Milliet/Macpherson).

Wednesday 13.10.2021 12-14

Fan Yang

Title: Intermediate logics in the team semantics setting

Abstract: Several authors have recently defined intuitionistic logic based on team semantics (tIPC). In this work in progress we provide two alternative approaches to intermediate logics in the team semantics setting. We do this by modifying tIPC with axioms written with the two different versions of disjunction in the logic, a local one \/ and a global one \\/. In the first approach we prove some completeness results by using the \\/-disjunctive normal form of the logic. In the second approach we generalize the standard team semantics by considering the powerset P(W) (i.e., the set of all teams in an Intuitionistic Kripke model) with an arbitrary partial order (instead of the standard superset relation between teams).

This is joint work with Nick Bezhanishvili.

Wednesday 20.10.2021 12-14

Daniel Palacin

*Title.* Stable relations and squares

*Abstract. *The goal of this talk is to present some recent interactions between stability theory and arithmetic combinatorics. In particular, we should explain how to adapt the local approach to stability, due to Hrushovski and Pillay, to obtain certain desired combinatorial patterns. This is joint work with Amador Martin-Pizarro and Julia Wolf.

Wednesday 27.10.2021 12-14

Exam week

Wednesday 3.11.2021 12-14

Andreas Weiermann, **by zoom**

Title: Phase transitions for Gödel incompleteness

Abstract: We first survey several results where for theories T like PRA or PA given true parameterized arithmetical assertions switch from T-provability to T-unprovability when the parameter passes beyond a threshold.

We cover examples from Ramsey theory, the theory of well-quasi orders, and the theory of well orders.

If time allows we cover in a second part recent results related to Friedman's miniaturization of the Bolzano Weierstrass theorem.

Wednesday 10.11.2021 12-14, C124 in Exactum

Otto Rajala: C* and short sequences of measures **IN PERSON**

Abstract: I will present Ur Ya'ar's recent paper titled "Models for short sequences of measures in the cofinality-\omega constructible model"

Wednesday 17.11.2021 12-14, C124 in Exactum

Petteri Kaski: Distributed proof systems that tolerate errors **IN PERSON**

Abstract: Can one distribute the task of preparing a proof for a hard claim to a community of provers prone to making errors, even adversarial errors?

This talk looks at an error-tolerant distributed proof system (with probabilistic soundness) for the #P-hard inference problem in graphical models.

(Based on joint work with Negin Karimi and Mikko Koivisto --- https://doi.org/10.1609/aaai.v34i06.6580 .)

Wednesday 24.11.2021 12-14, C124 in Exactum

Boban Velickovic **IN PERSON**

Title: **Non vanishing higher derived limits **

Abstract: In the study of strong homology Mardesic and Prasolov isolated a certain inverse system of abelian groups A indexed by functions from \omega to \omega.

They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits lim^n A must vanish for n >0.

They also proved that under the Continuum Hypothesis lim^1 A does not vanish. On the other hand Down, Simon and Vaughan showed that under PFA lim^1 A=0

The question whether lim^n A vanishes higher n has attracted considerable attention recently. First, Bergfalk shows that it was consistent lim^2 A does not vanish.

Later Bergfalk and Lambie-Hanson showed that, assuming modest large cardinal axioms, lim^n A vanishes for all n. The large cardinal assumption was later removed by Bergfalk, Hrusak and Lambie-Hanson. We complete the picture by showing that, for any n>0, it is relatively consistent with ZFC that lim^n A is non zero.

This is joint work with Alessandro Vignati.

Wednesday 1.12.2021 12-14, C124 in Exactum

Aleksi Anttila: Modal team logics for modelling Free Choice inference (joint work with Maria Aloni and Fan Yang) **IN PERSON**

Abstract: Maria Aloni's Bilateral State-Based Modal Logic is a modal team logic introduced to account for linguistic phenomena such as Free Choice inference and epistemic contradictions. I present an axiomatization for this logic and for several variants.

Wednesday 8.12.2021 12-14

Ur Yaar **by zoom**

Title: Iterating the cofinality-$\omega$ constructible model

Abstract:

Consider C* - the inner model constructed in an L-like fashion, but using first order logic augmented with the "cofinality $\omega$'' quantifier. C* has some canonicity properties similar to L, but a notable difference is that C* does not necessarily satisfy the axiom V=C*, that is - constructing the C* of C* may result in a smaller inner model. If this happens, one can iterate this construction, taking intersections at limit stages and ask at what stage it stabilizes or "breaks" (i.e. that the result is no longer a model of ZFC). This type of construction was considered with regards to HOD, where it was shown by McAaloon, Harrington, Jech and Zadrozny that "everything is possible" - on one hand there are models with iterated HODs of any ordinal length (and even of length Ord), and on the other hand it is possible that after $\omega$ many stages the intersection either satisfies ZF without AC or even doesn't satisfy ZF at all.

In this talk we will discuss iterating the C* construction, and show that under ZFC alone we can only reach finitely many steps, while a sequence of length $\omega$ is equiconsistent with an inner model with a measurable cardinal.

Wednesday 15.12.2021 12-14, C124 in Exactum

Arnaud Durand: A quick tour on the complexity of query answering : survey and new results **IN PERSON**

Abstract : There are various tasks related to query answering such as deciding if a Boolean query is true or not, counting the size of the answer set or enumerating the results. In this talk I will make a is a survey of some of the many tools from complexity measures trough algorithmic methods to conditional lower bounds that have been designed in the domain over the last years.

At the end of the talk, if time permits, I will present a work in progress based on using circuits (and first order logic) to define enumeration processes.

Wednesday 22.12.2021 12-14

Exam week