# Logic Seminar

The Logic seminar is held on Wednesdays, usually at 12-14. During the spring 2023 we will have both on-site and online talks, and we try to keep the page updated on when we have which kind.

The permanent Zoom room for the seminar is: https://helsinki.zoom.us/j/62891400777?pwd=UldCeThTaTJVQjUzUFo4S2ErcndNQT09 (Meeting ID: 628 9140 0777, Passcode: 164195)

**PLEASE NOTE: Due to the prevalence of zoom talks given by scholars based in various time zones, the times that the seminar meets will occasionally ****change from the usual 12-14 slot. **Also, if not separately specified, the talks always start at a quarter past.

The seminar is led by prof. Juha Kontinen and Ulla Karhumäki.

## Schedule of the spring term 2023

Wed 18.1.2023 12-14, C124

Vadim Weinstein: 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.

Wed 25.1.2023 12-14, C124 (and zoom)

Jonathan Kirby: 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.

Wed 1.2.2023 12-14, Zoom

Christian** **d'Elbée: Fields with a generic multiplicative endomorphism

Abstract: The theory of fields expanded by a unary function symbol for a multiplicative endomorphism admits a model-companion, we denote it ACFH. It is a new example of an NSOP1 theory which is not simple. I will mention some features that appear in ACFH, notably a conjecturally suboptimal description of Shelah's forking and the ubiquity of definable groups whiches theories are pseudofinite-cyclic, that is, elementary equivalent to an ultraproduct of finite cyclic groups.

Wed 8.2.2023 12-14, C124

William Mance: Descriptive complexity in number theory and dynamics

Abstract: Informally, a real number is normal in base $b$ if in its $b$-ary expansion, all digits and blocks of digits occur as often as one would expect them to, uniformly at random. Kechris asked several questions involving descriptive complexity of sets of normal numbers. The first of these was resolved in 1994 when Ki and Linton proved that the set of numbers normal in base $b$ is $\Pi_3^0$-complete. Further questions were resolved by Becher and Slaman. Many of the techniques used in these proofs can be used elsewhere. We will discuss recent results where similar techniques were applied to solve a problem of Sharkovsky and Sivak and a question of Kolyada, Misiurewicz, and Snoha. Furthermore, we will discuss a recent result where the set of numbers that are continued fraction normal, but not normal in any base $b$, was shown to be complete at the expected level of $D_2(\Pi_3^0)$. An immediate corollary is that this set is uncountable, a result (due to Vandehey) only known previously assuming the generalized Riemann hypothesis.

Wed 15.2.2023 12-14, C124

William Mance: Part II of the talk `Descriptive complexity in number theory and dynamics' last week

Abstract: Same abstract as last week.

Wed 22.2.2023 12-14, C124

Max Sandström: The Why and What of Linear Temporal Logic under Team Semantics

Abstract: In this talk I will give a thorough introduction to and motivation for research into the team semantics of linear temporal logic. I will present some recent positive complexity results for the asynchronous team semantics of linear temporal logic, which are based on joint work with Juha Kontinen and Jonni Virtema.

Wed 1.3.2023 12-14, C124

Cancelled

Wed 8.3.2023 12-14, C124

Exam week

Wed 15.3.2023 12-14, C124

Åsa Hirvonen: Crash course on quantum computation

Abstract: The talk is based on a tutorial I gave last autumn. I'll cover the basics on quantum computation, and talk a bit about complexity and error correction. I don't look at physical realizations, but concentrate on the theoretical questions about what is known to work, what is not known, and what is known to be hard, even if physical realizations get good enough for practical purposes.

Wed 22.3.2023 12-14, Zoom

Tingxiang Zou: Number of rational points of difference varieties in finite difference fields

Abstract: A difference field is a field with a distinguished automorphism. Automorphisms of a finite field are powers of the Frobenius map. In this talk, I will discuss how to estimate the number of rational points of a difference variety, namely a system of difference equations, in a finite field with a distinguished power of Frobenius. Like algebraic geometry, one can assign a dimension, called transformal dimension, to a difference variety. I will present a result which is a difference version of the Lang-Weil estimate with respect to the transformal dimension. This is joint work with Martin Hils, Ehud Hrushovski and Jinhe Ye.

Wed 29.3.2023 12-14, C124

Tapani Hyttinen: Model theory of path integrals

Abstract: Almost 10 years ago in Jouko’s 60th birthday meeting B. Zilber suggested use of model theory to study mathematical enigmas from quantum physics. One of the best known of these is Feynman path integrals. In this talk I will look where these path integrals come from, what is the intuition behind them why they are interesting and finish by looking possibilities of model theory saying something about them.

Wed 5.4.2023 12-14, C124

Tuomo Lehtonen: Computing inferences in abstract rule-based argumentation

Abstract:

Formal argumentation is an approach to defeasible reasoning with intuitive dialectical semantics. For example, various forms of non-monotonic reasoning, such as default logic, autoepistemic logic and logic programming, have been given argumentative interpretations. I will give an introduction to formal argumentation, in particular to the influential abstract rule-based argumentation (ASPIC+). The semantics of ASPIC+ are originally defined via constructing a set of arguments whose size is not bounded polynomially. I will present recent work (joint with Matti Järvisalo and Johannes P. Wallner) giving new characterizations of various ASPIC+ semantics avoiding the exponential blow up. These characterizations enable proofs of central complexity results and development of efficient algorithms based on reductions to answer set programming.

Wed 12.4.2023 12-14, C124

Nicolás Nájar: Abstract Elementary Classes from a different point of view.

The Eventual Categoricity Conjecture of Shelah for Abstract Elementary Classes (AECs) promoted the development of a strong (semantic) superstability-like theory for them which generalizes some notions of superstability in first order model theory. All the tools for that line were developed in a semantic way (with the help of Shelah's Presentation Theorem); all of these tools are included in the paper of Vasey and Shelah (Categoricity and Multidimensional Diagrams) from 2019. There they gave a positive answer to the conjecture, with some set theoretical assumptions. In 2022 Shelah an Villaveces gave an optimal axiomatization of an AEC by a sentence in some infinitary logic and in a new work in progress from them, the concept of NIP AECs was introduced.

In this talk we introduce an application of the axiomatization of Shelah and Villaveces, introduce the general framework of NIP AECs and ask some questions about these two new lines.

Wed 19.4.2023 12-14, C124

Reijo Jaakkola: First-Order Logic with Game-Theoretic Recursion

In [1] a natural Turing-complete logic (in the sense of descriptive complexity theory) was introduced, which extends first-order logic with two natural features. The first one is the ability to modify the underlying structures, while the second one is the ability to use recursion (looping) via self-reference. Syntactically speaking, the self-referential statements in the Turing-complete logic are very analogous to the (in)famous goto statements that are permitted in several programming languages. The purpose of this presentation is to present recent work on an extension of first-order logic with this type of self-reference (without the ability to modify the underlying model). This talk is based on joint work with Antti Kuusisto.

[1] Kuusisto, Antti. "Some Turing-Complete Extensions of First-Order Logic.", GandALF 2014

Wed 26.4.2023 12-14, C124

Anna Dmitrieva: Quasiminimality of a correspondence between two elliptic curves

One of the well-known accomplishments of model theory is the study of the field of complex numbers. Its complete theory possesses numerous nice properties, including quantifier elimination. Moreover, using quantifier elimination, one can see that any definable subset is finite or cofinite, i.e. the theory is strongly minimal. However, adding the exponential map to the structure makes it possible to define the ring of integers, preventing minimality and many other properties. Nevertheless, there is still hope that the theory is somewhat well-behaved. For instance, Zilber’s quasiminimality conjecture states that the complex exponential field is quasiminimal, i.e. every definable subset is countable or co-countable. Analogous conjectures were made, replacing the exponential map with other interesting analytical functions. In this talk we will focus on a reduct of one of these conjectures, which involves a correspondence between two elliptic curves, and discuss the quasiminimality of the structure in question.

Wed 3.5.2023 12-14, C124

Tapio Saarinen: Second-order categoricity in L^mu

Abstract: We discuss the question of categoricity of complete second-order theories, focusing on L^mu, the inner model for a measurable cardinal. We briefly discuss the background of the question, define L^mu and recollect its basic properties, and then present the known categoricity and non-categoricity results that hold in it.

Wed 10.5.2023 12-14, C124

Tapio Saarinen: P_max forcing and the axiom (*)

Abstract: Hugh Woodin introduced P_max forcing and the axiom (*) in the early 1990s, arising from investigations into the Axiom of Determinacy (AD) and the nonstationary ideal I_NS on omega_1. P_max is a curious forcing poset, intended to be applied over models of AD, yet forcing the Axiom of Choice to hold in the resulting generic extension. Meanwhile the axiom (*) states that L(P(omega_1)) is a P_max extension of L(R), and one consequence of (*) is that any Pi_2 sentence satisfied by (H(omega_2), \in, I_NS) in a generic extension of the universe is already implied by (*)

Wed 17.5.2023 12-14, C124

Jouko Väänänen: 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.