2021 EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC LOGIC COLLOQUIUM ’21 Adam Mickiewicz University Poznań, Poland July 19–24, 2021

Abstract

of the invited 31st Annual Gödel Lecture ELISABETH BOUSCAREN, The ubiquity of configurations in model theory. CNRS—Université Paris-Saclay, Gif-sur-Yvette, France. E-mail: elisabeth.bouscaren@universite-paris-saclay.fr. Originally in Classification Theory, then in Geometric Stability, and now, beyond Stability, in Tame Model Theory, one common essential feature is the identification and study of some geometric configurations, of combinatorial and dimensional theoretic nature. They can witness the combinatorial and the model theoretic complexity of a theory or indicate the existence of specific definable algebraic structures. This enables model theory to tackle questions from very diverse subjects. We will attempt to illustrate the importance of these configurations through some examples. Abstract of invited tutorialsof invited tutorials KRZYSZTOF KRUPIŃSKI, Topological dynamics in model theory. University of Wrocław, Wrocław, Poland. E-mail: kkrup@math.uni.wroc.pl. Some fundamental notions and methods of topological dynamics were introduced to model theory by Newelski in the mid-2000s. In the first part of my tutorial, I will recall some basic notions of topological dynamics, discuss the flows which appear naturally in model theory (as various spaces of types), and give applications of basic topological dynamics to some group covering results of Newelski such as: if an א0-saturated group is covered by countably many 0-type-definable sets Xn , n ∈ , then for some finite A ⊆ G and n ∈ , G = AXnX –1 n . In the second part, I will define the Ellis semigroup and Ellis group of a flow, and focus on connections between the Ellis groups of natural flows in model theory and certain invariants of definable groups (quotients by model-theoretic connected components) or first order theories (Galois groups of first order theories as well as spaces of strong types). In particular, I will discuss the results of Pillay, Rzepecki, and myself which present certain invariants of this kind as quotients of compact (Hausdorff) groups (which are canonical Hausdorff quotients of Ellis groups). This has various consequences obtained by Pillay, Rzepecki, and myself, e.g., it leads to a general result that model-theoretic type-definability of a bounded invariant equivalence relation defined on a single complete type over ∅ is equivalent to descriptive set theoretic smoothness of this relation. 270 LOGIC COLLOQUIUM ’21 In the last part, I will discuss a definable variant of Kechris–Pestov–Todorčević (KPT) theory, developed by Lee, Moconja, and myself. KPT theory studies relationships between dynamical properties of the groups of automorphisms of Fraïssé structures and Ramseytheoretic (so combinatorial) properties of the underlying Fraïssé classes. In our research, the idea is to find interactions between dynamical properties of first order theories (i.e., properties related to the actions of the automorphism group of a sufficiently saturated model on various types spaces over this model) and definable versions of Ramsey-theoretic properties of the theory. This leads to analogs of various results of KPT theory (i.e., a combinatorial characterization of the definable extreme amenability of a theory), but also to some rather novel theorems, e.g., yielding criteria for profiniteness of the Ellis group of a first order theory. The author is supported by National Science Center, Poland, grants 2015/19/B/ST1/ 01151, 2016/22/E/ST1/00450, and 2018/31/B/ST1/00357. ANDREW MARKS, Characterizing Borel complexity and an application to decomposability. University of California Los Angeles, Los Angeles, CA, USA. E-mail: marks@math.ucla.edu. We give a new characterization of when sets in the Borel hierarchy are Σn hard. This characterization is proved using Antonio Montalban’s true stages method for conducting priority arguments in computability theory. We use this to prove the decomposability conjecture, assuming projective determinacy. The decomposability conjecture describes what Borel functions are decomposable into a countable union of partial continuous functions with Πn domains. This is joint work with Adam Day. Abstracts of invited keynote lecturess of invited keynote lectures ARTEM CHERNIKOV, Measures in model theory. Department of Mathematics, University of California Los Angeles, Los Angeles, CA 900951555, USA. E-mail: chernikov@math.ucla.edu. URL Address: http://www.math.ucla.edu/~chernikov/. In model theory, a type is an ultrafilter on the Boolean algebra of definable sets in a structure, which is the same thing as a finitely additive {0, 1}-valued measure. This is a special kind of a Keisler measure, which is just a finitely additive real-valued probability measure on the Boolean algebra of definable sets. Introduced by Keisler in the late 80s, Keisler measures became a central object of study in the last decade. This is motivated by several intertwined lines of research. One of them (and perhaps the oldest one) is the development of probabilistic and continuous logics. Another is the study of definable groups in o-minimal, and more generally in NIP theories, leading to interesting connections with topological dynamics. Further motivation comes from applications in additive and in extremal combinatorics, uniting the aforementioned directions. I will survey some of the recent developments in the subject. [1] A. Chernikov, Model theory, Keisler measures and groups, this Journal, vol. 24 (2018), no. 3, pp. 336–339. [2] A. Chernikov and K. Gannon, Definable convolution and idempotent Keisler measures. Israel Journal of Mathematics, to appear, 2021, arXiv:2004.10378. [3] A. Chernikov, E. Hrushovski, A. Kruckman, K. Krupinski, S. Moconja, A. Pillay, and N. Ramsey, Invariant measures in simple and in small theories, preprint, 2021, arXiv:2105.07281. [4] A. Chernikov and P. Simon, Definably amenable NIP groups. Journal of the American Mathematical Society, vol. 31 (2018), no. 3, pp. 609–641. [5] A. Chernikov and S. Starchenko, Regularity lemma for distal structures. Journal of the European Mathematical Society, vol. 20 (2018), no. 10, pp. 2437–2466. [6] A. Chernikov and H. Towsner, Hypergraph regularity and higher arity VC-dimension, preprint, 2020, arXiv:2010.00726. LOGIC COLLOQUIUM ’21 271 VERA FISCHER, Combinatorial sets of reals. University of Vienna, Vienna, Austria. E-mail: vera.fischer@univie.ac.at. Infinitary combinatorial sets of reals, such as almost disjoint families, cofinitary groups, independent families, and towers, occupy a central place in the study of the set-theoretic properties of the real line. Of particular interest are such extremal sets of reals, i.e., combinatorial sets which are maximal under inclusion with respect to a desired property, their possible cardinalities, definability properties, as well as the existence or non-existence of ZFC dependences. The study of such combinatorial sets of reals is closely connected with the development of a broad spectrum of forcing techniques. In this talk we will see some recent advances in the subject and point towards interesting remaining open questions. NOAM GREENBERG, The information common to relatively random sequences. Victoria University of Wellington, Wellington, New Zealand. E-mail: noam.greenberg@vuw.ac.nz. If X and Y are relatively random, what common information can X and Y have? We use algorithmic randomness and computability theory to make sense of this question. The answer involves some unexpected ingredients, such as the Lebesgue density theorem, and linear programming, and reveals a rich hierarchy of Turing degrees within the K -trivial degrees. BENOÎT MONIN, The computational content of Miliken’s tree theorem. Créteil University, Créteil, France. E-mail: benoit.monin@computability.fr. The Milliken’s tree theorem is an extension of Ramsey’s theorem to trees. It implies for instance that if we assign to all the sets of two strings of the same length, one among k colors, there is an infinite binary tree within which every pair of strings of the same height has the same color. We are going to present some results on Milliken’s tree theorem from the viewpoint of computability theory and reverse mathematics. LUCA MOTTO ROS, Generalized descriptive set theory for all cofinalities, and some applications. University of Turin, Turin, Italy. E-mail: luca.mottoros@unito.it. Generalized descriptive set theory is nowadays a very active field of research. The idea is to develop a higher analogue of classical descriptive set theory in which is systematically replaced with an uncountable cardinal κ. With a few exceptions, papers in this area tend to concentrate on the case of regular cardinals. This is because under such an assumption one can easily generalize a number of basic facts and techniques from the classical setup, but from the theoretical viewpoint the choice is indeed not fully justified. In this talk I will survey some recent work in which the theory is instead developed in a uniform and cofinality-independent way, thus naturally including the case of singular cardinals. I will also consider some interesting applications connecting generalized descriptive set theory to Shelah’s stability theory (in the case of regular cardinals), and to the study of nonseparable complete metric spaces under Woodin’s axiom IO (in the case of singular cardinals of countable cofinality). FRANK PFENNING, Adjoint logic. Carnegie Mellon University, Pittsburgh, PA, USA. E-mail: fp@cs.cmu.edu. We introduce adjoint logic as a general framework for integrating logics with different structural properties, that is, admitting or denying exchange, weakening, or contraction among the hypotheses. We investigate its proof-theoretic properties from two angles: proof construction and proof reduction. The former is the basis for applications in logical 272 LOGIC COLLOQUIUM ’21 frameworks and logic programming, while the latter provides computational interpretations in functional and concurrent programming.