Logic Archives for Fall 2025 to Spring 2026
Invariant random expansions and invariant Keisler measures
When: Tue, September 3, 2024 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Samuel Braunfeld (Charles University) -
Abstract: We study how a homogeneous structure M can be randomly expanded to a larger language in an Aut(M)-invariant manner. We show that under certain conditions, such an expansion is not just Aut(M)-invariant but fully Aut((N, =))-invariant, which allows us to classify such expansions. The problem of classifying Aut(M)-invariant Keisler measures may be seen as a special case of this problem, and the resulting classifications of Aut(M)-invariant Keisler measures yield many natural examples of very tame simple theories with non-forking formulas that are universally measure zero. This is joint work with Colin Jahel and Paolo Marimon, based on https://arxiv.org/abs/2408.08370
All These Approximate Ramsey Properties (and other stories…)
When: Tue, September 10, 2024 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Aristomenis Papadopoulos (University of Maryland, College Park) -
Abstract: Ramsey’s theorem is an often overlooked result in mathematical logic which resolves a special case of the Entscheidungsproblem. That being said, some somewhat more influential “theorems on combinations" appear in the first section of Ramsey’s paper, and from these “theorems on combinations” a beautiful branch of pure mathematics (Ramsey theory) was formed (largely by Erdős, at least according to Graham and Nešetřil). Now that I (hopefully) have your attention, this will broadly be a survey talk, in which I will discuss some results that appear in joint works with Meir (https://arxiv.org/abs/2212.08027 and https://arxiv.org/abs/2307.14468), and Meir and Touchard (https://arxiv.org/abs/2307.14468). The goal of the talk is to illustrate some rather interesting connections between model theory and (structural) Ramsey theory.
Borel complexity of theories of finite equivalence relations via large cardinals
When: Tue, September 17, 2024 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Danielle Ulrich (University of Maryland) -
Tannaka Krein duality for Roelcke-precompact non-archimedean Polish groups
When: Tue, September 24, 2024 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Rémi Barritault (University of Maryland, College Park) - https://rbarritault.github.io/
Abstract: Tannaka and Krein independently established around 1940 a duality theory in abstract harmonic analysis: any compact group can be fully recovered from the data contained in its unitary representations. These results can be seen as an analogue of the Pontryagin--van-Kampen duality for locally compact abelian groups.
After an introduction on abstract harmonic analysis, where I will go over the mentioned duality theories, I will explain how Tannaka's and Krein's results can be extended to Roelcke-precompact non-archimedean Polish groups (think Aut(M) for an ω-categorical first order structure M). Along the way, we obtain two realizations of the Hilbert compactification of such a group.
Almost homomorphisms and measurable cocycles
When: Tue, October 8, 2024 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Christian Rosendal (University of Maryland) - https://sites.google.com/view/christian-rosendal
Abstract: In response to a question of Yongle Jiang, we show that a measurable map between Polish groups that is multiplicative on a large set must be identical to an actual homomorphism on a large set. We also discuss an automatic continuity result for Baire measure cocycles, which is analogous with a recent result of Meyerovich and Solan for Haar measurable cocycles.
Merges of smooth classes and their model theoretic, Ramsey, and EPPA properties
When: Tue, October 15, 2024 - 3:30pmWhere: CSIC 4122
Speaker: Morgan Bryant (University of Maryland) -
Abstract: Given two Fraisse-like classes with generic limits, we ask whether we can merge the two classes into one class with a generic limit. We then study the properties of these merges and their generics, as well as their connections to structural Ramsey theory and the Hrushovski property (EPPA).
Zarankiewicz’s Problem and Model Theory
When: Tue, October 22, 2024 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Aris Papadopoulos (University of Maryland) - https://arispapadopoulos.github.io/
Abstract: A shower thought that anyone interested in graph theory must have had at some point in their lives is the following: `How “sparse" must a given graph be, if I know that it has no “dense” subgraphs?’. This curiosity definitely crossed the mind of Polish mathematician K. Zarankiewicz, who asked a version of this question formally in 1951. In the years that followed, many central figures in the development of extremal combinatorics contemplated this problem, giving various kinds of answers. Some of these will be surveyed in the first part of my talk.
So far so good, but this is a logic seminar and the title says the words “Model Theory"… In the second part of my talk, I will discuss how the celebrated Szemerédi-Trotter theorem gave a starting point to the study of Zarankiewicz’s problem in “geometric” contexts, and how the language of model theory has been able to capture exactly what these contexts are. I will then ramble about improvements to the classical answers to Zarankiewicz’s problem, when we restrict our attention to semilinear/semibounded o-minimal structures, Presburger arithmetic, and various kinds of Hrushovski constructions.
The new results that will appear in the talk were obtained jointly with Pantelis Eleftheriou.
Weak A2 spaces, the Kastanas game and strategically Ramsey sets
When: Tue, October 29, 2024 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Clement Yung (University of Toronto) - https://clementyung.github.io/
Abstract: We introduce the notion of a weak A2 space (or just wA2-space). wA2 spaces satisfy a part of Todorčević's axioms for topological Ramsey spaces, and is also a generalisation of countable vector spaces. It turns out that the abstract Kastanas game (introduced in 2023 by Cano and Di Prisco) on wA2-spaces serves as an intersection of Ramsey theory of topological Ramsey spaces, and the study of strategically Ramsey subsets of countable vector spaces. In this talk, I will discuss some properties of Kastanas Ramsey subsets of wA2-spaces, and use them to give quick proofs of classical results of Ramsey subsets of topological Ramsey spaces, and of strategically Ramsey subsets of countable vector spaces.
Asymptotically spherical groups
When: Tue, November 5, 2024 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Jenna Zomback (University of Maryland) - https://sites.google.com/umd.edu/zomback
Abstract: In joint work with Christian Rosendal, we investigate a notion of asymptotically spherical topological groups, which says that spheres of large radius with respect to any maximal length function are still spherical with respect to any other maximal length function. This is a strengthening of a related condition introduced by Sebastian Hurtado, which we call bounded eccentricity. Our main result is a partial characterization of which groups are asymptotically spherical, and we also give an example of a discrete, bounded eccentric group who fails to be asymptotically spherical.
In this first of two talks (the second of which will be given by Christian Rosendal next week), we will motivate and define the aforementioned notions, present the main results, and begin to discuss the proofs.
Asymptotically spherical groups, part II
When: Tue, November 12, 2024 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Christian Rosendal (University of Maryland) - https://sites.google.com/view/christian-rosendal
Abstract: This is the continuation of Jenna Zomback's talk on our joint work on asymptotically spherical groups.
The fundamental observation of geometric group theory states that a large class of topological groups ,such as f.g. discrete, compactly generated locally compact or monogenic Polish groups, carry an inherent quasimetric structure. This means that the various metrics defining this structure are biLipschitz for large distances. However, for the subclass of asymptotically spherical groups defined in the first talk, the different metrics are asymptotically dilations of each other.
We will present a sufficient condition for a topological group to be asymptotically spherical and also present a partial converse that indicates that this is close to being necessary.
PAC-learning, Differential Privacy, and Model Theory
When: Tue, November 19, 2024 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Vince Guingona (Towson University) - https://tigerweb.towson.edu/vguingona/
Abstract: In this talk, we discuss some applications of model theory to machine learning. In particular, we explore PAC-learning and differential privacy. It has recently been shown (Alon, Bun, Livni, Malliaris, Moran) that a concept class admits a differentially private PAC-learning algorithm if and only if it has finite Littlestone dimension. That is, if and only if it is a definable family in a model of a stable theory. The best known bounds for the sample complexity of a differentially private PAC-learning algorithm on a concept class with Littlestone dimension d is roughly O(d^6) (Ghazi, Golowich, Kumar, Manurangsi). In our current work, we are looking for an improvement to this bound. Towards this end, we are examining special cases of stable theories, like equational theories (as was recommended to us by Artem Chernikov).
This work is joint with Alexei Kolesnikov, Miriam Parnes, and Natalie Piltoyan.
The delightful theories of product modules
When: Tue, December 3, 2024 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Chris Laskowski (UMCP) -
Abstract: I will present Danielle Ulrich's proof of the Borel completeness of theories of R-modules when the theory of R (as an R-module) is not omega-stable. The proof exploits properties of pp-constructible models over countable sets, which simultaneously act like countable, atomic models and countable saturated models, although they are neither.
Two cases of Wilson's conjecture for omega-categorical Lie algebras
When: Tue, January 28, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Christian de'Elbee (University of Leeds)
Abstract: Recall that a structure (group, Lie algebra, associative algebra, etc) M is omega-categorical if there is a unique countable model of its first-order theory, up to isomorphism. This model theoretic notion has a dynamical definition: M is omega-categorical if and only if there are only finitely many orbits in the component-wise action of Aut(M) on the cartesian power M^n, for all natural number n. In 1981, Wilson conjectured that any omega-categorical locally nilpotent group is nilpotent. If true, a quite satisfactory decomposition of omega-categorical groups would follow. This conjecture is very much open more than 40 years later. The analogue statement for Lie algebras (every locally nilpotent omega-categorical Lie algebra is nilpotent) is also open and, as it turns out, it reduces to proving that for each n and prime p, every omega-categorical n-Engel Lie algebra over F_p is nilpotent. As for associative algebras, the analogous question was already answered by Cherlin in 1980: every locally nilpotent omega-categorical ring is nilpotent. We see the Wilson conjecture for Lie algebra as a bridge between the result of Cherlin and the original question of Wilson for omega-categorical groups. The question of Wilson, for groups, for Lie algebras or for associative algebras are connected to classical nilpotency problems such as the Burnside problem, the Kurosh problem or the problem of local nilpotency of n-Engel groups. Using a classical result of Zelmanov, the Wilson conjecture for omega-categorical Lie algebras is true asymptotically in the following sense: for each n, every n-Engel Lie algebra over F_p is nilpotent for all but finitely many p's. The situation for small values of the pair (n,p) is as follows: . Every 2-Engel Lie algebra is nilpotent (Higgins 1954), . Every 3-Engel Lie algebra over F_p with p\neq 2,5 is nilpotent (Higgins 1954), . Every 4-Engel Lie algebra over F_p is nilpotent for p\neq 2,3,5. (Higgins 1954, Kostrikin 1959), . Every 5-Engel Lie algebra over F_p is nilpotent for p\neq 2,3,5,7 (Vaughan-Lee, 2024). In other words, for (n,p) = (3,2), (3,5), (4,2), (4,3), (4,5),... It is known that n-Engel Lie algebras of char p are not globally nilpotent. Our goal, on the long run, is to prove that for those values of (n,p), omega-categorical n-Engel Lie algebra of characteristic p are nilpotent. We have recently dealt with the cases (n,p) = (3,5) and (n,p) = (4,3), and the proofs are different both in taste and method. The goal of the talk is to present a proof that every omega-categorical 4-Engel Lie algebras of characteristic 3 is nilpotent. Our solution of the case at hand consists in adapting in the definable context some classical tools for studying Engel Lie algebras, appearing earlier in the work of Higgins, Kostrikin, Zelmanov, Vaughan-Lee, Traustason and others. Our solution involves the use of computer algebra.
Organizational meeting
When: Tue, February 4, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: () -
Paradoxical Games and the Axiom of Choice
When: Tue, February 11, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Daniel Velleman (Amherst College) - https://www.amherst.edu/people/facstaff/djvelleman
Abstract: Alice and Bob are playing a guessing game. A room contains infinitely many boxes, labeled with the positive integers. Each box contains a real number. Alice will go into the room, open some but not all boxes, and then guess the contents of an unopened box. Then she leaves, the boxes are closed, and Bob enters, opens some but not all boxes, and guesses the contents of an unopened box. Alice and Bob can plan their strategies before the game, but once the game starts they cannot communicate. Paradoxical Theorem: There are strategies that Alice and Bob can follow that guarantee that at least one of them will guess correctly. The proof uses the axiom of choice, but a recent theorem of Elliot Glazer calls into question whether the axiom of choice can be blamed for the paradox.
Model theory of the Farey Graph via Smooth Classes
When: Tue, February 18, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Connor Lockhart (University of Maryland) - https://sites.google.com/umd.edu/connor/home
Abstract: We study the model theory of the Farey graph $F$ by realizing it as the generic of a smooth class $(K,\leq)$. By varying the relation $\leq$ we may obtain distinct generics that are either atomic or saturated. This will allow us to demonstrate a quantifier elimination for $Th(F)$. The Farey graph is also the simplest nontrivial curve complex of a surface, where $F=C(\Sigma_{1,1})$. Modifications of this technique to obtain results for the general model theory of the curve complex $C(\Sigma_{g,n})$ will be discussed.
Paradoxical Games and the Axiom of Choice
When: Tue, February 25, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Daniel Velleman (Amherst College) - https://www.amherst.edu/people/facstaff/djvelleman
Abstract: Alice and Bob are playing a guessing game. A room contains infinitely many boxes, labeled with the positive integers. Each box contains a real number. Alice will go into the room, open some but not all boxes, and then guess the contents of an unopened box. Then she leaves, the boxes are closed, and Bob enters, opens some but not all boxes, and guesses the contents of an unopened box. Alice and Bob can plan their strategies before the game, but once the game starts they cannot communicate. Paradoxical Theorem: There are strategies that Alice and Bob can follow that guarantee that at least one of them will guess correctly. The proof uses the axiom of choice, but a recent theorem of Elliot Glazer calls into question whether the axiom of choice can be blamed for the paradox.
On the model theory of free generalized polygons
When: Tue, March 4, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Katrin Tent (University of Münster) - https://ivv5hpp.uni-muenster.de/u/tent/
Abstract: We show that for $n\geq 3$ the theory of free generalized n-gons is complete, strictly stable and strictly 1-ample, yielding a new and easily accessible class of examples in the zoo of stable theories. The construction proceeds via Hrushovski amalgamation. (Joint with A.-M. Ammer)
Failure of the amalgamation property for definable types
When: Tue, March 11, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Martin Hils (University of Münster) - https://www.uni-muenster.de/Logik/en/hils/
Abstract: In recent joint work with Pablo Cubides Kovacsics and Jinhe Ye on beautiful pairs in the unstable context, the amalgamation property (AP) for the class of global definable types plays a key role. In the talk, we will first indicate some important cases in which AP holds, and we will then present the construction of examples of theories - some even NIP - obtained in joint work with Rosario Mennuni, where AP fails.
Topological Groups with Tractable Minimal Dynamics
When: Tue, March 25, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Andy Zucker (University of Waterloo) - https://ajzucker.wordpress.com/
Abstract: In joint work with Gianluca Basso, we explore the class of Polish groups whose universal minimal flows admit a comeager orbit. By work of Ben Yaacov, Melleray, and Tsankov, this class contains all Polish groups with metrizable universal minimal flow, and by an example of Kwiatkowska, this inclusion is strict. We isolate the correct generalization of this class of Polish groups to the class of all topological groups. We call these the topological groups with "tractable minimal dynamics (TMD)." One way of phrasing what makes this class "tractable" is an "abstract Kechris-Pestov-Todorcevic correspondence" which characterizes membership in TMD using a Ramsey-theoretic property of the group. In particular, this implies that TMD is absolute between models of set theory. We also state some conjectures to the effect that any topological group not in TMD has "wild" minimal dynamics.
Hyperhyperfiniteness and complexity
When: Tue, April 1, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Forte Shinko (UC Berkeley) - https://math.berkeley.edu/~forte/
Abstract: There is an array of long-standing open problems in the theory of countable Borel equivalence relations (CBER), all of which state that the class of hyperfinite CBERs is nice in some way. For instance, the unresolved Union Problem asks whether the class of hyperfinite CBERs is closed under increasing unions, and in a different direction, it is also open whether the hyperfinite CBERs form a $\mathbf{\Pi}^1_1$ set, which would be nicer than the naive complexity of $\mathbf{\Sigma}^1_2$. There are many other such problems, and it is widely believed that if one of them is false, then most of the others will be false as well, although there is no formal statement to this effect. To this end, we show an implication between the two aforementioned problems: precisely, we show that if the Union Problem has a negative answer, then the Borel complexity of the class of hyperfinite CBERs is as high as possible, namely $\mathbf{\Sigma}^1_2$-complete. This is joint with Joshua Frisch and Zoltan Vidnyanszky.
Where does homogeneity come from?
When: Tue, April 8, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Mervyn Tong (University of Leeds) - https://eps.leeds.ac.uk/maths/pgr/11722/mervyn-tong
Abstract: Everyone loves a good decomposition. How can we break down a mathematical object --- a graph, a group, or a function --- efficiently into well-behaved (or regular) parts? And what conditions can we place on these objects to guarantee a higher degree of regularity, such as homogeneity? It turns out an excellent source of such conditions is model-theoretic dividing lines, that is, tameness properties of (first-order) structures. This is not a coincidence. In this talk, I will dive into the deep theory of these dividing lines in search of the source of homogeneity.
TBA
When: Tue, April 15, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Artem Chernikov (University of Maryland) - https://chernikov.umd.edu/
Automorphisms with Knight rank and bounds on cardinalities of potential Scott sentences
When: Tue, April 22, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Shaun Allison (University of Toronto) - https://math.huji.ac.il/~sallison/
Abstract: In recent work I developed the Knight model rank of a countable structure M, inspired by a construction by Julia Knight called "Knight's model". The Knight model rank identifies when the structure's automorphism group Aut(M) involves S_\infty and thus has "maximal classification strength" in the sense of invariant descriptive set theory. We develop this rank further to show that given a bound on the Knight model rank of M, we can bound the cardinality of any potential Scott sentence for a model extending M. The talk will include a brief review of Scott sentences and their potential versions, as well as some historical context behind Knight's original construction that significantly motivated this work.
Poisson–Voronoi tessellations and fixed price in higher rank
When: Tue, April 29, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Amanda Wilkens (Carnegie Mellon University) - https://www.math.cmu.edu/~awilkens/
Abstract: We briefly define and motivate the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discuss the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk and Sam Mellick, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group. We give some intuition for the proof. No prior knowledge on fixed price or higher rank will be assumed.
Orbit equivalence of Baumslag-Solitar groups and measure-class preserving relations
When: Tue, May 6, 2025 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Antoine Poulin (McGill University) - https://antoinegpoulin.github.io/
Abstract: In this talk, we present the full classification of the Baumslag-Solitar family of groups under orbit equivalence. The techniques are based in studying measure-class preserving relations and treeings of. We sketch how to bridge the gap between probability measure preserving and measure-class preserving, and the combinatorial aspects of the proof. Based on work joint with Damien Gaboriau, Anush Tserunyan, Robin Tucker-Drob and Konrad Wróbel.