Abstract: Motivated by the study of mapping class groups, we discuss the proper homotopy equivalence relation for locally finite graphs. We show that the Borel complexity of this equivalence relation is bireducible with the homeomorphism relation on closed subsets of the Cantor set, so by work of Camerlo and Gao, it is Borel complete. On the other hand, we will see that there is a generic equivalence class. We construct Polish spaces of locally finite graphs in such a way that we can extend these results to the homeomorphism relation for noncompact surfaces with pants decompositions. This is joint work with Jenna Zomback.
Abstract: In continuous logic, there are plenty of examples of interesting stable metric structures. On the other side of the SOP line, there are only a few metric structures where order is relevant, and order appears in a different way in each of them. In this talk, joint work with Diego Bejarano, we present a unified approach to linear and cyclic orders in continuous logic.
We axiomatize these theories, and find generic completions in the ultrametric case, analogous to the complete theory DLO. We study some expansions of these theories, including real closed metric valued fields, from this perspective, and characterize which expansions of metric linear orders should be considered o-minimal.
Abstract: In the classification of first-order theories, many dividing lines are defined by forbidding specific configurations of definable sets (e.g. Stability = No Order Property, Simplicity = No Tree Property, etc). In this talk, we abstract the combinatorial nature of these properties; using the concept of patterns of consistency and inconsistency, we give a general framework for studying this kind of dividing lines. Taking this idea to its limit, we review a notion of maximal complexity (SM) in the context of patterns. Weakening SM, we define new dividing lines (PM and the PM^{(k)} hierarchy), provide examples separating these properties from previously defined dividing lines, and prove various results about them.
Abstract: Based on recent work by Domat-Iyer-Shinko, and independently by Long, we give geometric criteria to determine when the universal minimal flow of a subgroup of the mapping class group of an infinite-type surface is non-metrizable.
Abstract: In this talk, I discuss how the study of Ramsey structures and generalized indiscernibles allows us to define new versions of well-known model-theoretic tree properties. Focusing attention on the colored linear order as an index structure, I prove that the so-called colored tree property gives a characterization of stability in terms of positive formulas, while colored versions of TP1 and TP2 give equivalent versions of TP1 and the Independence Property, respectively.
Abstract: After being open for 50 years, the Connes Embedding Problem (CEP) in operator algebras was settled several years ago as a consequence of the quantum complexity result MIP*=RE. One equivalent formulation of the CEP is that the group $F_2\times F_2$ is residually finite-dimensional (RFD), where $F_2$ is the free group on 2 generators. In their 2012 paper, Fritz, Netzer, and Thom proved that any RFD group $G$ is such that the standard presentation of the universal group C*-algebra $C^*(G)$ is computable and thus raised the question as to whether or not the standard presentation of the universal group C*-algebra $C^*(F_2\times F_2)$ is computable, for a negative answer to this question would refute the CEP. While MIP*=RE settled the CEP, it failed to resolve the question of Fritz, Netzer, and Thom. In this talk, I will show that the standard presentation of the universal group C*-algebra $C^*(F_2\times F_2)$ is not computable, using an even more recent quantum complexity result known as MIP^{co}=coRE. Time permitting, I will discuss related results. The work presented in this talk is joint with Thomas Sinclair.
Abstract: Reading a theorem of Puritz from the early 70s through our modern model-theoretic glasses we see a characterisation of tensor products of types in the full theory of the naturals. This (re-)reading was the starting point of joint work (still in progress) with R. Mennuni, in which we extract several candidate definitions generalising the proof of Puritz’s theorem. In the talk I’ll discuss some of these definitions and various contexts in which each of them holds/does not hold.
Abstract: The titular question occurred to me while learning Rosendal’s groundbreaking framework for understanding coarse geometries of topological groups, especially as they apply to questions surrounding what geometric group theorists know as the Milnor–Schwarz Lemma. The key case to understand is for connected groups, where I remain mostly mystified. In fact, the goal of the talk is mostly to situation the question, explain why I think it is exciting, and some neat, if confounding, tools and examples arising from Brazas’s work on free (Graev) topological groups.
Abstract: A classical result of Ruzsa and Szemeredi shows that there is no subset A of the integers such that both the sumset A+A and the productset AA are small. Breuillard, Katz and Tao found a similar result for the subsets of finite prime fields of “medium size”. We show that under certain mild assumptions on a dimension theory , a “medium sized” subset of a field with small sumset and productset in the sense of not expanding in dimension indicates the existence of a subfield of the same dimension. This is joint work with Sergei Starchenko.
Abstract: I will discuss some classes of mathematical structures that can be reconstructed from their automorphism groups, starting with sets and ending with symplectic manifolds.
Abstract: Abstract elementary classes (AECs) provide an extension of first order model theory in which we can still attempt a classification theory. The question of when limit models (a kind of surrogate for saturated models for AECs) are isomorphic has connections to important open problems in AECs, such as Shelah's categoricity conjecture. Most work in this area is towards 'positive' results - that is, showing limit models are isomorphic. The question of when limit models are not isomorphic is less explored.
In this talk we give a full characterisation of the spectrum of limit models under general assumptions in a stable AEC - that is, describe completely which models are isomorphic and which are not. Given time we will discuss applications, a more general result in the 'positive' direction, and touch on a recent result which says that all high cofinality limit models are disjoint amalgamation bases. Based partly on joint work with Marcos Mazari-Armida.
Abstract: Grothendieck compactness criterion, and the stability theory from model theory, show that the inner product on the unit ball over Hilbert spaces is stable. We study this phenomenon quantitatively. We prove that the inner product is $(k,\epsilon)$-stable for all $k\geq \exp(\pi/\epsilon)$, and it is not $(k,\epsilon)$-stable for $k\leq \exp(\log 2/\epsilon)$, showing that the growth is necessarily exponential in $1/\epsilon$. This answers a question of Conant. We also analyze how stability scales under nonlinear connectives applied to the inner product, and general binary formulas over Hilbert spaces expanded by unitary or normal operators.
Abstract: We prove that the theory of the Farey graph is pseudofinite by constructing a sequence of finite structures that satisfy increasingly large subsets of its first-order axiomatization. The Farey graph was recently axiomatized by Tent and Mohammadi. We show that while no finite planar graph can satisfy these axioms for sufficiently large substructures, they can be satisfied by triangulations densely embedded on orientable surfaces of higher genus. By applying a result of Archdeacon, Hartsfield, and Little on the existence of triangulations with representativity and connectedness, we establish that every finite subset of the theory of the Farey graph has a finite model.
Abstract: Let $X$ be a set and let $\mathcal{C}\subset \mathcal{P}(X)$ be a family of subsets of $X$, viewed as possible “concepts” to be learned from labeled samples (i.e., elements of $X$ labeled by $0$ or $1$, depending on whether they belong to some $c^*\in \mathcal{C}$). A class $\mathcal{C}$ is PAC-learnable if there is an algorithm which, given sufficiently many labeled samples, outputs with high probability a subset of $X$ that has a small error with respect to $c^*$. Recent work by Ghazi et.al and Alon et. al. has shown that $\mathcal{C}$ is privately PAC-learnable if and only if it has a finite Littlestone dimension (or, in model-theoretic terms, is stable). This invites a question whether one can exploit additional structure, beyond stability, to obtain better private learning bounds. Following Artem Chernikov’s suggestion, we explore equational classes. I will briefly review the learning-theoretic set up and will describe a natural private learning algorithm for equational classes. This is joint work with Vince Guingona, Miriam Parnes, and Natalie Piltoyan.
Abstract: The mapping class groups of locally finite infinite graphs are non-Archimedean topological groups that directly generalize the outer automorphism groups of free groups. I will discuss the sphere complex associated with a graph and outline my own work as well as joint work with Hill-Kopreski-Rechkin-Shaji which together show that the automorphism group of the sphere complex is naturally isomorphic to the mapping class group of the associated graph. With whatever time is remaining, I'll introduce a sort of "metaconjecture" which allows one to translate many results about mapping class groups of surfaces to mapping class groups of graphs, and list some consequences.
Abstract: Superstructures provide the essential framework required to formalize non-standard analysis. This presentation will cover the fundamental construction of superstructures and their core properties, culminating in a streamlined, non-standard proof of Tychonoff's Theorem.
Abstract: Algorithmic meta-theorems are uniform results in complexity theory that simultaneously yield an efficient solution to a wide class of computational problems, typically those expressible by a sentence of a certain logic. In the context of graphs, various research programmes have identified combinatorial conditions that provide dividing lines for the existence of uniform algorithmic results, such as the graph minors theory of Robertson and Seymour, the sparsity theory of Nesetril and Ossona de Mendez, and the twin-width theory of Bonnet et al. Recent years have seen the emergence of model-theoretic techniques in unifying these programmes and understanding the most general conditions that imply algorithmic tractability. This talk will survey the structural tractability programme, and present recent progress towards a conjecture that places monadic NIP as the limit to algorithmic tractability for first-order expressible problems. This is based on joint work with Jan Dreier, Nikolas Mahlmann, Rose McCarty, Michal Pilipczuk, and Szymon Torunczyk (FOCS 2024).
Abstract: Recall that a subset X of a group is said to be an approximate group if the product set XX:=\{ab:a,b\in X\} can be covered by finitely many translates of X. I will discuss a work-in-progress which attempts to give a structure theory for approximate groups definable in NIP theories. First I will present an example of an approximate group X definable in the universal cover of SL(2,R) such that has no type-definable subgroup of bounded index; this gives a NIP counterexample to a conjecture of Massicot-Wagner, first disproved by Hrushovski, Krupiński, and Pillay. I will then discuss some tentative positive structural results, some of which are of interest even in the case of definable groups. Among other things I will discuss Hrushovski’s “generalized locally compact models” in the NIP setting, and give a new proof of Gismatullin’s result on the existence of G^{000} for NIP groups G.
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