Past Logic Seminar

In this talk, I will introduce the frameworks of globally valued fields (Ben Yaacov-Hrushovski) and adelic curves (Chen-Moriwaki). Both of these frameworks aim at understanding the arithmetic of fields sharing common features with global fields. A lot of examples fit in this scope (e.g. global fields, finitely generated extension of the prime fields, fields of meromorphic functions) and we will try to describe some of them.

Although globally valued fields and adelic curves came from different motivations and might seem quite different, they are related (and even essentially equivalent). This relation opens the door for new methods in the study of global arithmetic. As an application, we will sketch the proof of an arithmetic analogue of Siu inequality in algebraic geometry (a fundamental tool to detect the existence of global sections of line bundles in birational geometry). This is a joint work with Michał Szachniewicz.

I am going to start by reviewing axioms of quantum mechanics, which in fact give a description of a Hilbert space. I will argue that the language that Dirac and his followers developed is that of continuous logic and the form of axiomatisation is that of "algebraic logic" in the sense of A. Tarski's cylindric algebras. In fact, Hilbert spaces can be seen as a continuous model theory version of cylindric algebras.

Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of uniform reflections. This result is commonly known as Feferman's completeness theorem. The talk aims to give one or two new proofs of Feferman's completeness theorem that, we hope, shed new light on this mysterious and often overlooked result.

Moreover, one of the proofs furnishes sharp bounds on the order types of well-orders necessary to attain completeness.

(This is joint work with Fedor Pakhomov and Dino Rossegger.)

In the theory of perfectoid fields, the tilting operation takes a perfectoid field K (a densely normed complete field of positive residue characteristic p for which the map which sends x to its p-th power is surjective as a self-map on O/pO where O is the ring of integers) to its tilt, which is computed as the limit in the category of multiplicative monoids of K under repeated application of the map sending x to its p-th power, and then a natural normed field structure is constructed. It may happen that two non-isomorphic perfectoid fields have isomorphic tilts. The family of characteristic zero untilts of a complete nontrivially normed complete perfect field of positive characteristic are parameterized by the Fargues-Fontaine curve.

Taking into account these parameters, we show that this correspondence between perfectoid fields of mixed characteristic and their tilts may be regarded as a quantifier-free bi-interpretation in continuous logic. The existence of this bi-interpretation allows for some soft proofs of some features of tilting such as the Fontaine-Wintenberger theorem that a perfectoid field and its tilt have isomorphic absolute Galois groups, an approximation lemma for the tilts of definable sets, and identifications of adic spaces.

This is a report on (rather old, mostly from 2016/7) joint work with Silvain Rideau-Kikuchi and Pierre Simon available at https://arxiv.org/html/2505.01321v1 .

I will sketch a proof of the statement in the title and outline how it is related to Ehrenfeucht–Fraïssé games on C*-algebras. I will provide the relevant background on C*-algebras (and descriptive set theory) and explain how to construct a standard Borel category X that can play a role of their `moduli'. The theorem from the title is an application of the compactness theorem, for a suitable first-order theory whose models correspond to functors from X. If time permits, I will mention some related problems and connections with conceptual completeness for infinitary logic. This talk is based on several discussions with Ehud Hrushovski, Jennifer Pi, Mira Tartarotti, and Stuart White after a reading group on the paper "Games on AF-algebras" by Ben De Bondt, Andrea Vaccaro, Boban Velickovic and Alessandro Vignati.

Let F(x,y) be a polynomial over the complex numbers. The Elekes-Ronyai theorem says that if F(x,y) is not essentially addition or multiplication, then F(x,y) exhibits expansion: for any finite subset A, B of complex numbers of size n, the size of F(A,B)={F(a,b):a in A, b in B} will be much larger than n. In fact, it is proved that |F(A,B)|>Cn^{4/3} for some constant C. In this talk, I will present a recent joint work with Martin Bays, which is an asymmetric and higher dimensional version of the Elekes-Rónyai theorem, where A and B can be taken to be of different sizes and y a tuple. This result is achieved via a generalisation of the Elekes-Szabó theorem.

A major open problem in the model theory of valued fields is to gain an understanding of the first-order theory of the power series field F((t)), where F denotes a finite field. For sufficiently "nice" henselian valued fields, the Ax-Kochen/Ershov philosophy allows to reduce questions of elementary equivalence and elementary embeddings to the analogous questions about the value group and residue field (or related structures). In my talk, I will present a new such principle which applies in particular to a large class of algebraic extensions of F((t)), albeit not to F((t)) itself. The talk is based on joint work with Konstantinos Kartas and Jonas van der Schaaf.

The open core of a structure is the reduct generated by the open definable sets. Tame topological structures (e.g. o-minimal) are often inter-definable with their open core. Structures such as M = (ℝ,<, +, ℚ) are wild in the sense that they define a dense co-dense set. Still, M is NIP and its open core is o-minimal. In this talk, we push forward the thesis that the open core of an NTP2 (a generalization of NIP) topological structure is tame. Our main result is that, under suitable conditions, the open core has quantifier elimination (every definable set is constructible), and its definable functions are generically continuous.

I will discuss several interesting examples of classes of structures for which there is a sensible first-order theory of "almost all" structures in the class, for certain notions of "almost all". These examples include the classical theory of almost all finite graphs due to Glebskij-Kogan-Liogon'kij-Talanov and Fagin (and many more examples from finite model theory), as well as more recent examples from the model theory of infinite fields: the theory of almost all algebraic extensions and the universal/existential theory of almost all completions of a global field (both joint work with Arno Fehm). Interestingly, such asymptotic theories are sometimes quite well-behaved even when the base theories are not.

Model-theoretic dividing lines have long been a source of tameness for various areas of mathematics, with combinatorics jumping on the bandwagon over the last decade or so. Szemerédi’s regularity lemma saw improvements in the realm of NIP, which were further refined in the subrealms of stability and distality. We show how relations satisfying the distal regularity lemma enjoy improved bounds for Zarankiewicz’s problem. We then pivot to arithmetic regularity lemmas as pioneered by Green, for which NIP and stability also imply improvements. Unsettled by the absence of distality in this picture, we discuss the role of distality in additive combinatorics, appealing to our result connecting distality with arithmetic tameness.

It is well known that the theory of differential-difference fields in characteristic zero has a model companion. Here by a differential-difference field I mean a field with a differential and a difference structure where the operators commute (in other words the difference structure is a differential-endomorphism). The theory DCFA_0 was studied in a series of papers by Bustamante. In this talk I will address the case of positive characteristic.

The monadic second-order theory S1S of (ℕ,<) is decidable (it essentially describes ω-automata). Undecidability of the monadic theory of (ℝ,<) was proven by Shelah. Previously, Rabin proved decidability if the monadic quantifier is restricted to Fσ-sets.
We discuss decidability for Borel sets, or even σ-combinations of analytic sets. Moreover, the Boolean combinations of Fσ-sets form an elementary substructure. Under determinacy hypotheses, the proof extends to larger classes of sets.

Ramification theory serves the dual purpose of a diagnostic tool and treatment by helping us locate, measure, and treat the anomalous behavior of mathematical objects. In the classical setup, the degree of a finite Galois extension of "nice" fields splits up neatly into the product of two well-understood numbers (ramification index and inertia degree) that encode how the base field changes. In the general case, however, a third factor called the defect (or ramification deficiency) can pop up. The defect is a mysterious phenomenon and the main obstruction to several long-standing open problems, such as obtaining resolution of singularities. The primary reason is, roughly speaking, that the classical strategy of "objects become nicer after finitely many adjustments" fails when the defect is non-trivial. I will discuss my previous and ongoing work in ramification theory that allows us to understand and treat the defect.

Von Neumann algebras which are not matrix algebras, yet still possess a unique trace, form a basic class called II$_1$ factors. The set of asymptotically commuting elements (or, the relative commutant of the algebra within its own ultrapower), dubbed the central sequence algebra, can take many different forms. In this talk, we discuss an elementary class of II$_1$ factors whose central sequence algebra is again a II$_1$ factor. We show that the class of infinitely generic II$_1$ factors possess this property, and ask some related questions about properties of other existentially closed II$_1$ factors. This is based on joint work with Isaac Goldbring, David Jekel, and Srivatsav Kunnawalkam Elayavalli.

Given a locally compact topological group, there is a correspondence between idempotent probability measures and compact subgroups. An analogue of this correspondence continues into the model theoretic setting. In particular, if G is a stable group, then there is a one-to-one correspondence between idempotent Keisler measures and type-definable subgroups. The proof of this theorem relies heavily on the theory of local ranks in stability theory. Recently, we have been able to extend a version of this correspondence to the abelian setting. Here, we prove that fim idempotent Keisler measures correspond to fim subgroups. These results rely on recent work of Conant, Hanson and myself connecting generically stable measures to generically stable types over the randomization. This is joint work with Artem Chernikov and Krzysztof Krupinski.

In the first half of the talk, I will briefly survey the theory of matroids with coefficients, which was introduced by Andreas Dress and Walter Wenzel in the 1980s and refined by the speaker and Nathan Bowler in 2016. This theory provides a unification of vector subspaces, matroids, valuated matroids, and oriented matroids. Then, in the second half, I will outline an intriguing connection between Lorentzian polynomials, as defined by Petter Brändén and June Huh, and matroids with coefficients.  The second part of the talk represents joint work with June Huh, Mario Kummer, and Oliver Lorscheid.

I will exposit some recent joint work with Paul Larson and Grigor Sargsyan that uses higher models of the Axiom of Determinacy---models with nontrivial structure above $\Theta$, the least ordinal which is not the surjective image of the reals---to show that instances of the fundamental problem of inner model theory, the iterability conjecture, consistently fail.

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, obtained in joint work with Rosario Mennuni, where AP fails.