Conference

An influential theorem of Elekes and Szabó indicates that the intersections of a given algebraic variety with large finite grids of points may have maximal size only for varieties that are closely connected to algebraic groups.  Techniques from model theory --- variants of Hrushovski's group configuration and of Zilber's trichotomy principle --- are very useful in recognizing these groups, and led to far reaching generalizations of Elekes-Szabó in the last decade. In this talk, focusing on the o-minimal case, we provide a generalization of our earlier result to arbitrary co-dimension, in particular obtaining explicit bounds in a theorem of Bays-Breuillard. Joint work with Kobi Peterzil and Sergei Starchenko.

Shift spaces of sequences (including those indexed by the natural numbers, the integers, binary trees, etc.) are the central object of study in symbolic dynamics, and those which correspond to finite automata have useful and compelling properties. One property of recent interest is the HomShift, a shift which corresponds to an undirected graph in a canonical way. We will discuss the decidability of when a one-sided edge shift is conjugate to a one-sided Hom-shift, and the decidability of whether a tree edge shift is conjugate to a Hom tree shift. By interpreting these shifts in the natural way vis-à-vis k-automatic structures, we obtain analogous decidability results for classifying certain kinds of k-automatic expansions.

The Pila--Wilkie Theorem is a seminal result in o-minimality that counts rational points (and, in later variants, algebraic and semi-rational points) on definable sets, which has played a key role in many applications to diophantine geometry. An effective version of this theorem in certain situations would give greater control over some applications, but the original proofs of this theorem and of its variants do not give an effective bound, so alternative approaches are required.

During the last few years there has been significant progress on this topic. This had led not only to a uniform, finely tuned, effective version of this theorem (and its variants) in certain settings, as well as effective diophantine applications thereof, for example to an effective, uniform Manin--Mumford statement for products of elliptic curves with complex multiplication (joint work with Binyamini, Jones and Schmidt), but also to the introduction of `sharply' and `effectively o-minimal structures', a new perspective on how to approach effective applications of o-minimality, and even improved counting bounds by combining and developing various of these ideas (due to Binyamini, Novikov and Zak). I will aim to survey these developments and this perspective, which has the potential to broaden the scope of effective applications of o-minimality.

By recent work of Dobrinen and Balko, Chodounsk\'y, Dobrinen, Hubi\v{c}ka, Kone\v{c}n\'y, Vena, L. and Zucker, we know that every finite $G$ in the Henson graph $\mathbb{H_{n+1}$ (the universal ultrahomogeneous $n$-clique free graph) has exact finite big Ramsey degree $k({G,n})$. That is, there is a positive integer $k({G,n})$ such that for all finite colorings $C$ of the copies of $G$ in $\mathbb{H}_{n+1}$, there is $\tilde{\mathbb{H}}$, a substructure of $\mathbb{H}_{n+1}$ and isomorphic to $\mathbb{H}_{n+1}$, such that in $\tilde{\mathbb{H}}$ at most $k({G,n})$ colors are used on the induced coloring from $\mathbb{H}_{n+1}$ on the copies of $G$ in $\tilde{\mathbb{H}}$. Moreover, for exactness,  for some coloring and all corresponding $\tilde{\mathbb{H}}$, all $k({G,n})$ colors are needed. One result by Cholak, Dobrinen, and McCoy, we will discuss here is that if $|G|\geq 2$, then there is a finite computable coloring $C$ such that, for  all such $\tilde{\mathbb{H}}$, we have that $\tilde{\mathbb{H}}$ computes  $\emptyset^{(|G|-1)}$ (and hence the halting set).  We will also provide a template so that this result should be duplicatable in other structures which have finite big Ramsey degrees. We will also briefly mention another recent result by Cholak, Dobrinen, and Townser, that $\tilde{\mathbb{H}}$ can always be found arithmetically from the coloring.  Also, initially, we will explain why these results might be interesting to someone not familiar with Structural Ramsey Theory. 

Scott analysis is a branch of computable structure theory and descriptive set theory that aims to calibrate the complexity of the isomorphism relation. Two of the most fundamental notions studied in Scott analysis are Scott rank and the ordinal indexed back-and-forth relations. The Scott rank of a countable structure tells you how difficult it is to understand the isomorphism type of that structure. Meanwhile, the ordinal indexed back-and-forth relations measure how difficult it is to tell different structures apart from each other. We introduce the concept of a Scott filtration, which connects these two fundamental concepts in Scott analysis. Given a structure M, a Scott filtration uses simpler structures (i.e., structures of smaller Scott rank) to concretely understand the back-and-forth classes of M. We will discuss the existence of Scott filtrations up to the Vaught ordinal along with some other related theorems.

This is joint work with Dino Rossegger and Dan Turetsky. The speaker has happily able to attend the first four weeks of the HIM DDC trimester, and thanks the institute for the support provided that helped make this project possible.

Scott’s Isomorphism Theorem shows that each countable structure can be uniquely defined, up to isomorphism, by a sentence of infinitary logic, now known as the Scott Sentence of the structure. The complexity of a structure’s Scott Sentence can then be viewed as a measure of complexity of the structure. In this talk, we will discuss the relationship of Scott complexity with other measures of complexity, as well as discuss the Scott Complexity of certain structures. 

Let $Q(x_1,\dots,x_n)$ be a given quadratic polynomial, and consider $\xi_1,..,\xi_n$ to be independent unbiased $\{1,-1\}$-valued random variables (i.e. each $\xi_i$ takes value $1$ with probability $1/2$ and value $-1$ with probability $1/2$). To what extent can $Q(\xi_1,\dots,\xi_n)$ concentrate on a single value? This question is a quadratic version of the classical Littlewood--Offord problem for linear polynomials, and was popularised by Costello, Tao and Vu in their study of symmetric random matrices. This talk will discuss joint work with Matthew Kwan, in which we obtained an essentially optimal bound for this question, conjectured by Nguyen and Vu. We will also briefly discuss a more general conjecture for subsets of $\mathbb{R}^d$ definable with respect to an o-minimal structure, due to Fox, Kwan and Spink.

Generalizing work of Poizat in the stable case, motivated by the definability status of important spaces of definable types in the theory ACVF of algebraically closed valued fields, Cubides Kovacsics, Ye and I introduced and studied beautiful pairs for unstable theories, as a more semantic way to approach such definability questions. As one of the main applications, we could establish the strict pro-definability of all definable types and of all bounded definable types, living on an algebraic variety V defined over the valued field - these are definable analogues of the Zariski-Riemann space and the Huber space associated to V.

In the talk, I will explain the main ideas behind this approach, which brings to bear an Ax-Kochen-Ershov type reduction. Moreover, I will report on some ongoing project, joint with Hrushovski, Ye and Zou, on versions of the results for ACVF where a non-standard Frobenius automorphism is added to the structure. We obtain in particular suprisingly simple axioms for the resulting pairs, as these turn out to be beautiful as well.

For an infinite algebraic field extension $E/F$, Calvert, Harizanov, and Shlapentokh described the automorphisms of $E$ over $F$ as the set of paths through a tree.  Under some basic computability assumptions about $E$ and $F$, this tree is computable, and composition and inversion of the paths are both computable by Turing functionals.  Thus we have an effective presentation of the Galois group $\operatorname{Gal}(E/F)$, despite its potentially-continuum size.

For finite fields, this presentation of the absolute Galois group is extremely nice, indeed as close to being a decidable structure as one can get in this cardinality.  We will explain the notion of ``tree-decidability,'' developed by Block and the speaker, that makes this rigorous.  Predictably, the absolute Galois group of $\mathbb Q$ is nastier:  we will quantify this nastiness in a few specific ways -- one of them joint with Kundu -- and pose several open questions about exactly how bad it gets.

A major theme in mathematics is the classification problem: given a class of objects and a notion of equivalence, classify the objects up to the equivalence. In algebra, this often takes the form of isomorphism problems. There are multiple ways to formalize isomorphism problems using computability theory, many of which are closely related to the definability of the structures. For instance, the index set of a structure is related to its infinitary definability, namely, Scott analysis; and a result of Harrison-Trainor, Melnikov, Miller, and Montalban relates computable functors between (the isomorphism classes) of structures and effective (infinitary) interpretability.

In this talk, we consider a variant of the isomorphism problem that originates in algebra, namely, the isomorphism problem of (algebraic) presentations. More precisely, given an algebraic variety (in the sense of universal algebra), we consider the isomorphism problem on the finitely generated c.e. presentations in the variety. We then compare the complexities of these isomorphism problems under computable reducibility, an effectivization of Borel reducibility. We first focus on the varieties of commutative monoids, commutative semigroups, abelian groups, and unary structures. We then show some general results connecting the algebraic properties of varieties and the complexities of their isomorphism problem. 

Hardy fields are one-dimensional relatives of o-minimal structures. We recently proved a theorem which permits the transfer of first-order statements between Hardy fields and related domains for “tame" asymptotic analysis. I will focus on the special role played by second-order linear differential equations in this story, as well as applications of our main result to such equations. (Joint work with L. van den Dries and J. van der Hoeven.)

The striking resemblance between the behaviour of pseudo-algebraically closed, pseudo real closed and pseudo p-adically fields has lead to numerous attempts at describing their properties in a unified manner. In this talk I will present one of these attempts: the class of pseudo-T-closed fields, where T is an enriched theory of fields. These fields verify a « local-global » principle with respect to models of T for the existence of points on varieties. Although it very much resembles previous such attempts, our approach is more model theoretic in flavour, both in its presentation and in the results we aim for.

The first result I would like to present is an approximation result, generalising a result of Kollar on PAC fields, respectively Johnson on henselian fields. This result can be rephrased as the fact that existential closeness in certain topological enrichments come for free from existential closeness as a field. The second result is a (model theoretic) classification result for bounded pseudo-T-closed fields, in the guise of the computation of their burden. One of the striking consequence of these two results is that a bounded perfect PAC field with n independent valuations has burden n and, in particular, is NTP2.

The recent rank stability results have implications for definability over big subrings of number fields and the number fields themselves.  We survey these implications.