While the natural model-theoretic ranks available in differentially closed fields of characteristic zero, namely Lascar and Morley rank, are known not to be definable in families of differential varieties; in this talk we show that the differential-algebraic rank given by the Kolchin polynomial is in fact definable. This result relies on a uniform bound on the Hilbert-Kolchin index. As a byproduct, we show that the property of being weakly irreducible for a differential variety is also definable in families. The question of full irreducibility remains open; it is known to be equivalent to the generalized Ritt problem. Slides.

This talk tries to outline a proof of the following result: in the first-order language of valued fields, every definable set in a non-trivially valued omega-free pseudo-algebraically closed field with a finite exponent of imperfectness, is dense in a definable set in the field-theoretic algebraic closure of said field, using only parameters from the smaller field, with respect to the topology induced by the valuation. Slides.

数学知识到底是先验的，还是后验的？这个问题或许可以算作数学哲学的基本问题之一。作为数学哲学的初学者，我希望以对如下几种具体过程发生的自然性和典型性的模拟来说明或解释数学知识的后验性。我们所关注的具体过程包括 (1) 几个具体的数学概念的提炼过程；(2) 几个具体的等式律与不等式律的发现过程；(3) 几个具体的外部操作被转化成内部操作的过程。通过对这些具体过程的自然性和典型性的模拟，我们希望揭示在从客观实在到观念存在再到数学理念存在以及有关这种存在的数学知识的多层次抽象过程中知觉与常识在数学内部的驱动力与外部的驱动力合成作用下所扮演的本性角色，以及经验事实与数学命题的相对真实性的密切关联。基于这些考量，我们愿意相信数学知识内在的后验性，就如同物理学知识所具有的内在的后验性那样。同时，我们也希望指出数学知识是一种由数学思考者在不断出现的有意义的问题牵引下为了满足解决那些自然而然的问题按照逻辑工具所提供的 假言推理的方式建立起来的在不断发展、不断协调、修正与完善起来的必然和有机的结构。从而，很难想象这样的一种知识结构是先验的。幻灯片。

Let $M$ be a weakly o-minimal non-valuational structure, and $N$ its canonical o-minimal extension (by Wencel). We prove that every group $G$ definable in $M$ is a dense subgroup of a group $K$ definable in $N$. As an application, we obtain that $G^{00}= G\cap K^{00}$, and establish Pillay's Conjecture in this setting: $G/G^{00}$, equipped with the logic topology, is a compact Lie group, and if $G$ has finitely satisfiable generics, then $\mathrm{dim}(G/G^{00})= \mathrm{dim}(G)$.   . Slides.

(Joint work with Itay Kaplan and Pierre Simon.) Compressibility is a certain isolation notion suited to NIP theories. A theory is distal if and only if every type is compressible. I will discuss some good properties of this notion and their consequences, in particular the existence of "compressibly atomic" models over arbitrary sets in countable NIP theories, and uniform honest definitions for an NIP formula. Notes.

If one asks a mathematician what she is working on, often the answer will be that they are seeking to prove or refute a certain conjecture. Equally as often, though, the answer will be that they are seeking to classify a certain kind of mathematical object. A chief aim of this talk is describe the means by which classification proceeds in mathematics and the intellectual ends which it serves. A secondary aim is to ask whether and to what extent classification theory in model theory can be understood to be an instance of the more quotidian notion of classification. This is based on Chapter 17 of the book Philosophy and Model Theory (OUP 2018), coauthored with Tim Button. Slides, Video.

A randomization of a first-order structure $M$, introduced by Keisler, is a structure $M^R$ in continuous logic whose elements are the "random" elements of $M$. One can think of it as a continuous structure whose types correspond to probability measures on the space of types of the original structure. Randomization preserves certain model-theoretic tameness properties, e.g. stability and NIP. The latter was demonstrated by Ben Yaacov via developing aspects of the VC-theory (Vapnik-Chervonenkis) in the continuous setting, connected to earlier work of Talagrand and others. A more general hierarchy of $n$-dependent theories was introduced by Shelah, with the case $n=1$ corresponding to NIP: a theory is $n$-dependent if the edge relation of an infinite generic $(n+1)$ -hypergraph is not definable. We will discuss n-dependence in continuous logic and demonstrate that $n$-dependence is also preserved by Keisler randomization. Our proof relies on structural Ramsey theory and multidimensional de Finetti-type results (and provides in particular a new proof in the NIP case). Joint work with Henry Towsner. Slides.

The notion of graphon was introduced by Lovász and Szegedy in 2006. A graphon is a limit of some graph sequences as the graph order goes to infinity. A graphon is not a graph but a symmetric Lebesgue measurable function from $[0,1]^2$ to $[0,1]$. In this talk, we use nonstandard analysis to approach graphons. Fix a hyperfinite set $N$. Consider internal graphs on $N$. Although graphons are not graphs, hyperfinite internal graphs are indeed graphs. We construct graphons as conditional probabilities of hyperfinite internal graphs. Slides.

(Joint work with Nicolas Chavarria Gomez.) We introduce a continuous logic formalism for a studying first order structures equipped with a family of maps to compact Hausdorff spaces. We apply this to the case of an abelian structure (abelian group $A$ together with a collection of subgroups of $A^{n}$) equipped with a homomorphism to a compact Hausdorff group. We adapt the usual "quantifier elimination down to positive primitive formulas" for modules, to the new context, deducing stability of the continuous first order theory. Slides.

The study of fundamental logical notions such as a formal language, theory, completeness, categoricity and other lead to an interaction with core mathematical theories at a very deep level. I will try to explain this phenomenon drawing also on some lessons from physics. Slides.

The program of higher degree theory, extending alpha-recursion theory, studies the various definability degree structures at uncountable cardinals (in particular singular cardinals of countable cofinality) in canonical inner models of set theory, focuses on the connection between the complexity of these degree structures and the large cardinal strength of the associated cardinals. We report recent progress in higher degree theory since the outbreak of COVID-19. Slides.

We study Whitehead groups and its variants in hereditary set universes and second-order arithmetic models and demonstrate the structural similarity between these two classes of foundations of mathematics. Slides.

The notion of effective computability was formulated on the basis that the set of natural numbers is effectively countable. Generalizing this notion to other structures is mathematically natural and desirable. More than 50 years ago, Kreisel already propounded some reasons for generalizing recursion theory. While some generalizations have been successful, there remain a number of foundational issues especially where they pertain to uncountable structures. In this talk, we review three examples of these generalizations and discuss some foundational questions arising therefrom. Slides.

Tennenbaum famously proved that there is no computable presentation of a nonstandard model of arithmetic or indeed of any model of set theory. In this talk, I shall discuss the Tennenbaum phenomenon as it arises for computable quotient presentations of models. Quotient presentations offer a philosophically attractive treatment of identity, a realm in which questions of identity are not necessarily computable. Objects in the presentation serve in effect as names for objects in the final quotient structure, names that may represent the same or different items in that structure, but one cannot necessarily tell which. Bakhadyr Khoussainov outlined a sweeping vision for quotient presentations in computable model theory and made several conjectures concerning the Tennenbaum phenomenon. In this talk, I shall discuss joint work with Michał Godziszewski that settles and addresses several of these conjectures. Slides.

(Joint work with Viktor Verboskiy.) We discuss Shelah’s dividing line methodology by reviewing three classifications: (neo)-stability, Keisler’s order, the universality-spectrum order. We say a property of a theory is virtuous if it has significant mathematical consequences for the theory or its models; a dividing line is a virtuous property whose negation is also virtuous. The dividing line strategy addresses a test problem by choosing successive dividing lines that converge on a solution. We explore to what extent these classifications are successful, fruitful (fecund) and robust. We consider the thesis that a dividing line strategy leads not merely to a classification but to a taxonomy. Slides.

(Joint work with Assaf Hasson and Kobi Peterzil) The study of definable (or interpretable) fields in various fields has a long history, though with relatively few results, in model theory. For interpretable fields the proof usually relies on elimination of imaginaries in some well understood language. In this talk we outline a proof that every definable field in a $dp$-minimal valued field $K$, of characteristic 0, with generic differentiability of definable functions is definably isomorphic to a finite extension of K. This latter condition holds, e.g., in $p$-adically closed fields, $t$-convex fields and algebraically closed valued fields (really in any $1$-$h$-minimal $dp$-minimal valued field).

If time permits, we will briefly outline a general method for the study of fields interpretable in $dp$-minimal valued fields (of characteristic 0) satisfying generic differentiability of definable functions, which bypasses elimination of imaginaries. More specifically, we show that in some situations the "interpretable" case reduces (locally) to the "definable" case. Slides.

Groups definable in o-minimal structures have been studied by many authors in the past thirty years, often in analogy with real Lie groups. Recall that a real Lie group is a real analytic manifold equipped with a real analytic group operation. Examples include (almost) all groups of real matrices you can think of. A natural question is whether the classical decompositions (Jordan-Chevalley decomposition, Cartan decomposition, Levi decomposition, Iwasawa decomposition etc.) can be found in the o-minimal setting too. In this talk some work in progress and some older results will be presented about these decompositions, and definability issues will be discussed. Slides.

Let $C$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Los-Tarski Theorem from classical model theory implies that $C$ is definable in first-order logic (FO) by a sentence $\varphi$ if and only if $C$ has a finite set of forbidden induced finite subgraphs. It provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms of forbidden induced finite subgraphs. Furthermore, by the Completeness Theorem, we can compute from $\varphi$ the corresponding forbidden induced subgraphs. We show that this machinery fails on finite graphs.

– There is a class $C$ of finite graphs which is definable in FO and closed under induced subgraphs but has no finite set of forbidden induced subgraphs.

– Even if we only consider classes $C$ of finite graphs which can be characterized by a finite set of forbidden induced subgraphs, such a characterization cannot be computed from an FO-sentence $\varphi$, which defines $C$, and the size of the characterization cannot be bounded by $f(|\varphi|)$ for any computable function $f$.

Besides their importance in graph theory, the above results also significantly strengthen similar known theorems for arbitrary structures.

This is joint work with Joerg Flum. Slides.