Tag - Model theory

Calum Hughes: An elementary theory of the 2-category of small categories

5th June, 2024

Lawvere’s Elementary Theory of the Category of Sets (ETCS) posits that the category Set is a well-pointed elementary topos with natural numbers object satisfying the axiom of choice. This provides a category theoretic foundation for mathematics which axiomatises the properties of function composition in contrast to Zermelo-Fraenkel set theory with the axiom of choice (ZFC), which axiomatises sets and their membership relation. Furthermore, ETCS augmented with the axiom schema of replacement can be shown to be equiconsistent with ZFC.

In this talk, I will present a categorification of ETCS which axiomatises the 2-category of small categories, functors and natural transformations; this is the elementary theory of the 2-category of small categories (ET2CSC) of the title. This extends Bourke’s characterisation of categories internal to a category E with pullbacks to the setting where E satisfies the extra properties of ETCS. Important 2-categorical definitions I will introduce are 2-well-pointedness, the full subobject classifier and the categorified axiom of choice. The main conclusion is that ET2CSC is 'Morita biequivalent’ with ETCS, meaning that the two theories have biequivalent 2-categories of models.

I will also describe how Shulman and Weber’s ideas on discrete opfibration classifiers can be used to incorporate replacement, in a way reminiscent of algebraic set theory.

Artem Chernikov: Recognizing groups in Erdős geometry and model theory

15th March, 2024

Erdős-style geometry is concerned with difficult questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be viewed as asking for the possible number of intersections of a given algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections 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. I will overview some of the recent developments in this area, in particular explaining how all of this is not just about polynomials and works for definable sets in o-minimal structures.

Andrei Yafaev: o-minimality and Diophantine geometry

4th December, 2023

In the recent years there have been some spectacular applications of the theory of o-minimality (a branch of Model Theory) to some problems in Diophantine Geometry. It culminated in the unconditional proof of the Andre-Oort conjecture on the Zariski closure of sets of special points on Shimura varieties. We will present ideas and methods surrounding this proof.

Andrey Nikolaev: Non-standard polynomials and non-standard free group

20th October, 2023

Interpretation and bi-interpretation offer a novel approach to studying all structures elementarily equivalent to a given one. We use this approach to describe and study non-standard models of the ring of polynomials, Laurent polynomials, and the group ring of a free group.

In the presence of interpretation but not bi-interpretation, this approach produces a family of structures elementarily equivalent to a given one. We exploit this to introduce non-standard models of a free group. As time permits, we discuss their main properties.

Artem Chernikov: Recognizing Groups in Erdős Geometry and Model Theory

28th March, 2023

Erdős-style geometry is concerned with combinatorial questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be typically viewed as asking for the possible number of intersections of a given (semi-)algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups.  Techniques from model theory - Hrushovski's group configuration and its variants - are very useful in recognizing these groups, and allow to obtain higher arity and dimension generalizations of the Elekes-Szabó theorem. In fact, all of this is not just about polynomials and works in the larger setting of definable sets in o-minimal structures.

Chieu-Minh Tran: Toward Classifying Reducts of the Complex Field

13th December, 2022

We discuss some recent progress on the model-theoretic problem of classifying the reducts of the complex field (with named parameters and up to interdefinability). The tools we use include Castle’s recent solution of the Restricted Trichotomy Conjecture in characteristic 0 and a generalized sumproduct result from additive combinatorics.

Giulio Lo Monaco: Vopěnka’s principle in ∞-categories

24th June, 2021

Vopěnka's principle has arisen as a model-theoretical statement, provably independent of ZFC set theory. However, there are a number of categorical ways of formulating it, preventing the existence of proper classes of objects with some conditions in presentable categories, and these are what our attention will be focused on. In particular, we will look at analogous statements in the context of ∞-categories and we will ask how these new statements interact with the older ones. Moreover, some of the consequences of Vopěnka's principle on classes of subcategories of presentable categories are investigated and to some extent generalized to ∞-categories. A parallel discussion is undertaken about the similar but weaker statement known as weak Vopěnka's principle.

Piotr Kowalski: Model Theory

1st May, 2021

This is a 22-lecture course, with each lecture being about 45 minutes or so, given online by Piotr Kowalski. It gives an introduction to model theory.

We state several classical results about fields (Ax’s theorem, Hilbert's Nullstellensatz), which have easy model-theoretic proofs and then introduce the necessary basic model-theoretic tools to describe those proofs. In this way we motivate and present basic definitions and results from model theory. We will also sketch at the end of the lecture some recent applications of model theory regarding non-existence of classical solutions of some differential equations, which was a problem considered by Painlevé in 1895 and an argument was found only recently.

Mark Kamsma: Independence Relations in Abstract Elementary Categories

15th April, 2021

In Shelah's classification of first-order theories we classify theories using combinatorial properties. The most well-known class is that of stable theories, which are very well behaved. Simple theories are more general, and then even more general are the NSOP1 theories. We can characterize those classes by the existence of a certain independence relation. For example, in vector spaces such an independence
relation comes from linear independence. Part of this characterisation is canonicity of the independence relation: there can be at most one nice enough independence relation in a theory.

Lieberman, Rosický and Vasey proved canonicity of stable-like independence relations in accessible categories. Inspired by this we introduce the framework of AECats (abstract elementary categories) and prove canonicity for simple-like and NSOP1-like independence relations. This way we reconstruct part of the hierarchy that we have for first-order theories, but now in the very general category-theoretic setting.

Jonathan Kirby: A model-theoretic look at exponential fields

25th March, 2021

An exponential function is a homomorphism from the additive group of a field to its multiplicative group. The most important examples are the real and complex exponentials, and these are naturally studied analytically.

However, one can also study the algebra of exponential fields and their logical theory. It turns out that the natural ways to do this take one outside the usual finitary classical logic of model theory and into positive/coherent logic, geometric logic, or other infinitary logics, or to the more algebraic and abstract setting of accessible categories.

I will describe some of this story, focusing on the more algebraic aspects of existentially closed exponential fields.