Forthcoming events in this series


Thu, 15 May 2025

11:00 - 12:00
C5

A criterion for sharpness of the Elekes-Szabó theorem in positive characteristic

Lucas Nistor
(École Normale Supérieure )
Abstract

We establish that a criterion based on ring-theoretic amenability is both necessary and sufficient for the abelian version of the Elekes-Szabó theorem to be sharp in the case of positive characteristic. Moreover, the criterion is always sufficient. We provide illustrative examples in the theories ACF_p and DCF_0.

Thu, 08 May 2025

16:00 - 17:00
C4

Globally Valued Fields: continuation

Michal Szachniewicz
Abstract

I will talk about intersection theory over any globally valued field and how it is connected to some model-theoretic problems.

Thu, 08 May 2025

11:00 - 12:00
C5

Simplicial reformulations of basic notions in model theory

Misha Gavrilovich
Abstract

We shall explain how to represent a couple of basic notions in model theory by standard simplicial diagrams from homotopy theory. Namely, we shall see that the notions of a {definable/invariant type}, {convergence}, and {contractibility} are defined by the same simplicial formula, and so are that of a {complete E-M type} and an {idempotent of an oo-category}.  The first reformulation makes precise Hrushovski's point of view that a definable/invariant type is an operation on types rather than a property of a type depending on the choice of a model, and suggests a notion of a type over a {space} of parameters. The second involves the nerve of the category with a single idempotent non-identity morphism, and leads to a reformulation of {non-dividing} somewhat similar to that of lifting idempotents in an oo-category. If time permits, I shall also present simplicial reformulations of distality, NIP, and simplicity.

We do so by associating with a theory the simplicial set of its n-types, n>0. This simplicial set, or rather its symmetrisation, appeared earlier in model theory under the names of {type structure}  (M.Morley. Applications of topology to Lw1w. 1974), {type category} (R.Knight, Topological Spaces and Scattered Theories. 2007), {type space functors} (Haykazyan. Spaces of Types in Positive Model Theory. 2019; M.Kamsma. Type space functors and interpretations in positive logic. 2022).

Thu, 01 May 2025

11:00 - 12:00
C5

Introduction to Arakelov theory

Michał Szachniewicz
(University of Oxford)
Abstract

I will talk about preliminaries in Arakelov geometry. Also, a historical overview will be provided. This talk will be the basis of a later talk about the theory of globally valued fields.

Thu, 06 Mar 2025

11:00 - 12:00
L5

Translation varieties (part 2)

Ehud Hrushovski
(University of Oxford)
Abstract

In algebraic geometry, the technique of dévissage reduces many questions to the case of curves. In difference and differential algebra, this is not the case, but the obstructions can be closely analysed. In difference algebra, they are difference varieties defined by equations of the form \si(𝑥)=𝑔𝑥\si(x)=gx, determined by an action of an algebraic group and an element g of this group. This is joint work with Zoé Chatzidakis.

Thu, 27 Feb 2025

11:00 - 12:00
C5

n-ampleness and pseudobuildings

Silke Meißner
(University of Münster)
Abstract
Zilber showed that a strongly minimal theory is 1-ample if and only if it interprets a pseudoplane. We will see a generalisation of this result to n-ample theories and define the notion of a pseudobuilding. This is joint work in progress with Katrin Tent.
Thu, 20 Feb 2025

11:00 - 12:00
C6

Translation varieties

Ehud Hrushovski
(University of Oxford)
Abstract

In algebraic geometry, the technique of dévissage reduces many questions to the case of curves. In difference and differential algebra, this is not the case, but the obstructions can be closely analysed. In difference algebra, they are difference varieties defined by equations of the form $\si(x)=g x$, determined by an action of an algebraic group and an element g of this group. This is joint work with Zoé Chatzidakis.

Thu, 13 Feb 2025

11:00 - 12:00
C5

Around Siu inequality

Michał Szachniewicz
(University of Oxford)
Abstract

I will talk about the connections between the Siu inequality and existence of the model companion for GVFs. The talk will be partially based on a joint work with Antoine Sedillot.

Thu, 23 Jan 2025

11:00 - 12:00
L5

A new axiom for Q_p^ab and non-standard methods for perfectoid fields

Leo Gitin
(University of Oxford)
Abstract

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of Q_p^ab, the maximal extension of the p-adic numbers Q_p with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of Q_p) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

Thu, 05 Dec 2024

11:00 - 12:00
C1

Local-Global Principles and Fields Elementarily Characterised by Their Absolute Galois Groups

Benedikt Stock
(University of Oxford)
Abstract

Jochen Koenigsmann’s Habilitation introduced a classification of fields elementarily characterised by their absolute Galois groups, including two conjecturally empty families. The emptiness of one of these families would follow from a Galois cohomological conjecture concerning radically closed fields formulated by Koenigsmann. A promising approach to resolving this conjecture involves the use of local-global principles in Galois cohomology. This talk examines the conceptual foundations of this method, highlights its relevance to Koenigsmann’s classification, and evaluates existing local-global principles with regard to their applicability to this conjecture.

Thu, 28 Nov 2024

11:00 - 12:00
TCC VC

Probability logic

Ehud Hrushovski
(University of Oxford)
Thu, 21 Nov 2024

11:00 - 12:00
C3

Almost sure convergence to a constant for a mean-aggregated term language

Sam Adam-Day
(University of Oxford)
Abstract
With motivation coming from machine learning, we define a term language on graphs generalising many graph neural networks. Our main result is that the closed terms of this language converge almost surely to constants. This probabilistic result holds for Erdős–Rényi graphs for a variety of sparsity levels, as well as the Barabási–Albert preferential attachment graph distribution. The key technique is a kind of almost sure quantifier elimination. A natural extension of this language generalises first-order logic, and a similar convergence result can be obtained there.
 
Thu, 17 Oct 2024

11:00 - 11:30
C2

Organisational meeting

Abstract

Please attend if you would like to give a talk in the Logic Advanced Class this term.

Thu, 13 Jun 2024

11:00 - 12:00
C3

The Ultimate Supercompactness Measure

Wojciech Wołoszyn
(University of Oxford)
Abstract

Solovay defined the inner model $L(\mathbb{R}, \mu)$ in the context of $\mathsf{AD}_{\mathbb{R}}$ by using it to define the supercompactness measure $\mu$ on $\mathcal{P}_{\omega_1}(\mathbb{R})$ naturally given by $\mathsf{AD}_{\mathbb{R}}$. Solovay speculated that stronger versions of this inner model should exist, corresponding to stronger versions of the measure $\mu$. Woodin, in his unpublished work, defined $\mu_{\infty}$ which is arguably the ultimate version of the supercompactness measure $\mu$ that Solovay had defined. I will talk about $\mu_{\infty}$ in the context of $\mathsf{AD}^+$ and the axiom $\mathsf{V} = \mathsf{Ultimate\ L}$.

https://woloszyn.org/

Thu, 06 Jun 2024

17:00 - 18:00
L3

Model theory of limits

Leo Gitin
(University of Oxford)
Abstract

Does the limit construction for inverse systems of first-order structures preserve elementary equivalence? I will give sufficient conditions for when this is the case. Using Karp's theorem, we explain the connection between a syntactic and formal-semantic approach to inverse limits of structures. We use this to give a simple proof of van den Dries' AKE theorem (in ZFC), a general AKE theorem for mixed characteristic henselian valued fields with no assumptions on ramification. We also recall a seemingly forgotten result of Feferman, that can be interpreted as a "saturated" AKE theorem in positive characteristic: given two elementarily equivalent $\aleph_1$-saturated fields $k$ and $k'$, the formal power series rings $k[[t]]$ and $k'[[t]]$ are elementarily equivalent as well. We thus hope to popularise some ideas from categorical logic.

Thu, 06 Jun 2024

11:00 - 12:00
C3

Demushkin groups of infinite rank in Galois theory

Tamar Bar-On
(University of Oxford)
Abstract
Demushkin groups play an important role in number theory, being the maximal pro-$p$ Galois groups of local fields containing a primitive root of unity of order $p$. In 1996 Labute presented a generalization of the theory for countably infinite rank pro-$p$ groups, and proved that the $p$-Sylow subgroups of the absolute Galois groups of local fields are Demushkin groups of infinite countable rank. These results were extended by Minac & Ware, who gave necessary and sufficient conditions for Demushkin groups of infinite countable rank to occur as absolute Galois groups.
In a joint work with Prof. Nikolay Nikolov, we extended this theory further to Demushkin groups of uncountable rank. Since for uncountable cardinals, there exists the maximal possible number of nondegenerate bilinear forms, the class of Demushkin groups of uncountable rank is much richer, and in particular, the groups are not determined completely by the same invariants as in the countable case.  
Additionally, inspired by the Elementary Type Conjecture by Ido Efrat and the affirmative solution to Jarden's Question, we discuss the possibility of a free product over an infinite sheaf of Demushkin groups of infinite countable rank to be realizable as an absolute Galois group, and give a necessary and sufficient condition when the free product is taken over a set converging to 1.
Thu, 30 May 2024

11:00 - 12:00
C3

Axiomatizing monodromy

Ehud Hrushovski
(University of Oxford)
Abstract

Consider definable sets over the family of finite fields $\mathbb{F}_q$. Ax proved a quantifier-elimination result for this theory, in a reasonable geometric language. Chatzidakis, Van den Dries and Macintyre showed that to a first-order approximation, the cardinality of a definable set $X$ is definable in a very mild expansion of Ax's theory.  Can such a statement be true of the next higher order approximation, i.e. can we write $|X(\mathbb{F}_q)| = aq^{d} + bq^{d-1/2} + o(q^{d-1/2})$, with $d,a,b$ varying definably with $X$ in a tame theory?    Here $b$ must be viewed as real-valued so continuous logic is needed. I will report on joint work in progress with Will Johnson.

Thu, 16 May 2024

11:00 - 12:00
C3

Basics of Globally Valued Fields and density of norms

Michał Szachniewicz
(University of Oxford)
Abstract

I will report on a joint work with Pablo Destic and Nuno Hultberg, about some applications of Globally Valued Fields (GVFs) and I will describe a density result that we needed, which turns out to be connected to Riemann-Zariski and Berkovich spaces.

Thu, 09 May 2024

11:00 - 12:00
C3

Skolem problem for several matrices

Emmanuel Breuillard
(University of Oxford)
Abstract

I will present a recent work with G. Kocharyan, where we show the undecidability of the following two problems: given a finitely generated subgroup G of GL(n,Q), a) determine whether G has a non-identity element whose (i,j) entry is equal to zero, and b) determine whether the stabilizer of a given vector in G is non-trivial. Undecidability of problem b) answers a question of Dixon from 1985. The proofs reduce to the undecidability of the word problem for finitely presented groups.

Thu, 02 May 2024

11:00 - 12:00
C3

Difference fields with an additive character on the fixed field

Stefan Ludwig
(École Normale Supérieure )
Abstract

Motivated by work of Hrushovski on pseudofinite fields with an additive character we investigate the theory ACFA+ which is the model companion of the theory of difference fields with an additive character on the fixed field. Building on results by Hrushovski we can recover it as the characteristic 0-asymptotic theory of the algebraic closure of finite fields with the Frobenius-automorphism and the standard character on the fixed field. We characterise 3-amalgamation in ACFA+. As cosequences we obtain that ACFA+ is a simple theory, an explicit description of the connected component of the Kim-Pillay group and (weak) elimination of imaginaries. If time permits we present some results on higher amalgamation.

Thu, 07 Mar 2024

11:00 - 12:00
C3

Model theory of Booleanizations, products and sheaves of structures

Jamshid Derakhshan
(University of Oxford)
Abstract

I will talk about some model-theoretic properties of Booleanizations of theories, subdirect products of structures, and sheaves of structures. I will discuss a result of Macintyre from 1973 on model-completeness, and more recent results jointly with Ehud Hrushovski and with Angus Macintyre.