Forthcoming events in this series

Thu, 02 Jun 2022

14:30 - 15:45
L4

### Non-elementary categoricity and projective o-minimal classes

Boris Zilber
(University of Oxford)
Abstract

Given a cover $U$ of a family of smooth complex algebraic varieties, we associate with it a class $C$ of structures locally definable in an o-minimal expansion of the reals, containing the cover $U$.  We prove that the class is $\aleph_0$-homogeneous over submodels and stable. It follows that $C$ is categorical in cardinality $\aleph_1$. In the one-dimensional case we prove that a slight modification of $C$ is an abstract elementary class categorical in all uncountable cardinals.

Thu, 02 Jun 2022
00:00

### (Postponed)

Tomás Ibarlucía
(Université Paris Cité)
Abstract

We call affine logic the fragment of continuous logic in which the connectives are limited to linear combinations and the constants (but quantification is allowed, in the usual continuous form). This fragment has been introduced and studied by S.M. Bagheri, the first to observe that this is the appropriate framework to consider convex combinations of metric structures and, more generally, ultrameans, i.e., ultraproducts in which the ultrafilter is replaced by a finitely additive probability measure. Bagheri has shown that many fundamental results of continuous logic hold in affine logic in an appropriate form, including Łoś's theorem, the compactness theorem, and the Keisler--Shelah isomorphism theorem.

In affine logic, type spaces are compact convex sets. In this talk I will report on an ongoing work with I. Ben Yaacov and T. Tsankov, in which we initiate the study of extremal models in affine logic, i.e., those that only realize extreme types.

Thu, 26 May 2022

11:30 - 12:45
L6

### Axiomatizing the existential theory of $F_p((t))$

Arno Fehm
(TU Dresden)
Abstract

From a model theoretic point of view, local fields of positive characteristic, i.e. fields of Laurent series over finite fields, are much less well understood than their characteristic zero counterparts - the fields of real, complex and p-adic numbers. I will discuss different approaches to axiomatize and decide at least their existential theory in various languages and under various forms of resolution of singularities. This includes new joint work with Sylvy Anscombe and Philip Dittmann.

Thu, 19 May 2022

14:30 - 15:45
L4

### Uniform families of definable sets in finite structures

Dugald Macpherson
(University of Leeds)
Abstract

A theorem of Chatzidakis, van den Dries and Macintyre, stemming ultimately from the Lang-Weil estimates, asserts, roughly, that if $\phi(x,y)$ is a formula in the language of rings (where $x,y$ are tuples) then the size of the solution set of $\phi(x,a)$ in any finite field $F_q$(where $a$ is a parameter tuple from $F_q$) takes one of finitely many dimension-measure pairs as $F_q$ and $a$ vary: for a finite set $E$ of pairs $(\mu,d)$ ($\mu$ rational, $d$ integer) dependent on $\phi$, any set $\phi(F_q,a)$ has size roughly $\mu q^d$ for some $(\mu,d) \in E$.

This led in work of Elwes, Steinhorn and myself to the notion of 'asymptotic class’ of finite structures (a class satisfying essentially the conclusion of Chatzidakis-van den Dries-Macintyre). As an example, by a theorem of Ryten, any family of finite simple groups of fixed Lie type forms an asymptotic class. There is a corresponding notion for infinite structures of  'measurable structure’ (e.g. a pseudofinite field, by the Chatzidakis-van den Dries-Macintyre theorem, or certain pseudofinite difference fields).

I will discuss a body of work with Sylvy Anscombe, Charles Steinhorn and Daniel Wolf which generalises this, incorporating a richer range of examples with fewer model-theoretic constraints; for example, the corresponding infinite 'generalised measurable’ structures, for which the definable sets are assigned values in some ordered semiring, need no longer have simple theory. I will also discuss a variant in which sizes of definable sets in finite structures are given exactly rather than asymptotically.

Thu, 19 May 2022

11:30 - 12:45
L6

### Skew-invariant curves and algebraic independence

Thomas Scanlon
(University of California, Berkeley)
Abstract
A $\sigma$-variety over a difference field $(K, \sigma)$ is a pair $(X, \varphi)$ consisting of an algebraic variety $X$ over $K$ and $\varphi : X \rightarrow X^{\sigma}$ is a regular map from $X$ to its transform $X^{\sigma}$ under $\sigma$. A subvariety $Y \subseteq X$ is skew-invariant if $\varphi(Y) \subseteq Y^{\sigma}$. In earlier work with Alice Medvedev we gave a procedure to describe skew-invariant varieties of $\sigma$-varieties of the form $(\mathbb{A}^n, \varphi)$ where $\varphi(x_1, \dots, x_n) = (P_1(x_1), \dots, P_n(x_n))$. The most important case, from which the others may be deduced, is that of $n=2$. In the present work we give a sharper description of the skew-invariant curves in the case where $P_2 = P_1^{\tau}$ for some other automorphism of $K$ which commutes with $\sigma$. Specifically, if $P \in K[x]$ is a polynomial of degree greater than one which is not eventually skew-conjugate to a monomial or $\pm$ Chebyshev (i.e. $P$ is "nonexceptional") then skew-invariant curves in $(\mathbb{A}^2, (P, P^{\tau}))$ are horizontal, vertical, or skew-twists: described by equations of the form $y = \alpha^{\sigma^n} \circ P^{\sigma^{n-1}} \circ \dots \circ P^{\sigma} \circ P(x)$ or $x = \beta^{\sigma^{-1}} \circ P^{\tau \sigma^{-n-2}} \circ P^{\tau \sigma^{-n-3}} \circ \dots \circ P^{\tau}(y)$ where $P = \alpha \circ \beta$ and $P^{\tau} = \alpha^{\sigma^{n+1}} \circ \beta^{\sigma^n}$ for some integer $n$.
We use this new characterization to prove that a function $f(t)$ which satisfies $p$-Mahler equation of nonexceptional polynomial type, by which we mean $f(t^p) = P(f(t))$ for $p \in \mathbb{Q}_{+} \setminus \{1\}$ and $P \in \mathbb{C}(t)[x]$ a nonexceptional polynomial, is necessarily algebraically independent from functions satisfying $q$-Mahler equations with $q$ multiplicatively independent from $p$.
This is a report on joint work with Khoa Dang Nguyen and Alice Medvedev available at arXiv:2203.05083.
Thu, 05 May 2022

14:30 - 15:45
L4

### Approaches to the Skolem Problem

James Worrell
(University of Oxford)
Abstract

The Skolem Problem asks to decide whether a linearly recurrent sequence (LRS) over the rationals has a zero term.  It is sometimes considered as the halting problem for linear loops.   In this talk we will give an overview of two current approaches to establishing decidability of this problem.  First, we observe that the Skolem Problem for LRS with simple characteristic roots is decidable subject to the $p$-adic Schanuel conjecture and the exponential-local-global principle.  Next, we define a set $S$ of positive integers such that (i) $S$ has positive lower density and (ii) The Skolem Problem is decidable relative to $S$, i.e., one can effectively determine the set of all zeros of a given LRS that lie in $S$.

The talk is based on joint work with Y. Bilu, F. Luca, J. Ouaknine, D. Pursar, and J. Nieuwveld.

Thu, 05 May 2022

11:30 - 12:45
L6

### Defining valuations in ordered fields

Franziska Jahnke
(University of Münster)
Abstract

We study the definability of valuation rings in ordered fields (in the language of ordered rings). We show that any henselian valuation ring that is definable in the language of ordered rings is already definable in the language of rings. However, this does not hold when we drop the assumption of henselianity.

This is joint work with Philip Dittmann, Sebastian Krapp and Salma Kuhlmann.

Thu, 03 Mar 2022
16:00
Virtual

### Existentially closed measure-preserving actions of universally free groups

Isaac Goldbring
(University of California Irvine)
Abstract

In this talk, we discuss existentially closed measure preserving actions of countable groups.  A classical result of Berenstein and Henson shows that the model companion for this class exists for the group of integers and their analysis readily extends to cover all amenable groups.  Outside of the class of amenable groups, relatively little was known until recently, when Berenstein, Henson, and Ibarlucía proved the existence of the model companion for the case of finitely generated free groups.  Their proof relies on techniques from stability theory and is particular to the case of free groups.  In this talk, we will discuss the existence of model companions for measure preserving actions for the much larger class of universally free groups (also known as fully residually free groups), that is, groups which model the universal theory of the free group.  We also give concrete axioms for the subclass of elementarily free groups, that is, those groups with the same first-order theory as the free group.  Our techniques are ergodic-theoretic and rely on the notion of a definable cocycle.  This talk represents ongoing work with Brandon Seward and Robin Tucker-Drob.

Thu, 03 Mar 2022
11:30
C6

Mikołaj Bojańczyk
(University of Warsaw/University of Oxford)
Abstract

MSO can be used not only to accept/reject words, but also to transform words into other words, e.g. the doubling function w $\mapsto$ ww. The traditional model for this is called MSO transductions; the idea is that each position of the output word is interpreted in some position of the input word, and MSO is used to define the order on output positions and their labels. I will explain that an extension, where output positions are interpreted using $k$-tuples of input positions, is (a) is also well behaved; and (b) this is surprising.

Thu, 24 Feb 2022
11:45
Virtual

### Absolute Model Companionship, the AMC-spectrum of set theory, and the continuum problem

Matteo Viale
(University of Torino)
Abstract

We introduce a classification tool for mathematical theories based on Robinson's notion of model companionship; roughly the idea is to attach to a mathematical theory $T$ those signatures $L$ such that $T$ as axiomatized in $L$ admits a model companion. We also introduce a slight strengthening of model companionship (absolute model companionship - AMC) which characterize those model companionable $L$-theories $T$ whose model companion is axiomatized by the $\Pi_2$-sentences for $L$ which are consistent with the universal theory of any $L$-model of $T$.

We use the above to analyze set theory, and we show that the above classification tools can be used to extract (surprising?) information on the continuum problem.

Thu, 17 Feb 2022
11:30
Virtual

### Higher-order generalisations of stability and arithmetic regularity

Julia Wolf
(University of Cambridge)
Abstract

Previous joint work with Caroline Terry had identified model-theoretic stability as a sufficient condition for the existence of strong arithmetic regularity decompositions in finite abelian groups, pioneered by Ben Green around 2003.
Higher-order arithmetic regularity decompositions, based on Tim Gowers’s groundbreaking work on Szemerédi’s theorem in the late 90s, are an essential part of today's arithmetic combinatorics toolkit.
In this talk, I will describe recent joint work with Caroline Terry in which we define a natural higher-order generalisation of stability and prove that it implies the existence of particularly efficient higher-order arithmetic regularity decompositions in the setting of finite elementary abelian groups. If time permits, I will briefly outline some analogous results we obtain in the context of hypergraph regularity decompositions.

Thu, 02 Dec 2021

11:30 - 12:45
C2

### Existential rank and essential dimension of definable sets

Philip Dittmann
(TU Dresden)
Abstract

Several natural measures of complexity can be attached to an
existentially definable ("diophantine") subset of a field. One of these
is the minimal number of existential quantifiers required to define it,
while others are of a more geometric nature. I shall define these
measures and discuss interesting interactions and behaviours, some of
which depend on properties of the field (e.g. imperfection and
ampleness). We shall see for instance that the set of n-tuples of field
elements consisting of n squares is usually definable with a single
quantifier, but not always. I will also discuss connections with
Hilbert's 10th Problem and a number of open questions.
This is joint work with Nicolas Daans and Arno Fehm.

Thu, 25 Nov 2021
11:30
C3

### Relating Structure to Power

Samson Abramsky
(University College London)
Further Information

This is an in-person seminar.

Abstract

In this talk, we describe some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics.

The motivations for this work come from Computer Science, but there may be something of interest for model theorists and other logicians.

The basic setting involves studying the category of relational structures via a resource-indexed family of adjunctions with some process category - which unfolds relational structures into treelike forms, allowing natural resource parameters to be assigned to these unfoldings.

One basic instance of this scheme allows us to recover, in a purely structural, syntax-free way:

- the Ehrenfeucht-Fraisse game

- the quantifier rank fragments of first-order logic

- the equivalences on structures induced by (i) the quantifier rank fragments, (ii) the restriction to the existential-positive part, and (iii) the extension with counting quantifiers

- the combinatorial parameter of tree-depth (Nesetril and Ossona de Mendez).

Another instance recovers the k-pebble game, the finite-variable fragments, the corresponding equivalences, and the combinatorial parameter of treewidth.

Other instances cover modal, guarded and hybrid fragments, generalized quantifiers, and a wide range of combinatorial parameters.

This whole scheme has been axiomatized in a very general setting, of arboreal categories and arboreal covers.

Beyond this basic level, a landscape is beginning to emerge, in which structural features of the resource categories, adjunctions and comonads are reflected in degrees of logical and computational tractability of the corresponding languages.

Examples include semantic characterisation and preservation theorems, Lovasz-type results on  isomorphisms, and classification of constraint satisfaction problems.

Thu, 18 Nov 2021
11:30
Virtual

### Some model theory of the curve graph

Javier de la Nuez González
(University of the Basque Country (UPV/EHU))
Abstract

The curve graph of a surface of finite type is a fundamental object in the study of its mapping class group both from the metric and the combinatorial point of view. I will discuss joint work with Valentina Disarlo and Thomas Koberda where we conduct a thorough study of curve graphs from the model theoretic point of view, with particular emphasis in the problem of interpretability between different curve graphs and other geometric complexes.

Thu, 04 Nov 2021
11:30
Virtual

### Martin's Maximum^++ implies the P_max axiom (*) -- Part II

David Aspero
(University of East Anglia)
Abstract

(This is Part II of a two-part talk.)

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and "consistent" needs to mean "consistent in a strong sense". It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. Our result builds upon earlier work of R. Jensen and (ultimately) Keisler's "consistency properties".

Thu, 28 Oct 2021
11:30
Virtual

### Martin's Maximum^++ implies the P_max axiom (*) -- Part I

Ralf Schindler
(University of Münster)
Abstract

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and "consistent" needs to mean "consistent in a strong sense". It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. Our result builds upon earlier work of R. Jensen and (ultimately) Keisler's "consistency properties".

(This is Part I of a two-part talk.)

Thu, 21 Oct 2021
15:00
Virtual

### The stable boundary

Maryanthe Malliaris
(University of Chicago)
Abstract

This talk will be about the stable boundary seen from different recent points of view.

Thu, 14 Oct 2021
11:30
Virtual

### Forking independence in the free group

Chloé Perin
(The Hebrew University of Jerusalem)
Abstract

Sela proved in 2006 that the (non abelian) free groups are stable. This implies the existence of a well-behaved forking independence relation, and raises the natural question of giving an algebraic description in the free group of this model-theoretic notion. In a joint work with Rizos Sklinos we give such a description (in a standard fg model F, over any set A of parameters) in terms of the JSJ decomposition of F over A, a geometric group theoretic tool giving a group presentation of F in terms of a graph of groups which encodes much information about its automorphism group relative to A. The main result states that two tuples of elements of F are forking independent over A if and only if they live in essentially disjoint parts of such a JSJ decomposition.

Thu, 17 Jun 2021
11:30
Virtual

### Compressible types in NIP theories

Itay Kaplan
(The Hebrew University of Jerusalem)
Abstract

I will discuss compressible types and relate them to uniform definability of types over finite sets (UDTFS), to uniformity of honest definitions and to the construction of compressible models in the context of (local) NIP. All notions will be defined during the talk.
Joint with Martin Bays and Pierre Simon.

Fri, 04 Jun 2021
11:30
Virtual

### Interpretable fields in certain expansions of valued fields

Kobi Peterzil
(University of Haifa)
Abstract

(Joint with Y. Halevi and A. Hasson.) We consider two kinds of expansions of a valued field $K$:

(1) A $T$-convex expansion of real closed field, for $T$ a polynomially bounded o-minimal expansion of $K$.

(2) A $P$-minimal field $K$ in which definable functions are PW differentiable.

We prove that any interpretable infinite field $F$ in $K$ is definably isomorphic to a finite extension of either $K$ or, in case (1), its residue field $k$. The method we use bypasses general elimination of imaginaries and is based on analysis of one dimensional quotients of the form $I=K/E$ inside $F$ and their connection to one of 4 possible sorts: $K$, $k$ (in case (1)), the value group, or the quotient of $K$ by its valuation ring. The last two cases turn out to be impossible and in the first two cases we use local differentiability to embed $F$ into the matrix ring over $K$ (or $k$).

Thu, 27 May 2021
11:30
Virtual

### Coarse approximate subgroups in weak general position and Elekes-Szabó problems for nilpotent groups

Zou Tingxiang
(University of Münster)
Abstract

The Elekes-Szabó's theorem says very roughly that if a complex irreducible subvariety V of X*Y*Z has ''too many'' intersection with cartesian products of finite sets, then V is in correspondence with the graph of multiplication of an algebraic group G. It was noticed by Breuillard and Wang that the algebraic group G must be abelian. There is a constraint for the finite sets witnessing ''many'' intersections with V, namely a condition called in general position, which plays a key role in forcing the group to be abelian.  In this talk, I will present a result which shows that in the case of the graph of complex algebraic groups, with a weaker general position assumption, nilpotent groups will appear. More precisely, for a connected complex algebraic group G the following are equivalent:

1. The graph of G has ''many'' intersections with finite sets in weak general position;

2. G is nilpotent;

3. The ultrapower of G has a pseudofinite coarse approixmate subgroup in weak general position.

Surprisingly, the proof of the direction from 2 to 3 invokes some form of generic Mordell-Lang theorem for commutative complex algebraic groups.

This is joint work with Martin Bays and Jan Dobrowolski.

Thu, 20 May 2021
11:30
Virtual

### Chromatic numbers of Stable Graphs

Yatir Halevi
(Ben Gurion University of the Negev)
Abstract
This is joint work with Itay Kaplan and Saharon Shelah.
Given a graph $(G,E)$, its chromatic number is the smallest cardinal $\kappa$ of a legal coloring of the vertices. We will mainly concentrate on the following strong form of Taylor's conjecture:
If $G$ is an infinite graph with chromatic number$\geq \aleph_1$ then it contains all finite subgraphs of $Sh_n(\omega)$ for some $n$, where $Sh_n(\omega)$ is the $n$-shift graph (which we will introduce).

The conjecture was disproved by Hajnal-Komjath. However, we will sketch a proof for a variant of this conjecture for $\omega$-stable\superstable\stable graphs. The proof uses a generalization of  Ehrenfeucht-Mostowski models, which we will (hopefully) introduce.
Thu, 13 May 2021
16:30
Virtual

### Applications of generalized indiscernible sequences

Lynn Scow
(California State University San Bernardino)
Abstract

We survey some of the applications of generalized indiscernible sequences, both in model theory and in structural Ramsey theory.  Given structures $A$ and $B$, a semi-retraction is a pair of  quantifier-free type respecting maps $f: A \rightarrow B$ and $g: B \rightarrow A$ such that $g \circ f: A \rightarrow A$ is quantifier-free type preserving, i.e. an embedding.  In the case that $A$ and $B$ are locally finite ordered structures, if $A$ is a semi-retraction of $B$ and the age of $B$ has the Ramsey property, then the age of $A$ has the Ramsey property.

Wed, 05 May 2021
17:00
Virtual

### Existential Closedness in Arithmetic Geometry

Sebastian Eterović
(University of California Berkeley)
Abstract

There are many open conjectures about the algebraic behaviour of transcendental functions in arithmetic geometry, one of which is the Existential Closedness problem. In this talk I will review recent developments made on this question: the cases where we have unconditional existence of solutions, the conditional existence of generic solutions (depending on the conjecture of periods and Zilber-Pink), and even a few cases of unconditional existence of generic solutions. Many of the results I will mention are joint work with (different subsets of) Vahagn Aslanyan, Jonathan Kibry, Sebastián Herrero, and Roy Zhao.

Thu, 11 Mar 2021
11:30
Virtual

### On pseudo-analytic and adelic models of Shimura curves (joint with Chris Daw)

Boris Zilber
((Oxford University))
Abstract

I will discuss the multi-sorted structure of analytic covers H -> Y(N), where H is the upper half-plane and Y(N) are the N-level modular curves, all N, in a certain language, weaker than the language applied by Adam Harris and Chris Daw.  We define a certain locally modular reduct of the structure which is called "pure" structure - an extension of the structure of special subvarieties.
The problem of non-elementary categorical axiomatisation for this structure is closely related to the theory of "canonical models for Shimura curves", in particular, the description of Gal_Q action on the CM-points of the Y(N). This problem for the case of curves is basically solved (J.Milne) and allows the beautiful interpretation in our setting:  the abstract automorphisms of the pure structure on CM-points are exactly the automorphisms induced by Gal_Q.  Using this fact and earlier theorem of Daw and Harris we prove categoricity of a natural axiomatisation of the pseudo-analytic structure.
If time permits I will also discuss a problem which naturally extends the above:  a categoricity statement for the structure of unramified analytic covers H -> X, where X runs over all smooth curves over a given number field.