### Some model theory of Quadratic Geometries

## Abstract

Forthcoming events in this series

Thu, 16 May 2024

17:00 -
18:00

L3

Charlotte Kestner

(Imperial College London)

I will introduce the theories of orthogonal spaces and quadratic geometries over infinite fields, giving some background on Lie coordinatisable structures, and bilinear forms over infinite fields. I will then go on to explain the quantifier elimination for these structures, and discuss the axiomatisation of their pseudo-finite completions and model companions. This is joint work in progress with Nick Ramsey.

Thu, 09 May 2024

17:00 -
18:00

L3

Jan Dobrowolski

(University of Manchester)

I will report on a joint work in progress with F. Gallinaro and R. Mennuni in which we aim to understand the (non-elementary) class of existentially closed valued difference fields (of equicharacteristic zero). As our approach relies on our earlier results with Mennuni about automorphisms of ordered abelian groups, I will start by briefly overviewing those.

Thu, 02 May 2024

17:00 -
18:00

L3

Silvain Rideau-Kikuchi

(École Normale Supérieure )

(Joint work with S. Montenegro)

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 another 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.

Thu, 25 Apr 2024

17:00 -
18:00

L3

Elena Bunina

(Bar-Ilan University)

We prove that any adjoint Chevalley group over an arbitrary commutative ring is regularly bi-interpretable with this ring. The same results hold for central quotients of arbitrary Chevalley groups and for Chevalley groups with bounded generation.

Also, we show that the corresponding classes of Chevalley groups (or their central quotients) are elementarily definable and even finitely axiomatizable.

Thu, 07 Mar 2024

17:00 -
18:00

Itay Glazer

(University of Oxford)

Let f:X-->Y be a polynomial map between smooth varieties, and let mu be a smooth, compactly supported measure on X(F), where F is a local field. An interesting phenomenon is that bad singularities of f manifest themselves in poor analytic behavior of the pushforward f_*(mu) of mu by f.

I will discuss this phenomenon in two settings; the first is when f:A^n-->A^m is a polynomial map between affine spaces and mu is the Haar measure on Z_p^n, and the second is when f:G^2-->G is a word map (e.g. the commutator map (g,h)-->ghg^(-1)h^(-1)) between simple algebraic groups, and mu is a Haar measure on G(Z_p).

In these cases (and in other "real life situations"), mu and consequently f_*(mu) are constructible measures in the sense of Cluckers-Loeser motivic integration. We utilize this fact to show that the analytic behavior of f_*(mu) cannot be too bad, leading to geometric and probabilistic applications.

Based on joint works with Yotam Hendel and Raf Cluckers.

Thu, 29 Feb 2024

17:00 -
18:00

Christian d'Elbée

(School of Mathematics, University of Leeds)

A structure is omega-categorical if its theory has a unique countable model (up to isomorphism). We will survey some old results concerning the Apps-Wilson structure theory for omega-categorical groups and state a conjecture of Wilson from the 80s on omega-categorical characteristically simple groups. We will also discuss the analogous of Wilson’s conjecture for Lie algebras and present some connections with the restricted Burnside problem.

Thu, 22 Feb 2024

17:00 -
18:00

Asaf Karagila (Leeds)

We can compare the relative sizes of sets by using injections or (partial) surjections, but without the axiom of choice we cannot prove that every two sets can be compared. We can use the ordinals to define a notion of size which allows us to determine whether a set is "large" or "small" relative to another. The first is defined by the Hartogs number, which is the least ordinal which does not inject into the set; the second is the Lindenbaum number of a set, which is the first ordinal which is not an image of the set. In this talk we will discuss some basic properties of these numbers and some basic historical results.

In a new work with Calliope Ryan-Smith we showed that given any pair of (infinite) cardinals, we can onstruct a symmetric extension in which there is a set whose Hartogs is the smaller and the Lindenbaum is the larger. Moreover, using the techniques of iterated symmetric extensions, we can realise all possible pairs in a single model.

This work appears on arXiv: https://arxiv.org/abs/2309.11409

Thu, 15 Feb 2024

17:00 -
18:00

Boris Zilber

(University of Oxford)

I am going to discuss main results of my paper "Physics over a finite field and Wick rotation", arxiv 2306.15698. It introduces a structure over a pseudo-finite field which might be of interest in Foundations of Physics. The main theorem establishes an analogue of the polar co-ordinate system in the pseudo-finite field. A stability classification status of the structure is an open question.

Thu, 01 Feb 2024

17:00 -
18:00

L3

Mark Kamsma

(Queen Mary University of London)

Positive logic is a generalisation of full first-order logic, where negation is not built in, but can be added as desired. In joint work with Jan Dobrowolski we succesfully generalised the recent development on Kim-independence in NSOP1 theories to the positive setting. One of the important theorems in this development is the independence theorem, whose statement is very similar to the well-known statement for simple theories, and allows us to amalgamate independent types. In this talk we will have a closer look at the proof of this theorem, and what needs to be changed to make the proof work in positive logic compared to full first-order logic.

Thu, 25 Jan 2024

17:00 -
18:00

L3

Margaret Bilu

(Institut de Mathématiques de Bordeaux)

The Hardy–Littlewood circle method is a well-known analytic technique that has successfully solved several difficult counting problems in number theory. More recently, a version of the method over function fields, combined with spreading out techniques, has led to new results about the geometry of moduli spaces of rational curves on hypersurfaces of low degree. I will explain how one can implement a circle method with an even more geometric flavour, where the computations take place in a suitable Grothendieck ring of varieties, leading thus to a more precise description of the geometry of the above moduli spaces. This is joint work with Tim Browning.

Thu, 30 Nov 2023

17:00 -
18:00

L3

Chris Daw

(University of Reading)

I will recall the Zilber-Pink conjecture for Shimura varieties and give my perspective on current progress towards a proof.

Thu, 23 Nov 2023

17:00 -
18:00

Jamshid Derakhshan

(Oxford)

In this talk I will present joint work with Ehud Hrushovski on imaginaries in the ring of adeles and more generally in products and restricted products of structures (including the generalised products of Feferman-Vaught).

We prove a general theorem on weak elimination of imaginaries in products with respect to additional sorts which we deduce from an elimination of imaginaries for atomic and atomless Booleanizations of a theory. This combined with uniform elimination of imaginaries for p-adic numbers in a language with extra sorts as p-adic lattices proved first by Hrushovski-Martin-Rideau and more recently by Hils-Rideau-Kikuchi in a slightly different language, yields weak elimination of imaginaries for the ring of adeles in a language with extra sorts as adelic versions of the p-adic lattices.

The proofs of the general results on products use Boolean valued model theory, stability theory, analysis of definable groups and liaison groups, and descriptive set theory of smooth Borel equivalence relations including Harrington-Kechris-Louveau and Glimm-Efros dichotomy.

Thu, 09 Nov 2023

17:00 -
18:00

L3

Gareth Jones

(Manchester)

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves contains a Zariski-dense set of special points if and only if it is special. I'll explain this result, and discuss an effective version that Gal Binyamini, Harry Schmidt, Margaret Thomas and I proved.

Thu, 02 Nov 2023

17:00 -
18:00

L3

Martin Bays (Oxford)

I will present a generalisation of the Elekes-Szabó result, that any ternary algebraic relation in characteristic 0 having large intersections with (certain) finite grids must essentially be the graph of a group law, to a version where one obtains an algebraic group action. In the end the conclusion will be similar, but with weaker assumptions. This is recent work with Tingxiang Zou.

Thu, 26 Oct 2023

17:00 -
18:00

L3

Alex Wilkie (Manchester/Oxford)

A key ingredient in the proof of the model completeness of the real exponential field was a valuation inequality for polynomially bounded o-minimal structures. I shall briefly describe the argument, and then move on to the complex exponential field and Zilber's quasiminimality conjecture for this structure. Here, one can reduce the problem to that of establishing an analytic continuation property for (complex) germs definable in a certain o-minimal expansion of the real field and in order to study this question I propose notions of "complex Hardy fields" and "complex valuations". Here, the value group is not necessarily ordered but, nevertheless, one can still prove a valuation inequality.

Thu, 15 Jun 2023

17:00

17:00

L4

Konstantinos Kartas

(IMJ-PRG/Sorbonne Université)

Given a perfectoid field, we find an elementary extension and a henselian defectless valuation on it, whose value group is divisible and whose residue field is an elementary extension of the tilt. This specializes to the almost purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified fields, which also uncovers some new model-theoretic phenomena in positive characteristic. Notably, we get that the perfect hull of Fp(t)^h is an elementary substructure of the perfect hull of Fp((t)). Joint work with Franziska Jahnke.

Thu, 08 Jun 2023

17:00

17:00

L4

Nadja Hempel

(HHU Düsseldorf)

(joint with Chernikov) 1-dependent theories, better known as NIP theories, are the first class of the strict hierarchy of n-dependent theories. The random n-hypergraph is the canonical object which is n-dependent but not (n−1)-dependent. We proved the existence of strictly n-dependent groups for all natural numbers n. On the other hand, there are no known examples of strictly n-dependent fields and we conjecture that there aren’t any.

We were interested which properties of groups and fields for NIP theories remain true in or can be generalized to the n-dependent context. A crucial fact about (type-)definable groups in NIP theories is the absoluteness of their connected components. Our first aim is to give examples of n-dependent groups and discuss a adapted version of absoluteness of the connected component. Secondly, we will review the known properties of NIP fields and see how they can be generalized.

Thu, 01 Jun 2023

17:00

17:00

L4

Gareth Jones

(University of Manchester)

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves is special if and only if it contains a Zariski-dense set of special points. I'll discuss joint work with Gal Binyamini, Harry Schmidt, and Margaret Thomas in which we use pfaffian methods to obtain an effective uniform version of Manin-Mumford for products of CM elliptic curves. Using this we then prove an effective version of Habegger's result.

Thu, 25 May 2023

17:00

17:00

L3

Sebastian Eterović

(University of Leeds)

The Zilber-Pink conjecture predicts that if V is a proper subvariety of an arithmetic variety S (e.g. abelian variety, Shimura variety, others) not contained in a proper special subvariety of V, then the “unlikely intersections” of V with the proper special subvarieties of S are not Zariski dense in V. In this talk I will present a strong counterpart to the Zilber-Pink conjecture, namely that under some natural conditions, likely intersections are in fact Euclidean dense in V. This is joint work with Tom Scanlon.

Thu, 18 May 2023

17:00

17:00

L3

Joel David Hamkins

(University of Notre Dame)

I shall present a new flexible method showing that every countable model of PA admits a pointwise definable end-extension, one in which every point is definable without parameters. Also, any model of PA of size at most continuum admits an extension that is Leibnizian, meaning that any two distinct points are separated by some expressible property. Similar results hold in set theory, where one can also achieve V=L in the extension, or indeed any suitable theory holding in an inner model of the original model.

Thu, 11 May 2023

17:00

17:00

L3

Francesco Gallinaro

(University of Freiburg)

A conjecture due to Zilber predicts that the complex exponential field is quasiminimal: that is, that all subsets of the complex numbers that are definable in the language of rings expanded by a symbol for the complex exponential function are countable or cocountable.

Zilber showed that this conjecture would follow from Schanuel's Conjecture and an existential closedness type property asserting that certain systems of exponential-polynomial equations can be solved in the complex numbers; later on, Bays and Kirby were able to remove the dependence on Schanuel's Conjecture, shifting all the focus to the existence of solutions. In this talk, I will discuss recent work about the quasiminimality of a reduct of the complex exponential field, that is, the complex numbers expanded by multivalued power functions. This is joint work with Jonathan Kirby.

Thu, 04 May 2023

17:00

17:00

L3

Shai Haran

(Technion – Israel Institute of Technology)

The usual language of algebraic geometry is not appropriate for Arithmetical geometry: addition is singular at the real prime. We developed two languages that overcome this problem: one replace rings by the collection of “vectors” or by bi-operads and another based on “matrices” or props. These are the two languages of [Har17], but we omit the involutions which brings considerable simplifications. Once one understands the delicate commutativity condition one can proceed following Grothendieck footsteps exactly. The square matrices, when viewed up to conjugation, give us new commutative rings with Frobenius endomorphisms.

Thu, 27 Apr 2023

17:00

17:00

L4

Tomás Ibarlucía

(Université Paris Diderot)

Affine logic is the fragment of continuous logic in which the connectives are limited to affine functions. I will discuss the basics of this logic, first studied by Bagheri, and present the results of a recent joint work with I. Ben Yaacov and T. Tsankov in which we initiate the study of extreme types and extremal models in affine logic.

In particular, I will discuss an extremal decomposition result for models of simplicial affine theories, which generalizes the ergodic decomposition theorem.

Thu, 16 Mar 2023

17:00

17:00

L3

Martin Bays

(Universitat Munster)

In their seminal 2012 paper, Elekes and Szabó found that a certain weak combinatorial non-expansion property of an algebraic relation suffices to trigger the group configuration theorem, showing that only (approximate subgroups of) algebraic groups can be responsible for it. I will discuss some more recent variations and elaborations on this result, focusing on the case of ternary relations on varieties of dimension >1.

Thu, 09 Mar 2023

17:00

17:00

L3

Philipp Hieronymi

(Universitat Bonn)

Let $k,l>1$ be two multiplicatively independent integers. A subset $X$ of $\mathbb{N}^n$ is $k$-recognizable if the set of $k$-ary representations of $X$ is recognized by some finite automaton. Cobham's famous theorem states that a subset of the natural numbers is both $k$-recognizable and $l$-recognizable if and only if it is Presburger-definable (or equivalently: semilinear). We show the following strengthening. Let $X$ be $k$-recognizable, let $Y$ be $l$-recognizable such that both $X$ and $Y$ are not Presburger-definable. Then the first-order logical theory of $(\mathbb{N},+,X,Y)$ is undecidable. This is in contrast to a well-known theorem of Büchi that the first-order logical theory of $(\mathbb{N},+,X)$ is decidable. Our work strengthens and depends on earlier work of Villemaire and Bès. The essence of Cobham's theorem is that recognizability depends strongly on the choice of the base $k$. Our results strengthens this: two non-Presburger definable sets that are recognizable in multiplicatively independent bases, are not only distinct, but together computationally intractable over Presburger arithmetic. This is joint work with Christian Schulz.

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