Forthcoming events in this series


Thu, 08 Feb 2024

11:00 - 12:00
C3

Model companions of fields with no points in hyperbolic varieties

Michal Szachniewicz
(University of Oxford)
Abstract

This talk is based on a joint work with Vincent Jinhe Ye. I will define various classes of hyperbolic varieties (Broody hyperbolic, algebraically hyperbolic, bounded, groupless) and discuss existence of model companions of classes of fields that exclude them. This is related to moduli spaces of maps to hyperbolic varieties and to the (open) question whether the above mentioned hyperbolicity notions are in fact equivalent.

Thu, 01 Feb 2024

11:00 - 12:00
C3

Non-archimedean equidistribution and L-polynomials of curves over finite fields

Francesco Ballini
(University of Oxford)
Abstract

Let q be a prime power and let C be a smooth curve defined over F_q. The number of points of C over the finite extensions of F_q are determined by the Zeta function of C, which can be written in the form P_C(t)/((1-t)(1-qt)), where P_C(t) is a polynomial of degree 2g and g is the genus of C; this is often called the L-polynomial of C. We use a Chebotarev-like statement (over function fields instead of Z) due to Katz in order to study the distribution, as C varies, of the coefficients of P_C(t) in a non-archimedean setting.

Thu, 25 Jan 2024

11:00 - 12:00
C3

Pre-seminar meeting on motivic integration

Margaret Bilu
(University of Oxford)
Abstract

This is a pre-seminar meeting for Margaret Bilu's talk "A motivic circle method", which takes place later in the day at 5PM in L3.

Thu, 30 Nov 2023

11:00 - 12:00
C6

Homotopy type of categories of models

Jinhe Ye
(University of Oxford)
Abstract

For a complete theory T, Lascar associated with it a Galois group which we call the Lacsar group. We will talk about some of my work on recovering the Lascar group as the fundamental group of Mod(T) and some recent progress in understanding the higher homotopy groups.

Thu, 16 Nov 2023

11:00 - 12:00
C6

On a proposed axiomatisation of the maximal abelian extension of the p-adic numbers

Leo Gitin
(University of Oxford)
Abstract

The local Kronecker-Weber theorem states that the maximal abelian extension of p-adic numbers Qp is obtained from this field by adjoining all roots of unity. In 2018, Koenigsmann conjectured that the maximal abelian extension of Qp is decidable. In my talk, we will discuss Koenigsmann's proposed axiomatisation. In contrast, the maximal unramified extension of Qp is known to be decidable, admitting a complete axiomatisation by an informed but simple set of axioms (this is due to Kochen). We explain how the question of completeness can be reduced to an Ax-Kochen-Ershov result in residue characteristic 0 by the method of coarsening.

Thu, 09 Nov 2023

11:00 - 12:00
C6

Unlikely Double Intersections in a power of a modular curve (Part 2)

Francesco Ballini
(University of Oxford)
Abstract

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

Thu, 02 Nov 2023

11:00 - 12:00
C6

Unlikely Double Intersections in a power of a modular curve (Part 1)

Francesco Ballini
(University of Oxford)
Abstract

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

Thu, 19 Oct 2023

11:00 - 12:00
C6

New ideas in Arakelov intersection theory

Michał Szachniewicz
(Mathematical Insitute, Oxford)
Abstract

I will give an overview of new ideas showing up in arithmetic intersection theory based on some exciting talks that appeared at the very recent conference "Global invariants of arithmetic varieties". I will also outline connections to globally valued fields and some classical problems.

Thu, 04 May 2017
11:00
C5

On fields with the absolute Galois group of Q

Jochen Koenigsmann
(Oxford)
Abstract

.. showing that a field K is isomorphic to Q if it has the same absolute Galois group and if it satisfies a very small additional condition (very similar to my talk 2 years ago).

Tue, 07 Mar 2017
11:00
C5

Unlikely Intersections in families of elliptic curves

Laura Capuano
(Oxford)
Abstract


What makes an intersection likely or unlikely? A simple dimension count shows that two varieties of dimension r and s are non "likely" to intersect if r < codim s, unless there are some special geometrical relations among them. A series of conjectures due to Bombieri-Masser-Zannier, Zilber and Pink rely on this philosophy. I will speak about a joint work with F. Barroero (Basel) in this framework in the special case of a curve in a family of elliptic curves. The proof is based on Pila-Zannier method, combining diophantine ingredients with a refinement of a theorem of Pila and Wilkie about counting rational points in sets definable in o-minimal structures.
   Everyone welcome!
 

Thu, 23 Feb 2017
11:00
C5

Non-reduced schemes and Zariski Geometries

Alfonso Ruiz
(Oxford)
Abstract

Using results by Eisenbud, Schoutens and Zilber I will propose a model theoretic structure that aims to capture the algebra (or geometry) of a non reduced scheme over an algebraically closed field. 

Thu, 02 Feb 2017
11:00
C4

Model Theoretic Aspects of Gelfand-Naimark duality.

Nicholas Wentzlaff-Eggebert
(Oxford)
Abstract


Abstract: We will consider a model theoretic approach to Gelfand-Naimark duality, from the point of view of (generalized) Zariski structures. In particular we will show quantifier elimination for compact Hausdorff spaces in the natural Zariski language. Moreover we may see a slightly unusual construction and tweak to the language, which improves stability properties of the structures.
 

Thu, 19 Jan 2017
11:00
C5

Towards a Ladder Theorem for Specialisations

Ugur Efem
Abstract


In this talk I will present some answers to the question when every specialisation from a \kappa-saturated extension of 
a Zariski structure is \kappa-universal? I will show that for algebraically closed fields, all specialisations from a \kappa-
saturated extension is \kappa-universal. More importantly, I will consider this question for finite and infinite covers of
Zariski structures. In these cases I will present a counterexample to show that there are covers of Zariski structures 
which have specialisations from a \kappa-saturated extension that are not \kappa-universal. I will present some natural 
conditions on the fibres under which all specialisations from a \kappa-saturated extension of a cover is \kappa-universal. 
I will explain how this work points towards a prospective Ladder Theorem for Specialisations and explain difficulties and 
further works that needs to be considered.
 

Thu, 16 Jun 2016

11:00 - 15:45
C3

'Around quantum j-mappings (model theory and sheaves)'.

Andres Villaveces
(Bogota)
Abstract
Abstract: finding a "non-commutative limit" of the j-invariant (to real numbers, in a way that captures reasonably well the connection with extensions of number fields) has prompted several approaches (Manin-Marcolli, Castaño-Gendron). I will describe one of these approaches in a brief way, and I will make some connections to the model theory of sheaves.
Thu, 28 Apr 2016
11:00
C5

"p-adica nova"

Jochen Koenigsmann
(Oxford)
Abstract

This will be a little potpourri containing some of the recent developments on the model theory of F_p((t)) and of algebraic extensions of Q_p.