Forthcoming events in this series


Thu, 06 Dec 2012

17:00 - 18:00
L3

An application of proof theory to lattice-ordered groups

George Metcalfe
(Bern)
Abstract

(Joint work with Nikolaos Galatos.) Proof-theoretic methods provide useful tools for tackling problems for many classes of algebras. In particular, Gentzen systems admitting cut-elimination may be used to establish decidability, complexity, amalgamation, admissibility, and generation results for classes of residuated lattices corresponding to substructural logics. However, for classes of algebras bearing a family resemblance to groups, such methods have so far met only with limited success. The main aim of this talk will be to explain how proof-theoretic methods can be used to obtain new syntactic proofs of two core theorems for the class of lattice-ordered groups: namely, Holland's result that this class is generated as a variety by the lattice-ordered group of order-preserving automorphisms of the real numbers, and the decidability of the word problem for free lattice-ordered groups.

Thu, 29 Nov 2012

17:00 - 18:00
L3

Valued difference fields and NTP2

Martin Hils
(Paris)
Abstract

(Joint work with Artem Chernikov.) In the talk, we will first recall some basic results on valued difference fields, both from an algebraic and from a model-theoretic point of view. In particular, we will give a description, due to Hrushovski, of the theory VFA of the non-standard Frobenius acting on an algebraically closed valued field of residue characteristic 0, as well as an Ax-Kochen-Ershov type result for certain valued difference fields which was proved by Durhan. We will then present a recent work where it is shown that VFA does not have the tree property of the second kind (i.e., is NTP2); more generally, in the context of the Ax-Kochen-Ershov principle mentioned above, the valued difference field is NTP2 iff both the residue difference field and the value difference group are NTP2. The property NTP2 had already been introduced by Shelah in 1980, but only recently it has been shown to provide a fruitful ‘tameness’ assumption, e.g. when dealing with independence notions in unstable NIP theories (work of Chernikov-Kaplan).

Thu, 22 Nov 2012

17:00 - 18:00
L3

A non-desarguesian projective plane of analytic origin

Boris Zilber
(Oxford)
Abstract
(This is a joint result with Katrin Tent.) We construct a series of new omega-stable non-desarguesian projective planes, including ones of Morley rank 2, 
avoiding a direct use of Hrushovski's construction. Instead we make use of the field of complex numbers with a holomorphic function  (Liouville function) which is an omega-stable structure by results of A.Wilkie and P.Koiran.  We first find a pseudo-plane interpretable in the above analytic structure and then "collapse" the pseudo-plane to a projective plane applying a modification of Hrushovski's mu-function. 
Thu, 08 Nov 2012

17:00 - 18:00
L3

Topological dynamics and model theory of SL(2,R)

Davide Penazzi
(Leeds)
Abstract

Newelski suggested that topological dynamics could be used to extend "stable group theory" results outside the stable context. Given a group G, it acts on the left on its type space S_G(M), i.e. (G,S_G(M)) is a G-flow. If every type is definable, S_G(M) can be equipped with a semigroup structure *, and it is isomorphic to the enveloping Ellis semigroup of the flow. The topological dynamics of (G,S_G(M)) is coded in the Ellis semigroup and in its minimal G-invariant subflows, which coincide with the left ideals I of S_G(M). Such ideals contain at least an idempotent r, and r*I forms a group, called "ideal group". Newelski proved that in stable theories and in o-minimal theories r*I is abstractly isomorphic to G/G^{00} as a group. He then asked if this happens for any NIP theory. Pillay recently extended the result to fsg groups; we found instead a counterexample to Newelski`s conjecture in SL(2,R), for which G/G^{00} is trivial but we show r*I has two elements. This is joint work with Jakub Gismatullin and Anand Pillay.

Thu, 18 Oct 2012

17:00 - 18:00
L3

Embeddings of the spaces of the form C(K)

Mirna Dzamonja (UEA)
Abstract

We discuss the question of the existence of the smallest size of a family of Banach spaces of a given density which embeds all Banach spaces of that same density. We shall consider two kinds of embeddings, isometric and isomorphic. This type of question is well studied in the context of separable spaces, for example a classical result by Banach states that C([0,1]) embeds all separable Banach spaces. However, the nonseparable case involves a lot of set theory and the answer is independent of ZFC.

Thu, 11 Oct 2012

17:00 - 18:00
L3

Plus ultra

Frank Wagner (Lyon)
Abstract

I shall present a very general class of virtual elements in a structure, ultraimaginaries, and analyse their model-theoretic properties.

Thu, 14 Jun 2012

17:00 - 18:00
L3

Algebraic closure in pseudofinite fields

Özlem Beyarslan (Bogazici)
Abstract

A pseudofinite field is a perfect pseudo-algebraically closed (PAC) field which

has $\hat{\mathbb{Z}}$ as absolute Galois group. Pseudofinite fields exists and they can

be realised as ultraproducts of finite fields. A group $G$ is geometrically

represented in a theory $T$ if there are modles $M_0\prec M$ of $T$,

substructures $A,B$ of $M$, $B\subset acl(A)$, such that $M_0\le A\le B\le M$

and $Aut(B/A)$ is isomorphic to $G$. Let $T$ be a complete theory of

pseudofinite fields. We show that, geometric representation of a group whose order

is divisibly by $p$ in $T$ heavily depends on the presence of $p^n$'th roots of unity

in models of $T$. As a consequence of this, we show that, for almost all

completions of the theory of pseudofinite fields, over a substructure $A$, algebraic

closure agrees with definable closure, if $A$ contains the relative algebraic closure

of the prime field. This is joint work with Ehud Hrushovski.

Thu, 24 May 2012

17:00 - 18:00
L3

S-independence in NIP theories

Pierre Simon (Ecole Normale Superiore)
Abstract

I will explain how to define a notion of stable-independence in NIP

theories, which is an attempt to capture the "stable part" of types.

Thu, 17 May 2012

17:00 - 18:00
L3

TBA

*Cancelled*
Thu, 10 May 2012

17:00 - 18:00
L3

Uniformly defining valuation rings in Henselian valued fields with finite and pseudo-finite residue field

Jamshid Derakhshan
Abstract
This is joint work with Raf Cluckers, Eva Leenknegt, and Angus Macintyre.

We give a first-order definition, in the ring language, of the ring of p-adic integers inside the field p-adic numbers which works uniformly for all p and for valuation rings of all finite field extensions and of all local fields of positive characteristic p, and in many other Henselian valued fields as well. The formula canbe taken existential-universal in the ring language. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula. For any fixed general p-adic field we give an existential formula in the ring language which defines the valuation ring.

We also state some connections to some open problems.

Thu, 26 Apr 2012

17:00 - 18:00
L3

Connecting Schanuel's Conjecture to Shapiro's Conjecture

Angus Macintyre (QMUL)
Abstract

Shapiro's Conjecture says that two classical exponential polynomials over the complexes can have infinitely many common zeros only for algebraic reasons. I will explain the history of this, the connection to Schanuel's Conjecture, and sketch a proof for the complexes using Schanuel, as well as an unconditional proof for Zilber's fields.

Thu, 08 Mar 2012

17:00 - 18:00
L3
Thu, 01 Mar 2012

17:00 - 18:00
L3

Imaginaries in valued fields with analytic structure

Dugald Macpherson (Leeds)
Abstract

I will give an overview of the description of imaginaries in algebraically closed (and some other) valued fields, and then discuss the related issue for valued fields with analytic structure (in the sense of Lipshitz-Robinson, and Denef – van Den Dries). In particular, I will describe joint work with Haskell and Hrushovski showing that in characteristic 0, elimination of imaginaries in the `geometric sorts’ of ACVF no longer holds if restricted exponentiation is definable.

Thu, 16 Feb 2012

17:00 - 18:00
L3

Topological Representations and the Model Theory of Abelian Group Rings

Peter Pappas (Oxford)
Abstract

This talk will be accessible to non-specialists and in particular details how model theory naturally leads to specific representations of abelian group rings as rings of global sections. The model-theoretic approach is motivated by algebraic results of Amitsur on the Semisimplicity Problem, on which a brief discussion will first be given.

Thu, 09 Feb 2012

17:00 - 18:00
L3

Categories of additive imaginaries and spectra

Mike Prest (Manchester)
Abstract

To each additive definable category there is attached its category of pp-imaginaries. This is abelian and every small abelian category arises in this way. The connection may be expressed as an equivalence of 2-categories. We describe two associated spectra (Ziegler and Zariski) which have arisen in the model theory of modules.

Thu, 26 Jan 2012

16:00 - 17:00
L3

New conjectures about zeros of Riemann’s zeta function

Yu V Matiyasevich
(Steklov Institute of Mathematics)
Abstract
In http://logic.pdmi.ras.ru/~yumat/personaljournal/artlessmethod/
artlessmethod.php the speaker described a surprising method for (approximate) calculation of the zeros of Riemann’s zeta function using terms of the divergent Dirichlet series.In the talk this method will be presented together with some heuristic “hints” explaining why the divergence of the series doesn’t spoil its use. Several conjectures about the zeros of Riemann’s zeta function will be stated including supposed new relationship between them and the prime numbers.
Thu, 19 Jan 2012

17:00 - 18:00
L3

Groups definable in ACFA

Zoe Chatzidakis (Paris)
Abstract

Recall that a difference field is a field with a distinguished automorphism. ACFA is the theory of existentially closed difference fields. I will discuss results on groups definable in models of ACFA, in particular when they are one-based and what are the consequences of one-basedness.

Thu, 17 Nov 2011

17:00 - 18:00
L3

Matroids and the Hrushovski constructions

David Evans (UEA)
Abstract

We give an exposition of some results from matroid theory which characterise the finite pregeometries arising from Hrushovski's predimension construction as the strict gammoids: a class of matroids studied in the early 1970's which arise from directed graphs. As a corollary, we observe that a finite pregeometry which satisfies Hrushovski's flatness condition arises from a predimension. We also discuss the isomorphism types of the pregeometries of countable, saturated strongly minimal structures in Hrushovski's 1993 paper and answer some open questions from there. This last part is joint work with Marco Ferreira, and extends results in his UEA PhD thesis.

Thu, 27 Oct 2011

17:00 - 18:00
L3

Geometric triviality of the general Painlev\'e equations

Anand Pillay (Leeds)
Abstract

(Joint with Ronnie Nagloo.) I investigate algebraic relations between sets of solutions (and their derivatives) of the "generic" Painlev\'e equations I-VI, proving a somewhat weaker version of ``there are NO algebraic relations".

Thu, 20 Oct 2011

17:00 - 18:00
L3

Homogeneous structures and homomorphisms

Deborah Lockett (Leeds)
Abstract

After a short introduction to homogeneous relational structures (structures such that all local symmetries are global), I will discuss some different topics relating homogeneity to homomorphisms: a family of notions of 'homomorphism-homogeneity' that generalise homogeneity; generic endomorphisms of homogeneous structures; and constraint satisfaction problems.

Thu, 30 Jun 2011
17:00
L3

tba

Thomas Scanlon
(Berkeley)
Thu, 23 Jun 2011
17:00
L3

Zariski Geometries

Tristram de Piro
Abstract

I will discuss the application of Zariski geometries to Mordell Lang, and review the main ideas which are used in the interpretation of a field, given the assumption of non local modularity. I consider some open problems in adapting Zilber's construction to the case of minimal types in separably closed fields.

Thu, 23 Jun 2011
17:00
L3

tba

Tristram de Piro
(Oxford)
Thu, 16 Jun 2011
17:00
L3

"Some model theory of the free group".

Rizos Sklinos
(Leeds)
Abstract

After Sela and Kharlampovich-Myasnikov independently proved that non abelian free groups share the same common theory model theoretic interest for the subject arose.

 In this talk I will present a survey of results around this theory starting with basic model theoretic properties mostly coming from the connectedness of the free group (Poizat).

Then I will sketch our proof with C.Perin for the homogeneity of non abelian free groups and I will give several applications, the most important being the description of forking independence.

 In the last part I will discuss a list of open problems, that fit in the context of geometric stability theory, together with some ideas/partial answers to them.

Thu, 02 Jun 2011
17:00
L3

"Generalized lattices over local Dedekind-like rings"

Carlo Toffalori - joint work with Gena Puninski
(Florence - Moscow)
Abstract

Recent papers by Butler-Campbell-Kovàcs, Rump, Prihoda-Puninski and others introduce over an order O over a Dedekind domain D a notion of "generalized lattice", meaning a D-projective O-module.

We define a similar notion over Dedekind-like rings -- a class of rings intensively studied by Klingler and Levy. We examine in which cases every generalized lattices is a direct sum of ordinary -- i.e., finitely generated -- lattices. We also consider other algebraic and model theoretic questions about generalized lattices.

Thu, 26 May 2011
17:00
L3

"Stability classes of partial types"

Enrique Casanovas
(Barcelona)
Abstract

"We will talk on stability, simplicity, nip, etc of partial types. We will review some known results and we will discuss some open problems."

Thu, 19 May 2011
17:00
L3

tba

Thu, 12 May 2011
16:00
L3

" Ribet points on semi-abelian varieties : a nest for counterexamples"

Daniel Bertrand
(Paris)
Abstract

The points in question can be found on  any semi-abelian surface over an elliptic curve with complex multiplication. We will show that they provide counter-examples to natural expectations in a variety of fields :  Galois representations (following K. Ribet's initial study from the 80's), Lehmer's problem on heights, and more recently, the relative  analogue of the Manin-Mumford conjecture. However, they do support Pink's general conjecture on special subvarieties of mixed Shimura varieties.

 

Thu, 05 May 2011
17:00
L3

"Model theory of local fields and counting problems in Chevalley groups"

Jamshid Derakhshan
(Oxford)
Abstract

This is joint with with Mark Berman, Uri Onn, and Pirita Paajanen.

 

Let K be a local field with valuation ring O and residue field of size q, and G a Chevalley group. We study counting problems associated with the group G(O). Such counting problems are encoded in certain zeta functions defined as Poincare series in q^{-s}. It turns out that these zeta functions are bounded sums of rational functions and depend only on q for all local fields of sufficiently large residue characteristic. We apply this to zeta functions counting conjugacy classes or dimensions of Hecke modules of interwining operators in congruence quotients of G(O). To prove this we use model-theoretic cell decomposition and quantifier-elimination to get a theorem on the values of 'definable' integrals over local fields as the local field varies.

Thu, 10 Mar 2011
17:00
L3

tba

Jonathan Kirby
(Norwich)
Thu, 10 Mar 2011
17:00
L3

First-order axioms for Zilber's exponential field

Jonathan Kirby
(University of East Anglia)
Abstract

Zilber constructed an exponential field B, which is conjecturally isomorphic to the complex exponential field. He did so by giving axioms in an infinitary logic, and showing there is exactly one model of those axioms. Following a suggestion of Zilber, I will give a different list of axioms satisfied by B which, under a number-theoretic conjecture known as CIT, describe its complete first-order theory

Thu, 24 Feb 2011
17:00
L3

An explicit preparation theorem for definable functions in some polynomially bounded o-minimal structures

Jean-Philippe Rolin
(Dijon)
Abstract

It is known that the expansion of the real field by some quasianalytic algebras of functions are o-minimal and polynomially bounded. We prove that, for these structures, the preparation theorem for definable functions proved by L. van den Dries and P. Speissegger has an explicit form, from which it is easy to deduce a quantifier elimination result.

Thu, 17 Feb 2011
16:00
L3

tba

Jan Denef
(Leuven)
Thu, 17 Feb 2011
16:00

Geometric proof of theorems of Ax-Kochen and Ersov

Jan Denef
(Leuven)
Abstract

We will sketch a new proof of the Theorem of Ax and Kochen that any projective hypersurface over the p-adic numbers has a p-adic rational point, if it is given by a homogeneous polynomial with more variables than the square of its degree d, assuming that p is large enough with respect to the degree d. Our proof is purely algebraic geometric and (unlike all previous ones) does not use methods from mathematical logic. It is based on a (small upgrade of a) theorem of Abramovich and Karu about weak toroidalization of morphisms. Our method also yields a new alternative approach to the model theory of henselian valued fields (including the Ax-Kochen-Ersov transfer principle and quantifier elimination).

Thu, 10 Feb 2011
17:00
L3

tba

Philip Welch
(Bristol)
Thu, 10 Feb 2011
17:00
L3

Games and Structures at aleph_2

Philip Welch
(Bristol)
Abstract

Games are ubiquitous in set theory and in particular can be used to build models (often using some large cardinal property to justify the existence of strategies). As a reversal one can define large cardinal properties in terms of such games.

We look at some such that build models through indiscernibles, and that have recently had some effect on structures at aleph_2.

Thu, 03 Feb 2011
17:00
L3

"C-minimal fields"

Francoise Delon
(Paris 7)
Abstract

A $C${\em -relation} is the ternary relation induced by an ultrametric distance, in particular a valuation on a field, when we only remember the relation:

$C(x;y,z)$

iff $d(x,y)

Thu, 27 Jan 2011
17:00
L3

Decidability of large fields of algebraic numbers

Arno Fehm
(Konstanz)
Abstract

   I will present a decidability result for theories of large fields of algebraic numbers, for example certain subfields of the field of totally real algebraic numbers. This result has as special cases classical theorems of Jarden-Kiehne, Fried-Haran-Völklein, and Ershov.

   The theories in question are axiomatized by Galois theoretic properties and geometric local-global principles, and I will point out the connections with the seminal work of Ax on the theory of finite fields.

Thu, 27 Jan 2011
17:00
L3

tba

Arno Fehm
(Konstanz)
Thu, 20 Jan 2011
17:00
L3

Tame measures

Professor Tobias Kaiser
Abstract

We are interested in measure theory and integration theory that ¯ts into the
o-minimal context. Therefore we introduce the following de¯nition:
Given an o-minimal structure M on the ¯eld of reals and a measure ¹ de¯ned on the
Borel sets of some Rn, we call ¹ M-tame if there is an o-minimal expansion of M such
that for every parameter family of functions on Rn that is de¯nable in M the family of
integrals with respect to ¹ is de¯nable in this o-minimal expansion.
In the ¯rst part of the talk we give the de¯nitions and motivate them by existing and
many new examples. In the second one we discuss the Lebesgue measure in this context.
In the ¯nal part we obtain de¯nable versions of important theorems like the theorem of
Radon-Nikodym and the Riesz representation theorem. These results allow us to describe
tame measures explicitly.
1

Thu, 20 Jan 2011
17:00
L3

tba

Tobias Kaiser
(Passau)