### S-independence in NIP theories

## Abstract

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

Forthcoming events in this series

Thu, 24 May 2012

17:00 -
18:00

L3

Pierre Simon (Ecole Normale Superiore)

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, 10 May 2012

17:00 -
18:00

L3

Jamshid Derakhshan

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 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

Angus Macintyre (QMUL)

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, 01 Mar 2012

17:00 -
18:00

L3

Dugald Macpherson (Leeds)

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

Peter Pappas (Oxford)

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

Mike Prest (Manchester)

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, 02 Feb 2012

17:00 -
18:00

L3

Thu, 26 Jan 2012

16:00 -
17:00

L3

Yu V Matiyasevich

(Steklov Institute of Mathematics)

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.

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

Zoe Chatzidakis (Paris)

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

David Evans (UEA)

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

Anand Pillay (Leeds)

(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

Deborah Lockett (Leeds)

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, 13 Oct 2011

17:00 -
18:00

L3

Tue, 28 Jun 2011

17:00

17:00

L3

Thu, 23 Jun 2011

17:00

17:00

L3

Tristram de Piro

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, 16 Jun 2011

17:00

17:00

L3

Rizos Sklinos

(Leeds)

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

17:00

L3

Carlo Toffalori - joint work with Gena Puninski

(Florence - Moscow)

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

17:00

L3

Enrique Casanovas

(Barcelona)

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

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