Forthcoming Seminars › Seminar Series

Logic Seminar

Thu, 30/04/2009
17:00 		Ivan Tomasic Logic Seminar Add to calendar L3
Etale cohomology of difference schemes
Difference schemes constitute important building blocks in the model-theoretic study of difference fields. Our goal is to pursue their number-theoretic aspects much further than required by model theory. Roughly speaking, a difference scheme (variety) is a scheme (variety) with a distinguished endomorphism. We will explain how to extend the methods of etale cohomology to this context and, time permitting, we will show the calculation of difference etale cohomology in some interesting cases.
Thu, 07/05/2009
17:00 		Dugald McPherson (Leeds) Logic Seminar Add to calendar L3
Automorphism groups of homogeneous structures
Thu, 14/05/2009
17:00 		TBA Logic Seminar Add to calendar L3
To Be Announced
Thu, 21/05/2009
11:00 		Margaret Thomas (Oxford) Logic Seminar Add to calendar L3
An o-minimal structure without wild parametrization.
Thu, 21/05/2009
17:00 		Moshe Jarden (Tel Aviv) Logic Seminar Add to calendar L3
Diamonds in Torsion of Abelian Varieties.
A theorem of Kuyk says that every Abelian extension of a Hilbertian field is Hilbertian. We conjecture that for an Abelian variety $ A $ defined over a Hilbertian field $ K $ every extension $ L $ of $ K $ in $ K(A_\tor) $ is Hilbertian. We prove our conjecture when $ K $ is a number field. The proofs applies a result of Serre about $ l $-torsion of Abelian varieties, information about $ l $-adic analytic groups, and Haran's diamond theorem.
Thu, 28/05/2009
17:00 		Tim Gendron (Mexico) Logic Seminar Add to calendar L3
Arithmetic and Geometric Irrationality via Substructures of Nonstandard Models
This purpose of this talk will be to introduce the idea that the spectrum of nonstandard models of a “standard” algebraic object can be used much like a microscope with which one may perceive and codify irrationality invisible within the standard model. This will be done by examining the following three themes:
  • Algebraic topology of foliated spaces We define the fundamental germ, a generalization of fundamental group for foliations, and show that the fundamental germ of a foliation that covers a manifold $ M $ is detected (as a substructure) by a nonstandard model of the fundamental group of $ M $.
  • Real algebraic number theory. We introduce the group $ (r) $ of diophantine approximations of a real number $ r $, a subgroup of a nonstandard model of the integers, and show how $ (r) $ gives rise to a notion of principal ideal generated by $ r $. The general linear group $ GL(2, \mathbb{Z}) $ plays here the role of a Galois group, permuting the real ideals of equivalent real numbers.
    • \item Modular invariants of a Noncommutative Torus. We use the fundamental germ of the associated Kronecker foliation as a lattice and define the notion of Eisenstein series, Weierstrass function, Weierstrass equation and j-invariant.
    Thu, 04/06/2009
    17:00     		Pietro dello Stritto Logic Seminar Add to calendar L3
    SUPERSIMPLE MOUFANG POLYGONS
    In this talk we discuss the main results of my PhD thesis. We begin by giving some background on Moufang polygons. This is followed by a short introduction of the basic model theoretic notions related to the thesis, such as asymptotic classes of finite structures, measurable structures, (superstable) supersimple theories and (finite Morley rank) S_1 rank. We also mention the relation between Moufang polygons and the associated little projective groups. Moufang polygons have been classified by Tits and Weiss, and a complete list is given in their book `Moufang polygons'. This work is inspired by a paper of Kramer, Tent and van Maldeghem called "Simple groups of finite Morley rank and Tits buildings". The authors work in a superstable context. They show that Moufang polygons of finite Morley rank are exactly Pappian polygons, i.e., projective planes, symplectic quadrangles and split Cayley hexagons, provided that they arise over algebraically closed fields. We work under the weaker assumption of supersimplicity. Therefore, we expect more examples. Indeed, apart from those already occuring in the finite Morley rank case, there are four further examples, up to duality, of supersimple Moufang polygons; namely, Hermitian quadrangles in projective dimension 3 and 4, the twisted triality hexagon and the (perfect) Ree-Tits octagon, provided that the underlying field (or `difference' field in the last case) is supersimple. As a result, we obtain the nice characterization that supersimple Moufang polygons are exactly those Moufang polygons belonging to families which also arise over finite fields. Examples of supersimple Moufang polygons are constructed via asymptoticity arguments: every class C of finite Moufang polygons forms an asymptotic class, and every non-principal ultraproduct of C gives rise to a measurable structure, thus supersimple (of finite S_1 rank). For the remaining cases one can proceed as follows: let \Gamma be any Moufang polygon belonging to a family which does not arise over finite fields, and call K its underlying field; then K is (first-order) definable in \Gamma, and by applying some model theoretic facts this definability is inconsistent with supersimplicity".
    Thu, 18/06/2009
    17:00     		David Evans (Norwich) Logic Seminar Add to calendar L3
    Finite covers
    I will talk about two pieces of work on finite covers: (i) Work of Hrushovski which, for a stable theory, links splitting of certain finite covers with higher amalgamation properties; (ii) Joint work of myself and Elisabetta Pastori which uses group cohomology to investigate some non-split finite covers of the set of k-sets from a disintegrated set.

    For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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