Forthcoming Seminars › Seminar Series

Logic Seminar

Wed, 10/10/2007
16:00 		Dr. Robin Knight (Oxford) Logic Seminar Add to calendar L3
Categories of Scott Spaces
Fri, 12/10/2007
15:15 		J. Koenigsmann (Oxford) Logic Seminar Add to calendar L3
AXIOMATIZING FIELDS VIA GALOIS THEORY
By classical results of Tarski and Artin-Schreier, the elementary theory of the field of real numbers can be axiomatized in purely Galois-theoretic terms by describing the absolute Galois group of the field. Using work of Ax-Kochen/Ershov and a p-adic analogue of the Artin-Schreier theory the same can be proved for the field $ \mathbb{Q}_p $ of p-adic numbers and for very few other fields. Replacing, however, the absolute Galois group of a field K by that of the rational function field $ K(t) $ over $ K $, one obtains a Galois-theoretic axiomatiozation of almost arbitrary perfect fields. This gives rise to a new approach to longstanding decidability questions for fields like $ F_p((t)) $ or $ C(t) $.
Thu, 18/10/2007
16:00 		I. Halupczok (Paris) Logic Seminar Add to calendar SR1
Motivic measure for pseudo-finite like fields
To understand the definable sets of a theory, it is helpful to have some invariants, i.e. maps from the definable sets to somewhere else which are invariant under definable bijections. Denef and Loeser constructed a very strong such invariant for the theory of pseudo-finite fields (of characteristic zero): to each definable set, they associate a virtual motive. In this way one gets all the known cohomological invariants of varieties (like the Euler characteristic or the Hodge polynomial) for arbitrary definable sets. I will first explain this, and then present a generalization to other fields, namely to perfect, pseudo-algebraically closed fields with pro-cyclic Galois group. To this end, we will construct maps between the set of definable sets of different such theories. (More precisely: between the Grothendieck rings of these theories.) Moreover, I will show how, using these maps, one can extract additional information about definable sets of pseudo-finite fields (information which the map of Denef-Loeser loses).
Fri, 02/11/2007
14:15 		Dugald Macpherson (Leeds) Logic Seminar Add to calendar L3
Some model-theoretic classes of permutation groups
Thu, 08/11/2007
10:00 		Assaf Hasson (Oxford) Logic Seminar Add to calendar L3
The classificatiion of structures interpretable in o-minimal theories
We survey the classification of structures interpretable in o-minimal theories in terms of thorn-minimal types. We show that a necessary and sufficient condition for such a structure to interpret a real closed field is that it has a non-locally modular unstable type. We also show that assuming Zilber's Trichotomy for strongly minimal sets interpretable in o-minimal theories, such a structure interprets a pure algebraically closed field iff it has a global stable non-locally modular type. Finally, if time allows, we will discuss reasons to believe in Zilber's Trichotomy in the present context
Fri, 09/11/2007
14:15 		Jonathan Kirby (Oxford) Logic Seminar Add to calendar L3
Schanuel's conjecture and dimension theory
I will push Schanuel's conjecture in four directions: defining a dimension theory (pregeometry), blurred exponential functions, exponential maps of more general groups, and converses. The goal is to explain how Zilber's conjecture on complex exponentiation is true at least in a "geometric" sense, and how this can be proved without solving the difficult number theoretic conjectures. If time permits, I will explain some connections with diophantine geometry.
Thu, 22/11/2007
10:00 		Alice Medvedev (UIC) Logic Seminar Add to calendar SR1
Minimal definable sets in difference fields.
I will speak about the Zilber trichotomy for weakly minimal difference varieties, and the definable structure on them. A difference field is a field with a distinguished automorphism $ \sigma $. Solution sets of systems of polynomial difference equations like $ 3 x \sigma(x) +4x +\sigma^2(x) +17 =0 $ are the quantifier-free definable subsets of difference fields. These difference varieties are similar to varieties in algebraic geometry, except uglier, both from an algebraic and from a model-theoretic point of view. ACFA, the model-companion of the theory of difference fields, is a supersimple theory whose minimal (i.e. U-rank $ 1 $) types satisfy the Zilber's Trichotomy Conjecture that any non-trivial definable structure on the set of realizations of a minimal type $ p $ must come from a definable one-based group or from a definable field. Every minimal type $ p $ in ACFA contains a (weakly) minimal quantifier-free formula $ \phi_p $, and often the difference variety defined by $ \phi_p $ determines which case of the Zilber Trichotomy $ p $ belongs to.
Fri, 30/11/2007
14:15 		Marcus Tressl (University of Manchester) Logic Seminar Add to calendar SR2
p-adically closed fields

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

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