Past Logic Seminar

22 November 2007
Alice Medvedev
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 \emph{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.
9 November 2007
Jonathan Kirby
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.
8 November 2007
Assaf Hasson
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
18 October 2007
I. Halupczok
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).
12 October 2007
J. Koenigsmann
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 $C(t)$.