# Past Logic Seminar

26 October 2007
15:15
A. Pillay
Abstract
• Logic Seminar
18 October 2007
16:00
I. Halupczok
Abstract
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).
• Logic Seminar
12 October 2007
15:15
J. Koenigsmann
Abstract
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 $C(t)$.
• Logic Seminar
10 October 2007
16:00
Dr. Robin Knight
Abstract
• Logic Seminar
5 October 2007
16:00
Janos Makowsky
Abstract
• Logic Seminar
5 October 2007
15:15
J. Makowsky
Abstract
• Logic Seminar
15 June 2007
15:15
• Logic Seminar
8 June 2007
15:15
Abstract
• Logic Seminar
1 June 2007
15:15
Abstract
&nbsp; Countable Borel equivalence relations arise naturally as orbit equivalence relations for countable groups. For each countable Borel equivalence relation E there is an infinitary sentence such that E is equivalent to the isomorphism relation on countable models of that sentence. For first order theories the question is open. &nbsp;
• Logic Seminar