Towards a model theory of global fields

Thursday, February 14, 20135:30 pmNYU, Courant Institute, Room 317

Ehud Hrushovski

Towards a model theory of global fields

Hebrew University of Jerusalem

The logic of diophantine or geometric structures appears to exhibit a sharp local-global dichotomy. For local or generic geometry, there are many quantifier-elimination results, describing an arbitrary definable set in geometric terms, and thus opening the way to a model-theoretic analysis. This includes Tarski’s theorem in real semi-algebraic geometry, and the Ax–Kochen theorem on the asymptotic theory of the p-adics. For global fields, all known results go in the opposite direction of undecidability; the first such statement is Godel’s interpretation of syntax (and finite mathematics in general) in the ring of integers. However the reason for the undecidability appears to reside in the accidental nature of finite sets, rather than with the adelic geometry that is generally used to analyze number- theoretic problems. One could conjecture that the parts of adelic geometry that are functorial with respect to finite field extensions should admit a model-theoretic analysis. The talk will report on a research program, joint with Itai Ben-Yaacov, with this aim. Existing results are mostly restricted to the function field version, and include the existential closedness of k(t)^alg as an adelic field in an appropriate language.

The talk is presented by the New York Joint Number Theory Seminar. It will be preceded by the RTG Seminar from 4:00-4:45 in Room 805 (aimed at graduate students and post-docs; regular faculty are requested not to attend.), and Tea from 5:00-5:30 in Room 317

Ehud Hrushovski is a Professor of Mathematics at the Hebrew University of Jerusalem. He is well known for his work in model theory, in particular in the branch that has become known as geometric model theory; and for the applications he has made of it to Diophantine geometry, including the Mordell–Lang conjecture. He is a fellow of the American Academy of Arts and Sciences (2007), and Israel Academy of Sciences and Humanities (2008).

Posted by on February 11th, 2013