Time

Sep 24 2009 - 3:30pm - 4:30 pm

Location

CC246

Speaker

Hans Schoutens (NYC College of Technology )

Abstract

The classical Grothendieck ring over an algebraically closed field $K$ is the Abelian group on isomorphism classes of varieties modulo the "scissor" relations $[X]=[Y]+[X-Y]$ for $Y\subset X$, with multiplication given by the Cartesian product of varieties. When Denef and Loeser in the late 90s generalized Konsevitch's motivic integration---itself a generalization of $p$-adic integration with values in the Grothendieck ring---using certain QE results in valued fields, many model-theorists (Scanlon, Haskell, Hrushovski, Cluckers, et al.) became very interested in the Grothendieck ring of an arbitrary first-order theory, the classical case (allegedly) corresponding to the theory ACF. For the last decade, I had been contemplating ways of incorporating schemes (=varieties with nilpotent structure) into a model-theoretic setup. Geometers, on the other hand, do not know how to treat schemes in a Grothendieck ring setup. I will simultaneously resolve both conundrums by restricting the class of formulae (schemic formulae) while working with a larger theory (the theory of local finite-dimensional $K$-algebras), thus obtaining the schemic Grothendieck ring (and some of its "infinitary" variants that are necessary to get a good notion of complement). The main motivation, however, for this new theory is to generalize the work of Denef-Loeser on the rationality of the Igusa-zeta series (a sort of abstract "counting" of approximate solutions). Whereas their proof relies heavily on resolution of singularities (together with the aforementioned QE results), and hence does not work in positive characteristic, the present theory is far more functorial, and preliminary calculations yield rationality in may instances (e.g., all Du Val singularities in any characteristic). Although many of these terms might seem frightening (what the heck is a scheme? a motive?) to a logician, they will sound soon very familiar from this new perspective, and require only a modest knowledge of first-order logic.