"Grothendieck rings, Euler characteristics, and Schanuel dimensions of models"
Abstract: A distributive category is a category C with finite limits and
colimits, an initial object and a final object satisfying the
distributivity condition that for any three objects A, B, and C, the
canonical map from (A x B) + (A x C) to A x (B + C) is an isomorphism. If C
is a small distributive category, then the set of isomorphism types of
objects in C carries a natural L_ring = L(+, *, 0, 1) structure, called the
rig of C. I will discuss this construction in the case that C is the
category of (parametrically) definable sets in some structure (with
definable functions as morphisms). These rigs impinge on many areas of
mathematics including o-minimal structures, field arithmetic, proof theory,
motivic integration, pseudofinite structures, p-adic analysis, and
stability theory. I will explain these connections and how in some cases
the formalism of rigs of definable sets can be used to solve outstanding
problems.
Back to Talks Page