### Ettore Aldrovandi

MATHEMATICS COLLOQUIUM
Stacks arise from the desire to parametrize species of mathematical objects, like all triangles in the plane, all quadruples of points on the sphere, "all" algebraic curves of a given genus; homotopy types arise from the desire to encode the connectivity information of spaces. Both kinds are usually equipped with some kind of algebraic structure, typically like that of a group or a ring, only "up to homotopy". After the introductory examples, I will illustrate some aspects of this "brave new algebra" and present some recent applications, results, and some work in progress, in intersection theory, generalized differential operators, and K-Theory (in collaboration with with N. Ramachandran, U. Bruzzo, and some of my students). |