16:00
When doing arithmetic geometry, it is helpful to have invariants of the objects which we are studying that see both the arithmetic and the geometry. Motivic homotopy theory allows us to produce new invariants which generalise classical topological invariants, such as the Euler characteristic of a variety. These motivic invariants not only recover the classical topological ones, but also provide arithmetic information. In this talk, I'll review the construction of a motivic Euler characteristic, then study its arithmetic properties, and mention some applications. I'll then talk about work in progress with Ran Azouri, Stephen McKean and Anubhav Nanavaty which studies a "higher Euler characteristic", allowing us to produce an invariant of automorphisms valued in an arithmetically interesting group. I'll then talk about how to relate part of this invariant to a more classical invariant of quadratic forms.