Departmental Colloquium

To a local system of complex vector spaces on a topological space one can naturally associate its Chern-Simons characteristic classes: they are odd degree cohomology classes with coefficients in C/Z. For example, the Chern-Simons class of an appropriate local system on a compact hyperbolic 3-manifold recovers the volume of the manifold. I will discuss an analog of this theory for local systems of vector spaces over the field of p-adic numbers Q_p, which, in contrast with the classical Chern-Simons theory, is defined not only for local systems on topological spaces, but also on arithmetic objects such as algebraic varieties over arbitrary fields. The output of the theory turns out to be most interesting for algebraic varieties over number fields and p-adic fields, for instance in the latter setting one finds a p-adic analog of the Milnor-Wood inequality. This is joint work with Lue Pan.