## Ziv Ran

We discuss some global and enumerative results on the Hilbert scheme parametrizing finite subschemes of a family of stable curves, including a method to compute arbitrary Chern numbers of the tautological bundle. In some important cases involving Hodge bundles and their associated degeneracy loci, the formal enumerative predictions or `virtual numbers’ from the Chern numbers are false. This is due to excess degeneracy at the boundary, corresponding to reducible curves. We describe an approach to resolving this excess degeneracy through boundary modifications of Hodge bundles. These mod- ifications involve some new structure at the boundary, which is closely related to the limit theory for linear systems (in this case, the canonical system). As an application, we give a method to compute the fundamental class on the moduli space of stable curves of the closure of the locus of smooth hyperelliptic curves.

## Greg Blekherman

A multivariate real polynomial $p$ is nonnegative if $p(x) \geq 0$ for all $x \in R^n$. I will review the history and motivation behind the problem of representing nonnegative polynomials as sums of squares. Such representations are of interest for both theoretical and practical computational reasons. However, much about the relationship between nonnegative polynomials and sums of squares remains unknown. In the smallest cases where there exist nonnegative polynomials that are not sums of squares I will present a complete classification of the differences between these sets. The fundamental reason that the set of sums of squares is smaller than the set of nonnegative polynomials is that polynomials of degree $d$ satisfy certain linear relations, known as the Cayley-Bacharach relations, which are not satisfied by polynomials of full degree $2d$.