Abstract: The main goal of this expository talk is to give a short introduction to equivariant intersection theory as it has been developed by B. Totaro, D. Edidin, and W. Graham.

We will start from recalling some basic facts from classical intersection theory and explain how they generalize in the equivariant world.

We will also discuss some of the fundamental examples such as projective spaces, grassmannians, and moduli spaces of curves.

Graduate students are particularly welcome to participate.

**Vance Blankers: ****The recursive structure of hyperelliptic loci in moduli spaces of curves**

Abstract: Moduli spaces of curves and their tautological rings enjoy a rich recursive combinatorial structure, which we describe in some detail. We then define a more generalized hyperelliptic locus \Hyp_{g,\ell,2m,n} in \Moduli_{g,\ell+2m+n}, requiring our hyperelliptic curves to have \ell marked Weierstrass points, m marked conjugate pairs, and n marked free points. Under the natural forgetful morphisms, these loci form a doubly-infinite family of tautological classes with its own recursive structure, which allows us to use an induction argument to show that any hyperelliptic class in genus two is extremal and rigid in the effective cone of codimension-(\ell+m) classes. Our main result extends work of Chen and Tarasca, who previously considered only Weierstrass point markings, and points towards a possible CohFT-like structure for hyperelliptic classes.

**Fumitoshi Sato: **** A construction of Moduli Spaces of Pointed Rational Curves/ Topological recursion relations via degree 2 maps**

In the first hour I will explain a (new?) construction of moduli spaces of pointed rational curves as a wonderful compactification of a product of projective lines. In the second hour, the talk will turn to studying tautological classes in the moduli spaces of curves.

Mumford started to study enumerative geometry of moduli of curves. I will give some generalizations of Mumford result.

**Linda Chen: ****Brill-Noether theory and determinantal formulas for degeneracy loci **

I will begin with an overview of degeneracy loci and applications to the Brill-Noether theory of special divisors, and then describe recent developments in Schubert calculus, including K-theoretic formulas for degeneracy loci and K-classes of Brill-Noether loci. These recover the formulas of Eisenbud-Harris, Pirola, and Chan-López-Pflueger-Teixidor for Brill-Noether curves. This is joint work with Dave Anderson and Nicola Tarasca.