Research Group:

## Virtual Classes

Topics:

1. A review of Fulton-MacPherson intersection theory.

2. Cones, normal cones, lci morphisms and reduced cotangent complex.

3. Groupoids, 2-categories and stacks.

4. Deligne-Mumford and Artin algebraic stacks.

5. Abelian cone stacks and derived category; normal cone stack.

6. Perfect obstruction theories and virtual classes.

7. Functorial properties of virtual classes and pullbacks.

8. Gromov-Witten invariants: definition and proof of the

Kontsevich-Manin axioms.

## Geometry of Character Varieties

The course will cover some of the work with Hausel and Letellier on the subject. Basic tools on finite fields, Weil conjectures, group representations and symmetric functions will be discussed as needed.