Motivic homotopy theory and refined enumerative geometry

Mini courses

Intersection theory and mixed motives with rational coefficients (Denis-Charles Cisinski)

We will describe mixed motives with rational coefficients in terms of Algebraic K-Theory, and study the motivic analog of the equivalence between the stable homotopy category with rational coefficients and the derived category of vector spaces over the field of rational numbers. For this purpose, despite what the title suggests, we shall review Morel’s theory of unramified sheaves and Milnor-Witt K-Theory within the motivic stable homotopy categories with integral coefficients.

Enumerative geometry via quadratic forms (Marc Levine)

Enumerative geometry is in rough outline based on three main ingredients:

1. The intersection theory of algebraic cycles

2. The construction of algebraic cycles via characteristic class of vector bundles

3. The counting of intersection products via the degree map

The foundational work of Barge-Morel, Fasel and others on the Chow-Witt groups, as well as recent works of Kass-Wickelgren exploring a number of geometric problems via A^1-homotopy theory, has opened the door to a refinement of classical enumerative geometry from an integer-valued theory to one that takes values in the Grothendieck-Witt ring of quadratic forms. We will discuss a number of constructions and applications of this emerging topic, including quadratic Euler characteristics, quadratic versions of Riemann-Hurwitz type formulas, quadratic virtual fundamental classes, and quadratic refinements of the classical theory of characteristic classes of vector bundles. We plan to illustrate the theory with a number of simple examples, such as counting lines on hypersurfaces and attaching quadratic invariants to singularities, as well as discussing several computational methods.

Introduction to Chow-Witt groups (Jean Fasel)

In this series of lectures, we will introduce Chow-Witt groups (of smooth varieties over a field of characteristic different from 2) focusing on the material needed to understand Marc Levine’s paper « Toward an enumerative geometry with quadratic forms ». This includes basic functorial properties as well as Euler classes of vector bundles. Time permitting, we will also discuss Chow-Witt of singular varieties.