Colloquium Mathematics, Dr. Noémie Combe (MPIM Bonn, Germany)

Title:
Geometric invariants of the configuration space of d marked points on the complex plane

Abstract:
A configuration space is a mathematical object related to state spaces in Physics. The most well known configuration space is the space Conf_d of d marked points on a Riemann surface, for example on the complex plane. Those configuration spaces are not homotopy invariant.

In the 1970s, Arnold and Fuchs calculated the cohomology groups of these spaces by using a given cellular decomposition. Although these spaces have been considered extensively in a given framework, a different approach brings out new insight on the structure of Conf_d.

We stratify this space, using its natural relation to the space of complex, monic, degree d polynomials. A stratum $A_{\sigma}$ is a set of polynomials indexed by a bi-colored chord diagram $\sigma$ (a superposition under constraints of two chord diagrams of different colors). These graphs are the isotopy classes of images of $P^(-1)(R U iR)$. A study of incidence relations between the strata gives a very detailed geometric description of this configuration space and a classification of polynomials in terms of graphs: each graph tells precisely the placement of roots, critical points, and critical values of the polynomials.

We show that:
1) this stratification is invariant under a finite Coxeter group, which defines new (geometric) invariants of Conf_d.
2) this decomposition forms a good cover in the sense of Cech (the strata are contractible, multiple intersections are contractible).
As an application of these results, one may derive a computer program to calculate explicitly the cohomology groups of braids