Do. 11.07.2019, Oberseminar Algebra

Thema: Geometric invariants of the configuration space of marked points on the complex plane. Referent: Noémie Combe (MPI Bonn), Raum: ENC-D 201, Zeit: 10:00 Uhr c.t.

Oberseminar Algebra

Geometric invariants of the configuration space of marked points on the complex plane.

Donnerstag, 11.07.2019, 10:00 Uhr c.t., Raum: ENC-D 201,

Referent: Noémie Combe (MPI Bonn)

Abstrakt:

Considering the space of monic degree d > 1 complex polynomials with distinct roots is equivalent to considering the configuration space of d marked points on the complex plane. A configuration space is a mathematical object, related to state spaces in physics. The most famous configuration space is the configuration space Conf_d of d marked points on a Riemann surface, for example the complex plane. Although these spaces have been extensively considered in a given framework (mainly using tools from Complex geometry) a different approach using real geometry brings out new insights on the structure of Conf_d . We show that a (real algebraic) topological stratification of this space can be given, using drawings of complex polynomials. A drawing is a system of colored curves in the complex plane, which are obtained by taking the inverse image of the real and imaginary axes, by a polynomial. A stratum is a set of polynomials, indexed by a (bi-colored) chord diagram, being an isotopy class of drawings relative to asymptotic directions. Studying the incidence relations between strata, gives a very detailed geometric description of this configuration space, and a classification of polynomials by graphs: each graph tells precisely the emplacement of the roots, the 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) in- variants of Conf_d . 2. this decomposition forms a good cover in the sense of Cech (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.