Niveau: Supérieur, Doctorat, Bac+8
Summer school on Moduli of curves and Gromov–Witten theory Institut Fourier, Grenoble, June 20th - July 1st, 2011 Abstracts of lectures Carel FABER and Dimitri ZVONKINE Title: Introduction to moduli spaces and their tautological cohomology and Chow ring Abstract: This introduction to the intersection theory on moduli spaces of curves is meant to be as elementary as possible, but still reasonably short. The intersection theory of an algebraic variety M looks for answers to the following questions: What are the interesting cycles (algebraic subvarieties) of M and what cohomology classes do they represent? What are the interesting vector bundles over M and what are their characteristic classes? Can we describe the full cohomology ring of M and identify the above classes in this ring? Can we compute their intersection numbers? In the case of moduli space, the full cohomology ring is still unknown. We are going to study its subring called the ”tautological ring” that contains the classes of most interesting cycles and the characteristic classes of most interesting vector bundles. To give a sense of purpose to the audience, we assume the following goal: after having followed the course, one should be able to write a computer program evaluating all intersection numbers between the tautological classes on the moduli space of stable curves. And to understand the foundation of every step of these computations. Gavril FARKAS Title: Birational geometry of moduli spaces of curves with level structure Abstract: I will discuss the problem of describing moduli spaces of curves with various level structure, concentrating on the case of (higher order)
- vector bundles
- theta-characteristics via nikulin surfaces
- moduli space
- called landau–ginzburg model
- symplectic geometry
- gromov–witten theory
- relations proposed
- gromov-witten theory