Many modern geometric constructions yield natural classes on the moduli space of curves. How can we compute these classes ? The genus 0 cases are the simplest and are often governed by essentially closed formulas. To make the jump from genus 0 to higher genus, a new route via the study of semisimple Cohomological Field Theories (CohFTs) and the Givental-Teleman classification can be used. I will discuss how the CohFT results lead to complete calculations in several cases (related to r-spin curves, Verlinde bundles, and Gromov-Witten theories). The talk represents joint work with several authors : F. Janda, A. Marian, D. Oprea, A. Pixton, H.-H. Tseng, and D. Zvonkine.
Un théorème direct sur un ensemble d’entiers utilise la définition et la structure de l’ensemble pour obtenir des propriétés intéressantes. Par exemple, le théorème de Lagrange, selon lequel tout entier positif est la somme de quatre carrés parfaits, est de ce genre. En revanche, un théorème inverse commence avec les propriétés d’un ensemble : le but est alors de découvrir la structure sous-jacente qui explique ces propriétés. Il y a plusieurs théorèmes inverses assez surprenants qui jouent un rôle très important dans la combinatoire additive.
In general relativity, the gravitational field is a lorentzian metric on a 4-dimensional space-time manifold. The Einstein field equations may be expressed, at least locally in time, as the trajectories of a hamiltonian system on a cotangent bundle T*R, where R is the (infinite dimensional) manifold of riemannian metrics on a 3-dimensional ``time slice", with initial values constrained to a certain submanifold C of T*R.
Geometric properties of C suggest that the constraints should be related to the symmetry group of the Einstein equations, consisting of the diffeomorphisms of space-time. But this group does not act on R, since it does not act on an individual time slice. Blohmann, Fernandes, and the speaker have shown that the algebraic structure of the constraints is in fact related to a groupoid of diffeomorphisms between pairs of time slices, but a direct connection between the constraints and this groupoid was not found.
I will report on ongoing work with Blohmann and Schiavina establishing this direct connection, centered around an extension to Lie algebroids (the infinitesimal version of Lie groupoids) of the theory of hamiltonian actions of Lie algebras. Aside from the application to relativity, the theory leads to interesting questions in symplectic topology. The talk will be aimed at a general audience, including graduate students, the only prerequisite being basic differential geometry.