|Speaker:||Leonardo Mihalcea (Virginia Tech)|
|Title:||QUANTUM K-THEORY AND THE GEOMETRY OF SPACES OF CURVES|
|Date:||Tue, Jul 31, 2012, 15:30 - 17:00|
The 3-point, genus 0, Gromov-Witten invariants on a flag manifold count rational curves of degree d satisfying certain incidence conditions - if the number of curves is expected to be finite. For infinitely many curves, Givental and Lee defined the K-theoretic Gromov-Witten invariants, which associates a measure to the space of rational curves in question, embedded in Kontsevich's moduli space of stable maps. The resulting quantum cohomology theory - the quantum K-theory - encodes the associativity relations satisfied by the K-theoretic Gromov-Witten invariants.
In this talk I will show how we can compute the K-theoretic Gromov-Witten in several cases, with an emphasis on the case of Grassmannians. In this case we can give a complete algorithm, therefore obtaining a description of the quantum K-theory algebra. The key is a "quantum=classical" phenomenon: the K-theoretic Gromov-Witten invariants for Grassmannians are equal to structure constants of the ordinary K-theory of certain two-step flag manifolds. My talk is based on several collaborations with A. Buch, P.E. Chaput, C. Li and N. Perrin.