March 25, 2014
Mathematics Colloquium lecture March 25
Alexander Givental, University of California, Berkeley, will present "The Hirzebruch-Riemann-Roch Theorem in Quantum K-theory" as part of the mathematics department Colloquium lecture series at 2:30 p.m. Tuesday, March 25, in 102 Cardwell Hall.
The abstract for the lecture is: The title theorem, which is a joint result of the speaker with Valentin Tonita, expresses genus-0 K-theoretic Gromov-Witten invariants in terms of cohomological ones. The former are holomorphic Euler characteristics of some interesting vector bundles over spaces of rational holomorphic curves in a given Kӓhler manifold, while the latter are suitable intersection indices in these spaces. The subject encompasses many previous developments in quantum cohomology theory, and is quite involved technically and conceptually. In this talk, we will try to focus on some relatively elementary aspect of the theory which, hopefully, has a general mathematical appeal. Namely, in contrast with the classical Hirzebruch-Riemann-Roch formula, the theorem in question is not a formula, but an example of what we call "adelic characterization". That is, generating-functions for K-theoretic Gromov-Witten invariants, which happen to have the form of Laurent polynomials in one variable, are completely characterized by interpreting their Laurent series expansions near the poles at the roots of unity as generating-functions for certain cohomological Gromov-Witten invariants.