January 28, 2015
Symplectic Topology, Geometry, and Gauge Theory Program


Newsletter, June 2001

Coxeter Lectures: Alexander Givental

In the middle of March Professor Alexander Givental (UC Berkeley and CalTech) gave the Coxeter Lecture Series, entitled "Gromov - Witten invariants in higher genus". In his lectures he outlined his recent spectacular proof of the so-called Virasoro conjecture for the Gromov-Witten potential for complex projective spaces. This conjecture claims, roughly speaking, that the Gromov-Witten potentials are annihilated by some special differential operators constituting half of the Virasoro algebra.

In the series of three lectures Givental introduced a formula expressing enumerative information about higher genus holomorphic curves in complex projective spaces (and conjecturally in many other complex algebraic manifolds) in terms of such information in genus zero. This subject intertwines the Witten-Kontsevich intersection theory on Deligne-Mumford moduli spaces of Riemann surfaces and axiomatic 2-dimensional topological fields theory with elementary representation theory of loop groups. Note that the results actually apply to the so-called virtual moduli space of holomorphic curves. The latter is a regularization of the ordinary moduli space done in such a way that the resulting space behaves as if it were Fredholm regular (i.e. as if its actual dimension coincided with the virtual dimension).

Addressed mostly to specialists, these lectures also served as a natural extension of another highlight of the Symplectic Topology, Geometry, and Gauge Theory Program, the intensive course on an intriguing new subject, symplectic field theory, given two weeks earlier by Yasha Eliashberg (Stanford University).

B. Khesin, University of Toronto and F. Lalonde, Université de Montréal