Joint IAS/Princeton University Symplectic Geometry Seminar

The Atiyah-Bott localization formula has become a valuable tool for computation of symplectic invariants given in terms of integrals on the moduli spaces of holomorphic stable maps. In contrast, the ``open'' moduli spaces, of stable maps of marked Riemann surfaces with boundary, have boundaries, and these must be taken into account in order to apply fixed point localization. Homological perturbation for twisted $A_{\infty}$ algebras allows one to write down expressions which effectively eliminate the boundaries in genus zero, so one can define equivariant invariants and compute them using localization. These invariants specialize to the open Gromov-Witten invariants, and in particular produce new combinatorial expressions for Welschinger's signed counts of real rational plane curves in terms of summation over certain even-odd diagrams. Time permitting, we'll discuss the two-sided information flow with the intersection theory of Riemann surfaces with boundary (mapping to a point), which lends evidence to a conjectural generalization of the localization formula to higher genus. Mostly joint work with Jake Solomon.