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The Atiyah-Bott localization formula has become a valuable tool forÌýcomputation of symplectic invariants given in terms of integrals onÌýthe moduli spaces of holomorphic stable maps. In contrast, the ``open''Ìýmoduli spaces, of stable maps of marked Riemann surfaces with boundary,Ìýhave boundaries, and these must be taken into account in order toÌýapply fixed point localization.ÌýHomological perturbation for twisted $A_{\infty}$ algebras allowsÌýone to write down expressions which effectively eliminate the boundariesÌýin genus zero, so one can define equivariant invariants and computeÌýthem using localization.ÌýThese invariants specialize to the open Gromov-Witten invariants,Ìýand in particular produce new combinatorial expressions for Welschinger'sÌýsigned counts of real rational plane curves in terms of summationÌýover certain even-odd diagrams.ÌýTime permitting, we'll discuss the two-sided information flow withÌýthe intersection theory of Riemann surfaces with boundary (mappingÌýto a point), which lends evidence to a conjectural generalizationÌýof the localization formula to higher genus. ÌýMostly joint work with Jake Solomon.