Elliptic PDEs and Bifurcations of Holomorphic Curves

The first part of this talk will be an overview of the analytic
 approach to Gromov-Witten invariants.    That analysis is
 particularly interesting when applied to holomorphic maps into a
 Calabi-Yau 3-fold.   I  will describe recent joint work with E. Ionel
 which, building on work of Taubes,  uses PDE methods to give a
 geometric accounting of the holomorphic curves, their covers,  and
 their bifurcations  in Calabi-Yau 3-folds.    The resulting picture
 gives a specific geometric interpretation, and  partially solves the
 'Gopakumar-Vafa conjecture'  which emerged from string theory.