This talk will develop part of the foundation needed to develop a partial Morse theory for conformal harmonic maps from a Riemann surface into a Riemannian manifold. Such maps are also called parametrized minimal surfaces. A partial Morse theory for such objects should parallel the well-known Morse theory of smooth closed geodesics.

The first step needed is a bumpy metric theorem which states that when a Riemannian manifold has a generic metric, all prime minimal surfaces are free of branch points and lie on nondegenerate critical submanifolds. (A parametrized minimal surface is prime if it does not cover a parametrized minimal surface of lower energy.)

We will present such a theorem and describe some applications.

Generating functions of Gromov-Witten invariants of compact
symplectic manifolds behave very much like tau-functions of Integrable
systems. It was conjectured by Eguchi-Hori-Xiong and S. Katz that
Gromov-Witten invariants of smooth projective varieties should
satisfy the Virasoro constraints, which also exist for many integrable
systems (e.g, Gelfand-Dickey hierarchies). It was conjectured by Witten
that the generating functions on moduli spaces of spin curves are
tau-functions of Gelfand-Dickey hierarchy. In a joint work with Kimura, we
showed that it is possible to use Virasoro constraints of a point and
the sphere to derive universal equtions for Gromov-Witten invariants of
all compact symplectic manifolds. Such equations can also be used to
compute certain intersection numbers on moduli spaces of spin curves which
coincide with predictions of Witten's conjecture.

The space-time monopole equation is obtained from a dimension reduction of the self-dual Yang-Mills field equation on R^{2,2}. It has a Lax pair, i.e., a linear system with a spectral parameter such that the equation is the condition that this linear system be solvable. The scattering data describe the singularities of the solutions of the linear system in the spectral parameter. The linear problem for the monopole equation is a family of d-bar operators, and we explain how to use loop group factorizations to solve the inverse problem and hence solve the Cauchy problem for the space-time monopole equation with small initial data. This is joint work with B. Dai and K. Uhlenbeck.

A Einstein metric is stable if the second variation of the total scalar curvature functional is nonpositive in the direction of changes in conformal structures. Using spin^c structure we prove that a compact Einstein metric with nonpositive scalar curvature admits a nonzero parallel spin$^c$ spinor is stable. In particular, all metrics with nonzero parallel spinor (these are Ricci flat with special holonomy such as Calabi-Yau and $G_2$) and Kahler-Einstein metrics with nonpositive scalar curvature are stable. In fact we show that metrics with nonzero parallel spinor are local maxima for the Yamabe invariant and any metric of positive scalar curvature cannot lie too close to them. Similar results also hold for Kahler-Einstein metrics with nonpositive scalar curvature. This is a joint work with Xianzhe Dai and Xiaodong Wang.