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.

Phase transitions in classical spin systems are well understood,
however phase transitions in quantum spin systems are not ... at least
that is what most mathematicians would say. (If anything, they might
question how well we even understand classical spin systems.) Physicists,
on the other hand, say that if the classical model has a phase transition,
then the quantum model does as well. We will prove that, for some special
models. The main tools are reflection positivity, coherent states, and a
new result which generalizes the Berezin-Lieb inequality to the level of
matrix elements.

We shall show a general mean value theorem on Riemannian manifold and how it leads to new monotonicity formulae for evloving metrics. As an application we show a local regularity theorem for Ricci flow.

We investigate the structure of complete Riemannian or Kaehler manifolds
that admit a weighted Poincare inequality and whose Ricci curvature tensor
is bounded from below in terms of the weight function. This subject has
been intensively studied recently by professors P. Li and J. Wang. We will
recall some of their fundamental results and discuss new ideas on the
problem.