The index theorem for an elliptic operator gives an integrality theorem; a characteristic integral over the manifold is an integer because it is the index of the operator. I will review some geometric examples: the Dirac operator on a spin manifold, the spin_C Dirac operator, and their analogues for complex manifolds.

When the manifold has no spin_C structure, these operators do not exist. Nevertheless one can define a 'projective' Dirac operator which has an analytic index with values in the rationals. This fractional analytic index can also be expressed as a characteristic integral. I'll describe a possible application to string theory. [Joint work with V. Mathai and R.B. Melrose, J. Diff Geom 74 no 2 (2006) math.DG/0206002]

We discuss some recent developments in the problem of Khler metrics of constant scalar curvature and stability in geometric invariant theory. In particular, we discuss various notions of stability, both finite and infinite-dimensional, and various analytic methods for the problem. These include estimates for energy functionals, density of states and Tian-Yau-Zelditch and Lu asymptotic expansions, geometric heat flows, and both a priori estimates and pluripotential theory for the complex Monge-Ampre equation.

Broadband, coherent array imaging can be made quite robust in random media by using interferometric
algorithms that tend to minimize the effect of random inhomogeneities. I will introduce and describe these algorithms in detail, and I will
show the results of several numerical simulations that assess their effectiveness.

Iwasawa theory, and especially its main conjectures, is the main tool for
studying the mysterious exact formulae in number theory linking the very
different mathematical worlds of purely arithmetic questions on the one
hand, with special values of complex L-functions on the other (typified by
the conjecture of Birch and Swinnerton-Dyer). My lecture will attempt to
explain how, in the special case of elliptic curves, non-commutative
phenomena which arise in each of these worlds lead to very unexpected
consequences in the other world.