The most natural formulation for the equations of elasticity is as a first order system, reflecting the very different nature of the equilibrium equation and the constitutive equation. Moreover this system applies more widely than second order formulations, for example to incompressible, plastic, or viscoelastic materials. The first-order system is captured variationally in the Hellinger-Reissner variational principle, which characterizes the symmetric stress tensor field and the displacement vector field as a saddle-point of a suitable functional. However it has proven extremely difficult to develop stable and effective finite element discretizations of this formulation--so called mixed finite elements for elasticity. Efforts to develop such methods go back to the earliest days of the finite element methods. However, stable mixed elasticity elements using polynomial shape functions have only been developed recently using the theory of finite element exterior calculus (FEEC). This talk will review the subject and especially recent progress connected to FEEC, which has led to very simple stable elements in two and three dimensions.

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.