Smoluchowski's equation is used to describe the coagulation-fragmentation
process macroscopically. Microscopically clusters of various sizes coalesce to
form larger clusters and fragment into smaller clusters. I formulate a conjecture about the nature of the fluctuations of the microscopic
clusters about the solutions to the Smoluchowski's equation. I also sketch the proof of the conjecture when the model is in equilibrium.

I will introduce a recent conjecture of Hayes concerning the value at s=0 of the
equivariant Artin L-function associated with an Abelian extension K/k of number fields. It
proposes a relationship between certain unramified Kummer extensions of K and the
denominators of the coefficients of this L-function value. The conjecture can be viewed as a
new generalization of the classical analytic class number formula.

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 have study the coupled ocean-ice dynamics in the marginal ice zone (MIZ) in the Bering Sea. First, sea ice motion and deformation have been analyzed using the high resolution synthetic aperture radar (SAR) images. Segmentation techniques and statistical methods have been used to derive ice motion and deformation maps. These techniques involve a dynamic local thresholding (DLT), which allows separation of sea ice into different classes of thickness and type. We observed two ice motion characteristics with one consisting of a translation and a rotation at scale larger than about 10km/day and the other being ice field deformations at spatial scale less than about 5km. The results were compared with low-resolution values derived from Advanced Microwave Scanning Radiometer for EOS (AMSR-E) data for consistency. In addition, we implemented a two-dimensional coupled ice-ocean model (with wave effects incorporated) and made a direct comparison between the model and remote sensing results. Preliminary results on the effects of waves on ice cover were also obtained.

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.