Self-organization is the process where particles or agents in a system with
seemingly simple rules of interactions exhibit ordering into coherent structures.
Controlled self-organization has a wide range of applications from manufacturing
to treatment of diseases and hence understanding the processes involved is of
great importance. This talk will present the derivation of a non-local
hydrodynamic theory to understand the self organization and order-disorder phase
transition in systems of interacting particles. Starting with a microscopic
description a kinetic theory will be identified as the equation of motion. Then
using a generalized Chapman-Enskog procedure a non-local hydrodynamics theory for
the phase transition will be derived. The so derived hydrodynamic model captures
atomistic length scale information of the particles with time scales comparable to
diffusion in the system. The general ideas and potential of this meso-scale
approach will be discussed in the context of a solid-liquid phase transitions.
Some numerical experiments to illustrate the potential of this approach and some
applications will also be discussed.

This work was done in collaboration with Aparna Baskaran (Brandeis University) and John Lowengrub (UC Irvine).

I will discuss some situations when uncertainty in model parameters motivates modeling in terms of random functions. Moreover, some about what is involved in the analysis of such problems. In the first example I consider a problem in mathematical finance, while in the second I consider a problem regarding waves propagating through very complex media.

Kloosterman sum is one of the most famous exponential sums
in number theory. It is defined using a prime p (and another number).
How do these sums vary with p? Ron Evans has made several conjectures
relating the moment of Kloosterman sums for p to the p-th Fourier
coefficient of certain modular forms. We sketch a proof of his
conjectures.