We study some problems relating to polynomials evaluated
either at random Gaussian or random Bernoulli inputs. We present a
structure theorem for degree-d polynomials with Gaussian
inputs. In particular, if p is a given degree-d polynomial, then p
can be written in terms of some bounded number of other polynomials
$q_1,...,q_m$ so that the joint probability density function of
$q_1(G),...,q_m(G)$ is close to being bounded. This says essentially
that any abnormalities in the distribution of $p(G)$ can be explained by
the way in which p decomposes into the $q_i$. We then present some
applications of this result.

I will present some new results on classical and quantum integrable systems, emphasizing the interactions between symplectic geometry and spectral theory. I will also briefly describe some recent advances in the study of invariants in symplectic geometry.

In this talk I will present results on a stochastic model of spatial evolution on a lattice, motivated by the process of carcinogenesis (or cancer initiation) from healthy epithelial tissue. Cancer often arises through a sequence of genetic alterations or mutations. Each of these alterations may confer a fitness advantage to the cell, resulting in a clonal expansion. To model this we will consider a generalization of the biased voter process which incorporates successive mutations modulating fitness. Under this model we will investigate a possible mechanism for the phenomenon of ``field cancerization," which refers to the clinical observation that multiple independent primary tumors often arise in the same region of tissue. (joint work w/ K. Leder, M. Ryser, and R. Durrett)