The space-time monopole equation is obtained from a dimension reduction of the self-dual Yang-Mills field equation on R^{2,2}. It has a Lax pair, i.e., a linear system with a spectral parameter such that the equation is the condition that this linear system be solvable. The scattering data describe the singularities of the solutions of the linear system in the spectral parameter. The linear problem for the monopole equation is a family of d-bar operators, and we explain how to use loop group factorizations to solve the inverse problem and hence solve the Cauchy problem for the space-time monopole equation with small initial data. This is joint work with B. Dai and K. Uhlenbeck.

Let N(f) denote the number of zeros of a sparse
multivariate polynomial f(x) over a finite field of
characteristic p. In this lecture, we discuss the
complexity and algorithms for computing the
reduction N(f) modulo a power of p.