Departmental Colloquium

I will discuss the singularities of the zero-locus of a complex valued polynomial equation. A particular focus will be payed to comparing different singularities. I will discuss two different approaches to this question, both analytic (characteristic zero) and algebraic (positive characteristic).

The theory of rings of differential operators and modules over them exists mostly in characteristic zero. Only some isolated results are known in characteristic p>0. Some characteristic zero results are known to be false in characteristic p>0. Even those results that hold both in characteristic zero and in characteristic p>0 often have very different proofs in these two cases. Very recently Vladimir Bavula found a characteristic-free definition of a fundamental characteristic zero concept - that of holonomic D-modules - and showed that his holonomic modules have some of the striking properties that make holonomic modules so useful in characteristic zero (in characteristic zero Bavula's definition coincides with the usual one). This discovery has since been used to give the first characteristic-free proof of a very basic fact about D-modules which had been proven earlier by two completely different methods in characteristic zero and in characteristic p>0. Study of Bavula's holonomic modules is ongoing. We will survey these developments in the talk.

We consider a nonlinear stochastic heat equation with multiplicative space-time white noise. It is now known that in many cases the solution has intermittent behavior at large times. Here we establish a number of fixed-time results about the solution; among other things these results imply that the behavior of the solution is highly sensitive to the structure of the initial function, as well as the nature of the nonlinear terms. We will argue that this "chaotic behavior" is partly responsible for large-time intermittency.