I will use concrete examples to argue why mathematics is even more important and powerful when computers become more and more powerful and to show what computational mathematics is about.

I will describe a circle of results on the Smoluchowski-Kramers limit in
systems of stochastic differential equations.  Some of them were directly
motivated by experiments, others suggest further laboratory work.  In each
case we identify a noise-induced drift which significantly changes the
observed properties of the system.  This is a joint work with experimental
physicists:  Giovanni Volpe (Bilkent University, Ankara) and Giuseppe
Pesce (University of Naples)  and with Scott Hottovy and Austin
McDaniel---graduate students at the University of Arizona.

Given an ordinary differential equation whose coefficients are
meromorphic functions of a complex variable, the only obstruction to
convergence of local solutions in a disc is the presence of
singularities within the disc. It was observed decades ago that this
fails if one replaces "complex" by "p-adic", e.g., consider the
exponential function. In recent work of Baldassarri, Poineau, Pulita,
and the speaker, it has emerged that the convergence properties of such
solutions in the p-adic case can be described quite simply in terms of
Berkovich analytic geometry. We will give this description (without
assuming any prior familiarity with Berkovich's theory) and mention some
applications to studying wild ramification of covers of p-adic curves.

The theory of polynomial functors allows one to make sense of the stable polynomial representation theory of the general linear group over a field of characteristic 0. It also has a good notion of specialization, so that calculations done in the "infinite limit" can be used to get information in the usual finite-dimensional siutation. (At the combinatorial level this is understood as the theory of symmetric functions vs. symmetric polynomials.) Such a theory is less well-understood in the other classical groups, but the analogous category has been introduced by Olshanskii and Penkov-Strykas. However, the specialization from the infinite case to the finite case (which is needed for applications) was not previously understood. This talk will explain joint work with Andrew Snowden and Jerzy Weyman where such a specialization is constructed and its properties analyzed. Time permitting, I will explain how this is intricately connected with the invariant theory and commutative algebra surrounding determinantal varieties. I will not assume prior knowledge of polynomial functors and representation theory of classical groups.

In this talk, I will report on a joint project with Yichao Tian. Let p be a prime unramified in a totally real field F. The Goren-Oort stratification is defined by the vanishing locus of the partial Hasse-invariants; it is an analog of the stratification of modular curve mod p by the ordinary locus and the supersingular locus. We give explicit global description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety for F.