varieties to p=2, using Dieudonné display theory.
An independent set in an $r$-uniform hypergraph is a subset of the vertices
that contains no edges. A container for the independent set is a superset
of it. It turns out to be desirable for many applications to find a small
collection of containers, none of which is large, but which between them
contain every independent set. ("Large" and "small" have reasonable
meanings which will be explained.)
Applications include giving bounds on the list chromatic number of
hypergraphs (including improving known bounds for graphs), counting the
solutions to equations in Abelian groups, counting Sidon sets,
establishing extremal properties of random graphs, etc.
The work is joint with David Saxton.
We study the nonlinear wave equations on a class of asymptotically flat Lorentzian manifolds $(\mathbb{R}^{3+1}, g)$ with time dependent inhomogeneous metric g. Based on a new approach for proving the decay of solutions of linear wave equations, we give several applications to nonlinear problems. In particular, we show the small data global existence result for quasilinear wave equations satisfying the null condition on a class of time dependent inhomogeneous backgrounds which do not settle to any particular stationary metric.
Joint work with Christophe Garban (Lyon) and Arnab Sen (Minnesota).
We prove convergence of random clusters, in the limit as the size of the individual aggregating particles tends to zero, to deterministic limits, provided the smoothing parameter does not tend to zero too fast. We also study scalings limit of the harmonic measure flow on the boundary, and show that it can be described in terms of stopped Brownian webs on the circle. In contrast to the case $\alpha=0$, the flow does not always collapse into a single Brownian motion, which can be interpreted as a random number of infinite branches being present in the clusters.