Stability of the Kerr Cauchy horizon
Abstract
The celebrated strong cosmic censorship conjecture in general relativity in particular suggests that the Cauchy horizon in the interior of the Kerr black hole is unstable and small perturbations would give rise to singularities. We present a recent result proving that the Cauchy horizon is stable in the sense that spacetime arising from data close to that of Kerr has a continuous metric up to the Cauchy horizon. We discuss its implications on the nature of the potential singularity in the interior of the black hole. This is joint work with Mihalis Dafermos.
14:15
Partition Regularity in the Naturals and the Rationals
Abstract
A system of linear equations is called partition regular if, whenever the naturals are finitely coloured, there is a monochromatic solution of the equations. Many of the classical theorems of Ramsey Theory may be phrased as assertions that certain systems are partition regular.
What happens if we are colouring not the naturals but the (non-zero) integers, rationals, or reals instead? After some background, we will give various recent results.
Construction of 2-adic integral canonical models of Hodge-type Shimura varieties
Abstract
We extend Kisin's construction of integral canonical models of Hodge-type Shimura
varieties to p=2, using Dieudonné display theory.
Understanding the Dynamics of Embryonic Stem Cell Differentiation: A Combined Experimental and Modeling Approach
Abstract
Pluripotency is a key feature of embryonic stem cells (ESCs), and is defined as the ability to give rise to all cell lineages in the adult body. Currently, there is a good understanding of the signals required to maintain ESCs in the pluripotent state and the transcription factors that comprise their gene regulatory network. However, little is known about how ESCs exit the pluripotent state and begin the process of differentiation. We aim to understand the molecular events associated with this process via an experiment-model cycle.
Containers for independent sets
Abstract
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.
Nonlinear wave equations on time dependent inhomogeneous backgrounds
Abstract
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.
Coalescing flows: a new approach
Abstract
The coalescing Brownian flow on $\R$ is a process which was introduced by Arratia (1979) and Toth and Werner (1997), and which formally corresponds to starting coalescing Brownian motions from every space-time point. We provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks. The invariance principle holds under a finite variance assumption and is thus optimal. In a series of previous works, this question was studied under a different topology, and a moment of order $3-\eps$ was necessary for the convergence to hold. Our proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work -- in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the non-crossing property.
Joint work with Christophe Garban (Lyon) and Arnab Sen (Minnesota).
Small-particle scaling limits in a regularized Laplacian growth model"
Abstract
With F. Johansson Viklund (Columbia) and A. Turner (Lancaster), we have studied a regularized version of the Hastings-Levitov model of random Laplacian growth. In addition to the usual feedback parameter $\alpha>0$, this regularized version of the growth process features a smoothing parameter $\sigma>0$.
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.