Tue, 21 Oct 2014

14:30 - 15:30
L6

Spanning Trees in Random Graphs

Richard Montgomery
(University of Cambridge)
Abstract
Given a tree $T$ with $n$ vertices, how large does $p$ need to be for it to be likely that a copy of $T$ appears in the binomial random graph $G(n,p)$? I will discuss this question, including recent work confirming a conjecture which gives a good answer to this question for trees with bounded maximum degree.
Mon, 10 Nov 2014

16:00 - 17:00
L1

Stability of the Kerr Cauchy horizon

Jonathan Luk
(University of Cambridge)
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.

Tue, 20 May 2014

14:30 - 15:30
L6

Partition Regularity in the Naturals and the Rationals

Imre Leader
(University of Cambridge)
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.

Tue, 11 Mar 2014

13:15 - 14:00
C4

Understanding the Dynamics of Embryonic Stem Cell Differentiation: A Combined Experimental and Modeling Approach

Stanley Strawbridge
(University of Cambridge)
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.

Tue, 15 Oct 2013

14:30 - 15:30
C2

Containers for independent sets

Andrew Thomason
(University of Cambridge)
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.

Thu, 24 Oct 2013

12:00 - 13:00
L6

Nonlinear wave equations on time dependent inhomogeneous backgrounds

Dr. Shiwu Yang
(University of Cambridge)
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.

Mon, 04 Nov 2013

14:15 - 15:15
Oxford-Man Institute

Coalescing flows: a new approach

Nathanael Berestycki
(University of Cambridge)
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).

Mon, 28 Oct 2013

15:45 - 16:45
Oxford-Man Institute

Small-particle scaling limits in a regularized Laplacian growth model"

Alan Sola
(University of Cambridge)
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.

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