Tue, 03 Mar 2015
14:30

Tiling the grid with arbitrary tiles

Vytautas Gruslys
(University of Cambridge)
Abstract

Suppose that we have a tile $T$ in say $\mathbb{Z}^2$, meaning a finite subset of $\mathbb{Z}^2$. It may or may not be the case that $T$ tiles $\mathbb{Z}^2$, in the sense that $\mathbb{Z}^2$ can be partitioned into copies of $T$. But is there always some higher dimension $\mathbb{Z}^d$ that can be tiled with copies of $T$? We prove that this is the case: for any tile in $\mathbb{Z}^2$ (or in $\mathbb{Z}^n$, any $n$) there is a $d$ such that $\mathbb{Z}^d$ can be tiled with copies of it. This proves a conjecture of Chalcraft.

Mon, 27 Oct 2014

15:45 - 16:45
Oxford-Man Institute

Phase transitions in Achlioptas processes

Lutz Warnke
(University of Cambridge)
Abstract

In the Erdös-Rényi random graph process, starting from an empty graph, in each step a new random edge is added to the evolving graph. One of its most interesting features is the `percolation phase transition': as the ratio of the number of edges to vertices increases past a certain critical density, the global structure changes radically, from only small components to a single giant component plus small ones.

In this talk we consider Achlioptas processes, which have become a key example for random graph processes with dependencies between the edges.

Starting from an empty graph these proceed as follows: in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. We discuss why, for a large class of rules, the percolation phase transition is qualitatively comparable to the classical Erdös-Rényi process.

                                                      

Based on joint work with Oliver Riordan.

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.

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