Tue, 26 Jan 2010

15:45 - 16:45
L3

(HoRSe seminar) Symmetric and reduced obstruction theories

Richard Thomas
(Imperial College London)
Abstract

I will describe some more of the deformation theory necessary for the first talk. This leads to a number of natural questions and counterexamples. This talk requires a strong stomach, or a fanatical devotion to symmetric obstruction theories.

Tue, 24 Nov 2009

14:30 - 15:30
L3

Dense $H$-free graphs are almost $(\chi(H)-1)$-partite

Peter Allen
(Warwick)
Abstract
Zarankiewicz showed that no $K_{r+1}$-free graph with minimum degree exceeding $(r-1)n/r$ can exist. This was generalised by Erdös and Stone, who showed that $K_{r+1}$ may be replaced by any graph $H$ with chromatic number $r+1$ at the cost of a $o(n)$ term added to the minimum degree.

Andr\'asfai, Erdös and S\'os proved a stability result for Zarankiewicz' theorem: $K_{r+1}$-free graphs with minimum degree exceeding $(3r-4)n/(3r-1)$ are forced to be $r$-partite. Recently, Alon and Sudakov used the Szemer\'edi Regularity Lemma to obtain a corresponding stability result for the Erdös-Stone theorem; however this result was not best possible. I will describe a simpler proof (avoiding the Regularity Lemma) of a stronger result which is conjectured to be best possible.
Tue, 01 Dec 2009
12:00
L3

On the classification of extremal black holes

James Lucietti
(Imperial)
Abstract

Extremal black holes are of interest as they are expected have simpler quantum descriptions than their non-extremal counterparts.  Any extremal black hole solution admits a well defined notion of a near horizon geometry which solves the same field equations. I will describe recent progress on the general understanding of such near horizon geometries in four and higher dimensions. This will include the proof of near-horizon symmetry enhancement and the explicit classification of near-horizon geometries (in a variety of settings). I will also discuss how one can use such results to prove classification/uniqueness theorems for asymptotically flat extremal vacuum black holes in four and five dimensions.

Tue, 17 Nov 2009
12:00
L3

Algebraically special solutions in more than four dimensions

Harvey Reall
(DAMTP Cambridge)
Abstract

Algebraic classification of the Weyl tensor is an important tool for solving the Einstein equation. I shall review the classification for spacetimes of dimension greater than four, and recent progress in using it to construct new exact solutions. The higher-dimensional generalization of the Goldberg-Sachs theorem will be discussed.

Tue, 17 Nov 2009

14:30 - 15:30
L3

Higher Order Tournaments

Imre Leader
(Cambridge)
Abstract
Given $n$ points in general position in the plane, how many of the triangles formed by them can contain the origin? This problem was solved 25 years ago by Boros and Furedi, who used a beautiful translation of the problem to a non-geometric setting. The talk will start with background, including this result, and will then go on to consider what happens in higher dimensions in the geometric and non-geometric cases.
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