Titles and abstracts of public and plenary talks
Sir Roger Penrose, University of Oxford 17:00-18:00
Title: Seeing Through the Big Bang into Another World: an Exercise in Conformal Geometry
Abstract: Einstein’s general theory of relativity describes space and time as a 4-dimensional curved space with a metric which determines the measures of time and distance. This metric is a quantity determined by 10 numbers per space-time point. Of these, 9 fix the light cone at each space-time point, determining causality and the propagation of light. The remaining number fixes the scale of space and time. The light cones themselves determine the conformal structure of space-time. The proposal of Conformal Cyclic Cosmology (abbreviated CCC) asserts that what we presently regard as the entire history of our universe, from its Big-Bang origin to its indefinitely expanding future, is but one aeon in an unending succession of similar such aeons, where the infinite future of each conformally matches to the big bang of the next. CCC predicts that supermassive black-hole encounters in the aeon prior to ours would be observable to us as families of concentric rings of unusual temperature structure in the cosmic microwave background. Recent analysis of data from the WMAP satellite has been argued to provide possible confirmation of this signal, allowing us to "see through" our Big Bang to such events occurring in the aeon prior to ours. The status of this controversial proposal will be discussed.
Monday, 18 April
Timothy Gowers, University of Cambridge 15:30-16:30
Title: Polymath and the density Hales-Jewett theorem
Abstract: The Hales-Jewett theorem is a natural combinatorial generalization of van der Waerden's theorem about arithmetic progressions. The density Hales-Jewett theorem generalizes Szemer\'edi's theorem in a similar way. Until recently, the only known proof, due to Furstenberg and Katznelson, was long and complex and made heavy use of ergodic theory. I shall talk about a new and elementary proof. This proof was discovered as the result of a multiple collaboration that was carried out in public with the help of blogs and a wiki. I shall spend part of the talk discussing this unusual proof discovery process.
Tuesday, 19 April
EPSRC lecture 11:30-12:30
David Harman, Head of the Mathematical Sciences Programme at EPSRC
Title: Presentation and Discussion on Mathematics and the EPSRC
Abstract: The last few months have seen the publication of the new EPSRC Delivery and Implementation Plans following the Government's Comprehensive Spending Review, as well as the report of the International Review of Mathematics held last winter. David Harman, Head of Mathematical Sciences at the EPSRC, will give a short presentation on the EPSRC plans followed by an open discussion.
Raphael Rouquier, University of Oxford 17:00-18:00
Title: Higher Representation Theory
Abstract: Classical representation theory studies symmetries of sets and vector spaces, whereas higher representation theory studies symmetries of categories and higher structures. There is a class of such higher symmetries encoded by simple Lie algebras, which can be viewed as the next step after quantization. We will discuss algebraic, geometrical and topological aspects.
Wednesday, 20 April
Elon Lindenstrauss, Hebrew University 11:30-12:40
and Princeton University
Title: From multiplying by 2 and 3 to quantum unique ergodicity
Abstract: In the 1960's Furstenberg noticed a curious fact: while the map x ->2x mod 1 on R/Z has plenty of invariant sets, there are very few sets that are simultaneously invariant under x2 and x3. Implicitly, this phenomenon was also obseved by Cassels and Swinnerton-Dyer in the 50's in the context of certain problems in the geometry of numbers. I will survey some recent development in this direction, and explain how it is related to a seemingly unrelated topic - the study of eigenfunctions of the Laplacian on hyperbolic surfaces.
Karen Vogtmann, Cornell University 17:00-18:00
Title: The topology and geometry of automorphism groups of free groups
Abstract: Free groups, free abelian groups and fundamental groups of closed orientable surfaces are the most basic and well-understood examples of infinite discrete groups. The automorphism groups of these groups, in contrast, are some of the most complex and intriguing groups in all of mathematics. In this lecture I will concentrate on groups of automorphisms of free groups, while drawing analogies with GL(n,Z) and surface mapping class groups. I will discuss modern techniques for studying automorphism groups of free groups, which include a mixture of topological, algebraic and geometric methods.
Thursday, 21 April
Franz Pedit, Tuebingen University and UMass Amherst 11:30-12:30
Title: Advances in surface geometry: theory and experiment
Abstract: Throughout the modern era some of the richest and most challenging problems in both differential geometry and geometric analysis have originated in the study of special surfaces in 3-space. Real dimension two has the virtue of being a natural setting for single variable complex analysis and the possible topological types are well understood. As a result, surface geometry has become an arena in which novel contributions have come from such diverse areas as integrable systems, moduli spaces of bundles, nonlinear variational problems, and computer visualization. In fact, it has often provided opportunities for the methods of one area to be brought to bear on others. This lecture will explore this phenomenon by focusing on the global theory of constant mean curvature surfaces and what role computer visualization plays in the formulation and testing of conjectures.