Oxford

We will demonstrate the following. Category theory, usually conceived as some very abstract form of metamathematics, is present everywhere around us. Explicitly, we show how it provides a kindergarten version of quantum theory, an how it will help Google to understand sentences rather than words.

Some references are:

-[light] BC (2010) "Quantum picturalism". Contemporary Physics 51, 59-83. arXiv:0908.1787 
-[a bit heavier] BC and Ross Duncan (2011) "Interacting quantum observables: categorical algebra and diagrammatics". New Journal of Physics 13, 043016. arXiv:0906.4725
-[light] New Scientist (8 December 2010) "Quantum links let computers understand language". www.cs.ox.ac.uk/people/bob.coecke/NewScientist.pdf
-[a bit heavier] BC, Mehrnoosh Sadrzadeh and Stephen Clark (2011) "Mathematical foundations for a compositional distributional model of meaning". Linguistic Analysis - Lambek Festschrift. arXiv:1003.439

Saturated fusion systems are a relatively new class of objects that are often described as the correct 'axiomatisation' of certain p-local phenomena in algebraic topology. Despite these geometric beginnings however, their structure is sufficiently rigid to afford its own local theory which in some sense mimics the local theory of finite groups. In this talk, I will briefly motivate the definition of a saturated fusion system and discuss a remarkable result of Michael Aschbacher which proves that centralisers of normal subsystems of a saturated fusion system, F, exist and are themselves saturated. I will then attempt to justify his definition in the case where F is non-exotic by appealing to some classical group theoretic results. If time permits I will speculate about a topological characterisation of the centraliser as the set of homotopy fixed points of a certain action on the classifying space of F.

I will present a history of the problem, relate it to other conjectures, and, with time permitting, indicate recent developments. The focus will primarily be group-theoretic and intended for the non-specialist.

The WZW functional for a map from a surface to a Lie group has a role in the theory of harmonic maps, and it also arises as the determinant of a d-bar operator on the surface, as the action functional for a 2-dimensional quantum field theory, as the partition function of 3-dimensional Chern-Simons theory on a manifold with boundary, and as the norm-squared of a state-vector. It is intimately related to the quantization of the symplectic manifold of flat bundles on the surface, a fascinating test-case for different approaches to geometric quantization. It is also interesting as an example of interpolation between commutative and noncommutative geometry. I shall try to give an overview of the area, focussing on the aspects which are still not well understood.

I will describe the recent construction of new supersymmetric compactifications of the heterotic string which yield just the spectrum of the MSSM at low energies. The starting point is the standard embedding solution on a Calabi-Yau manifold with Euler number -6 with various choices of Wilson lines, i.e., various choices of discrete holonomy for the gauge bundle. Although they yield three net generations of standard model matter, such models necessarily have a larger gauge group than the standard model, as well as exotic matter content. Families of stable bundles can be obtained by deformation of these simple solutions, the deformation playing the dual role of partially breaking the gauge group and reducing the matter content, and in this way we construct more realistic models. The moduli space breaks up into various branches depending on the initial choice of Wilson lines, and on eight of these branches we find models with exactly the standard model gauge group, three generations of quarks and leptons, two Higgs doublets, and no other massless charged states. I will also comment on why these are possibly the unique models of this type.

I will discuss an efficient algorithm for computing certain special values of p-adic L-functions, giving an application to the explicit construction of

rational points on elliptic curves.