Oxford

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.

DNA double strand breaks (DSB) are the most deleterious type of DNA damage induced by ionizing radiation and cytotoxic agents used in the treatment of cancer. When DSBs are formed, the cell attempts to repair the DNA damage through activation of a variety of molecular repair pathways. One of the earliest events in response to the presence of DSBs is the phosphorylation of a histone protein, H2AX, to form γH2AX. Many hundreds of copies of γH2AX form, occupying several mega bases of DNA at the site of each DSB. These large collections of γH2AX can be visualized using a fluorescence microscopy technique and are called ‘γH2AX foci’. γH2AX serves as a scaffold to which other DNA damage repair proteins adhere and so facilitates repair. Following re-ligation of the DNA DSB, the γH2AX is dephosphorylated and the foci disappear.

We have developed a contrast agent, 111In-anti-γH2AX-Tat, for nuclear medicine (SPECT) imaging of γH2AX which is based on an anti-γH2AX monoclonal antibody. This agent allows us to image DNA DSB in vitro in cells, and in in vivo model systems of cancer. The ability to track the spatiotemporal distribution of DNA damage in vivo would have many potential clinical applications, including as an early read-out of tumour response or resistance to particular anticancer drugs or radiation therapy.

The imaging tracer principle states that a contrast agent should not interfere with the physiology of the process being imaged. Therefore, we have investigated the influence of the contrast agent itself on the kinetics of DSB formation, repair and on γH2AX foci formation and resolution and now wish to synthesise these data into a coherent kinetic-dynamic model.