Oxford

I will discuss how \zeta(3) occurs in quantum corrections to the Einstein action, and how this causes \zeta(3) to be seen in the moduli space of CY manifolds. I will also draw attention to the fact that the dependence of the moduli space on \zeta(3) has a p-adic analogue.

Higgs bundles have a rich structure and play a role in many different areas including gauge theory, hyperkähler geometry, surface group representations, integrable systems, nonabelian Hodge theory, mirror symmetry and Langlands duality. In this introductory talk I will explain some basic notions of G-Higgs – including the Hitchin fibration and spectral data - and illustrate how this relates to mirror symmetry.

 "We give a diophantine criterion for a polynomial with rational coefficients not to have any
rational zero, i.e. an existential formula in terms of the coefficients expressing this property. This can be seen as a kind of restricted
model-completeness for Q and answers a question of Koenigsmann."

It is unknown whether a bound on axion field ranges exists within quantum gravity. We study axion field ranges using extended supersymmetry, in particular allowing an analysis within strongly coupled regions of moduli space. We apply this strategy to Calabi-Yau compactifications with one and two Kähler moduli. We relate the maximally allowable decay constant to geometric properties of the underlying Calabi-Yau geometry. In all examples we find a maximal field range close to the reduced Planck mass (with the largest field range being 3.25 $M_P$). On this perspective, field ranges relate to the intersection and instanton numbers of the underlying Calabi-Yau geometry.

A large number of examples of compact G2 manifolds, relevant to supersymmetric compactifications of M-Theory to four dimensions, can be constructed by forming a twisted connected sum of two appropriate building blocks times a circle. These building blocks, which are appropriate K3-fibred threefolds, are shown to have a natural and elegant construction in terms of tops, which parallels the construction of Calabi-Yau manifolds via reflexive polytopes.

  
The Cartan model computes the equivariant cohomology of a smooth manifold X with 
differentiable action of a compact Lie group G, from the invariant functions on 
the Lie algebra with values in differential forms and a deformation of the de Rham 
differential. Before extracting invariants, the Cartan differential does not square 
to zero. Unrecognised was the fact that the full complex is a curved algebra, 
computing the quotient by G of the algebra of differential forms on X. This 
generates, for example, a gauged version of string topology. Another instance of 
the construction, applied to deformation quantisation of symplectic manifolds, 
gives the BRST construction of the symplectic quotient. Finally, the theory for a 
X point with an additional quadratic curving computes the representation category 
of the compact group G.

Recent results showed that the low energy expansion of closed superstring amplitudes can be expressed in terms of

single-valued multiple elliptic polylogarithms. I will explain how these functions may be defined as iterated integrals on the torus and

sketch how they arise from Feynman integrals.