Oxford

Topological Quantum Field Theories are functors from a category of bordisms of manifolds to (usually) some categorification of the notion of vector spaces. In this talk we will first discuss why mathematicians are interested in these in general and an overview of the relevant notions. After this we will have a closer look at the example of functors from the bordism category of 1-, 2- and 3-dimensional manifolds equipped with principal G-bundles, for G a finite group, to nice categorifications of vector spaces.

The formal degree is a fundamental invariant of a discrete series representation which generalizes the notion of dimension from finite dimensional representations. For discrete series with unipotent cuspidal support, a formula for formal degrees, conjectured by Hiraga-Ichino-Ikeda, was verified by Opdam (2015). For split exceptional groups, this formula was previously known from the work of Reeder (2000). I will present a different interpretation of the formal degrees of unipotent discrete series in terms of the nonabelian Fourier transform (introduced by Lusztig in the character theory of finite groups of Lie type) and certain invariants arising in the elliptic theory of the affine Weyl group. This interpretation relates to recent conjectures of Lusztig about `almost characters' of p-adic groups. The talk is based on joint work with Eric Opdam.

Fang, Lu and Yoshikawa conjectured a few years ago that a certain string-theoretic invariant (originally introduced by the physicists M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa) of Calabi-Yau threefolds is a birational invariant. This conjecture can be viewed as a "secondary" analog (in dimension three) of the birational invariance of Hodge numbers of Calabi-Yau varieties established by Batyrev and Kontsevich. Using the arithmetic Riemann-Roch theorem, we prove a weak form of this conjecture. 

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.