Oxford

String compactifications incorporating non-vanishing H-flux have received increased attention over the past decade for their potential relevance to the moduli stabilization problem. Their internal spaces are in general not Kähler and, therefore, not Calabi-Yau. In the heterotic string an important technical problem is to construct gauge bundles on such spaces. I will present a method of how to explicitly construct gauge bundles over homogeneous nearly-Kähler manifolds of dimension six and discuss some of the arising implications for model building.

Generalized Donaldson-Thomas invariants $\bar{DT}^\alpha(\tau)$ defined by Joyce and Song are rational numbers which 'count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern character $\alpha$ on a Calabi-Yau 3-fold X, where $\tau$ denotes Gieseker stability for some ample line bundle on X. The theory of Joyce and Song is valid only over the field $\mathbb C$. We will extend it to algebraically closed fields $\mathbb K$ of characteristic zero.

We will describe the local structure of the moduli stack $\mathfrak M$ of coherent sheaves on X, showing that an atlas for $\mathfrak M$ may be written locally as the zero locus of an almost closed 1-form on an \'etale open subset of the tangent space of $\mathfrak M$ at a point, and use this to deduce identities on the Behrend

function $\nu_{\mathfrak M}$ of $\mathfrak M$. This also yields an extension of generalized Donaldson-Thomas theory to noncompact Calabi-Yau 3-folds.

Finally, we will investigate how our argument might yield generalizations of the theory to a even wider context, for example the derived framework using Toen's theory and to motivic Donaldson-Thomas theory in the style of Kontsevich and Soibelman.

The study of superconformal Chern-Simons theories has led to a deeper understanding of M-theory and a new example of the AdS/CFT correspondence. In this talk, I will give an overview of superconformal Chern-Simons theories and their gravity duals. I will also describe some recent work on scattering amplitudes in these theories.

A polar space $\Pi$ is a geometry whose elements are the totally isotropic subspaces of a vector space $V$ with respect to either an alternating, Hermitian, or quadratic form. We may form a new geometry $\Gamma$ by removing all elements contained in either a hyperplane $F$ of $\Pi$, or a hyperplane $H$ of the dual $\Pi^*$. This is a \emph{biaffine polar space}.

We will discuss two specific examples, one with automorphism group $q^6:SU_3(q)$ and the other $G_2(q)$. By considering the stabilisers of a maximal flag, we get an amalgam, or "glueing", of groups for each example. However, the two examples have "similar" amalgams - this leads to a group recognition result for their automorphism groups.

I shall discuss recent work in which bounds are obtained, generalising/specialising earlier work for general linear groups