Oxford

We introduce the notion of a real cubing. Roughly speaking, real cubings are to CAT(0) cube complexes what real trees are to simplicial trees. We develop an analogue of the Rips’ machine and establish the structure of groups acting nicely on real cubings.

The induced graph removal lemma states that for any fixed graph $H$ on $h$ vertices and any $e\textgreater 0$ there exists $d\textgreater0$ such that any graph $G$ with at most $d n^h$ induced copies of $H$ may be made $H$-free by adding or removing atmost $e n^2$ edges. This fact was originally proven by Alon, Fischer, Krivelevich and Szegedy. In this talk, we discuss a new proof and itsrelation to various regularity lemmas. This is joint work with Jacob Fox.

We discuss conjectures and results concerning small gaps between primes. In particular, we consider the work of Goldston, Pintz and Yildrim which shows that infinitely often there are gaps which have size an arbitrarily small proportion of the average gap.

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.