New ambitwistor string models are presented for a variety of theories and older models are shown to work at 1 loop and perhaps higher using a simpler formulation on the Riemann sphere.

Many people talk about properties that you would expect of a group. When they say this they are considering random groups, I will define what it means to pick a random group in one of many models and will give some properties that these groups will have with overwhelming probability. I will look at the proof of some of these results although the talk will mainly avoid proving things rigorously.

Following an idea of Bridgeland, we study the operator on the K-group of algebraic stacks, which takes a stack to its inertia stack.  We prove that the inertia operator is diagonalizable when restricted to nice enough stacks, including those with algebra stabilizers.  We use these results to prove a structure theorem for the motivic Hall algebra of a projective variety, and give a more conceptual definition of virtually indecomposable stack function.  This is joint work with Pooya Ronagh.

Let $G$ be a compact Lie group and $k$ be a field of characteristic $p\ge 0$ such that $H^*(G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category over $k$ if and only if it represents a non-zero object of that summand. Our result is based on: an explicit understanding of the wrapped Fukaya category through Koszul twisted complexes involving the zero-section and a cotangent fibre; and a functor canonically associated to the Hamiltonian $G$-action on $X$. Several examples can be studied in a uniform manner including toric Fano varieties and certain Grassmannians. 

Calabi-Yau manifolds without flux are perhaps the best-known
supergravity backgrounds that leave some supersymmetry unbroken. The
supersymmetry conditions on such spaces can be rephrased as the
existence and integrability of a particular geometric structure. When
fluxes are allowed, the conditions are more complicated and the
analogue of the geometric structure is not well understood.

In this talk, I will define the analogue of Calabi-Yau geometry for
generic D=4, N=2 backgrounds with flux in both type II and
eleven-dimensional supergravity. The geometry is characterised by a
pair of G-structures in 'exceptional generalised geometry' that
interpolate between complex, symplectic and hyper-Kahler geometry.
Supersymmetry is then equivalent to integrability of the structures,
which appears as moment maps for diffeomorphisms and gauge
transformations. Similar structures also appear in D=5 and D=6
backgrounds with eight supercharges.

As a simple application, I will discuss the case of AdS5 backgrounds
in type IIB, where deformations of these geometric structures give
exactly marginal deformations of the dual field theories.

I will explain how to relate the center of a cyclotomic quiver Hecke algebras to the cohomology of Nakajima quiver varieties using a current algebra action. This is a joint work with M. Varagnolo and E. Vasserot.
 

In 1964 Golod and Shafarevich discovered a powerful tool that gives a criteria for when a certain presentation defines an infinite dimensional algebra. In my talk I will assume the main machinery of the Golod-Shafarevich inequality for graded algebras and use it to provide counter examples to certain analogues of the Burnside problem in infinite dimensional algebras and infinite groups. Then, time dependent, I will define the Tarski number for groups relating to the Banach-Tarski paradox and show that we can using the G-S inequality show that the set of Tarski numbers is unbounded. Despite the fact we can only find groups of Tarski number 4, 5 and 6.