SR1

Consider the action of a complex reductive group on a complex projective variety X embedded in projective space. Geometric Invariant Theory allows us to construct a 'categorical' quotient of an open subset of X, called the semistable set. If in addition X is smooth then it is a symplectic manifold and in nice cases we can construct a moment map for the action and the Marsden-Weinstein reduction gives a symplectic quotient of the group action on an open subset of X. We will discuss both of these constructions and the relationship between the GIT quotient and the Marsden-Weinstein reduction. The quotients we have discussed provide a quotient for only an open subset of X and so we then go on to discuss how we can construct quotients of certain subvarieties contained in the complement of the semistable locus.

I will first introduce and motivate the notion of 'homological stability' for a sequence of spaces and maps. I will then describe a method of proving homological stability for configuration spaces of n unordered points in a manifold as n goes to infinity (and applications of this to sequences of braid groups). This method also generalises to the situation where the configuration has some additional local data: a continuous parameter attached to each point.

However, the method says nothing about the case of adding global data to the configurations, and indeed such configuration spaces rarely do have homological stability. I will sketch a proof -- using an entirely different method -- which shows that in some cases, spaces of configurations with additional global data do have homological stability. Specifically, this holds for the simplest possible global datum for a configuration: an ordering of the points up to even permutations. As a corollary, for example, this proves homological stability for the sequence of alternating groups.

This talk will be an introduction to the space of Bridgeland stability conditions on a triangulated category, focussing on the case of the derived category of coherent sheaves on a curve. These spaces of stability conditions have their roots in physics, and have a mirror theoretic interpretation as moduli of complex structures on the mirror variety.

For curves of genus g > 0, we will see that any stability condition comes from the classical notion of slope stability for torsion-free sheaves. On the projective line we can study the more complicated behaviour via a derived equivalence to the derived category of modules over the Kronecker quiver.

The main aim of this talk will be to present a proof of the Tsen-Lang theorem on the existence of points on complete intersections and sketch a proof of the Grabber-Harris-Starr theorem giving the existence of a section to a fibration of a rationally connected variety over a curve. Time permitting, recent work of de Jong and Starr on rationally simply connected varieties will be discussed with applications to the number theory of hypersurfaces.

An old conjecture of Hardy and Littlewood posits that on average, the number of representations of a positive integer N as a sum of k, k-th powers is "very small." Recently, it has been observed that this conjecture is closely related to properties of a discrete fractional integral operator in harmonic analysis. This talk will give a basic introduction to the two key problems, describe the  correspondence between them, and show how number theoretic methods, in particular the circle method and mean values of Weyl sums, can be used to say something new in abstract harmonic analysis.

$\AA^1$-homotopy theory is the homotopy theory for smooth algebraic varieties which uses the affine line as a replacement for the unit interval. The stable $\AA^1$-homotopy category is a generalisation of the topological stable homotopy category, and in particular, gives a setting where algebraic cohomology theories such as motivic cohomology, and homotopy invariant algebraic $K$-theory can be represented. We give a brief overview of some aspects of the construction and some properties of both the topological stable homotopy category and the new $\AA^1$-stable homotopy category.

We will state a theorem of Shouhei Ma (2008) relating the Cusps of the Kaehler moduli space to the set of Fourier--Mukai partners of a K3 surface. Then we explain the relationship to the Bridgeland stability manifold and comment on our work relating stability conditions "near" to a cusp to the associated Fourier--Mukai partner.



I will sketch a method to prolong certain classes of differential equations on manifolds using Lie algebra cohomology. The talk will be based on articles by Branson, Cap, Eastwood and Gover (arXiv:math/0402100 and ESI preprint 1483).

All of Joyce's constructions of compact manifolds with special holonomy are in some sense generalisations of the Kummer construction of a K3 surface. We will begin by reviewing manifolds with special holonomy and the Kummer construction. We will then describe Joyce's constructions of compact manifolds with holonomy G_2 and Spin(7).