# Research Areas

Here is a flavour of some of areas our group is working in.

### Moduli Spaces and Homological Methods

A very important general theme in algebraic geometry is to study varieties which parameterise algebro-geometric structures of some kind. Examples include the moduli space of vector bundles on an algebraic curve, the moduli space of algebraic curves or the moduli space of representations of a quiver. The phrase homological methods refers to the technique of studying a variety via its (derived) category of coherent sheaves. This is analogous to the idea of studying a ring via its category of modules, or a group via its category of representations.

Moduli spaces and homological techniques are closely entwined. On the one hand, if a variety X is a moduli space of coherent sheaves on another variety Y, then the universal sheaf gives a correspondence relating sheaves on X and sheaves on Y, which in certain situations can even be an equivalence. On the other hand, considering moduli spaces of sheaves allows one to derive geometric consequences from homological results: for example, an equivalence of categories of coherent sheaves on two varieties X and Y immediately tells us that the moduli spaces of sheaves on the two varieties are isomorphic.

### Mirror Symmetry

Mirror symmetry was invented by theoretical physicists studying string theory in the early 1990s. They were working with models based on maps from topological surfaces into space-time. It was discovered that the theory only worked when space-time had 10 dimensions. Since the world we see has 3 dimensions of space and 1 of time, they suggested that the remaining 6 dimensions are curled up so tightly that they are not directly visible. The particular shape of these compactifying dimensions controls the behaviour of the theory, and in particular, for the theory to work well they are required to have the structure of Calabi-Yau manifolds of complex dimension 3.

Thanks to the fundamental theorem of Yau, it is easy to write down complex varieties whose underlying complex manifolds can be given the structure of Calabi-Yau threefolds: any smooth quintic hyper surface in 4-dimensional projective space gives an example. String theory allows physicists to associate a super-symmetric conformal field theory to such a variety. This is a hugely non-trivial structure, which cannot yet be given a mathematically-rigorous definition, but which nonetheless allows physicists to make very precise and surprising mathematical predictions. Mirror symmetry is one of the most striking of these; it predicts that every Calabi-Yau threefold X should have a mirror partner Y, which is related in a very strange way: for example counts of rational curves on X can be computed by calculating Taylor expansions of integrals of differential forms on Y, and vice versa.

### Hall algebras and Donaldson-Thomas Theory

Donaldson-Thomas invariants were originally conceived as numbers encoding topological invariants of moduli spaces of coherent sheaves on Calabi-Yau threefolds. Nowadays they are applied in a wide range of situations, for example the theory of representations of quivers, which have no direct relation to coherent sheaves.

The key requirement is that the objects parameterised by the moduli space should live in a category with the three-dimensional Calabi-Yau property. This has the consequence that the moduli spaces are morally zero-dimensional, even if non-transversal intersections lead to positive-dimensional spaces in practice. The Donaldson-Thomas invariant should be thought of as the number of points of the mythical zero-dimensional moduli space; it is defined precisely as a virtual Euler characteristic. Donaldson-Thomas invariants associated to moduli spaces of rank one sheaves on a Calabi-Yau threefold are related to Gromov-Witten invariants by the the famous conjectures of Maulik, Okounkov, Nekrasov and Pandharipande, now mostly proved by Pandharipande and Pixton.

A very general feature of Donaldson-Thomas invariants is that they obey a wall-crossing formula under changes of stability condition. For example, if one is considering moduli spaces of stable bundles on a Calabi-Yau threefold, one can consider the way the resulting numbers depend on the choice of ample line bundle used to define stability. Work of Joyce and Kontsevich-Soibelman led to a complete answer to this problem. One crucial ingredient in their theory is the use of Hall algebras of coherent sheaves. These are defined by using the stack of short exact sequences in an abelian category to define a product on the cohomology of the stack of objects.

This sounds very abstract, but in special circumstances one can give very down-to-earth realisations which amount to a convolution product on the set of all complex-valued functions on the isomorphism classes of the category. A famous example is Ringel’s theorem, which constructs the quantised enveloping algebra of a simple Lie algebra as the Hall algebra of the category of representations of the quiver obtained by orienting the associated Dynkin diagram.