My research is mostly in Differential Geometry, with occasional forays into some more esoteric areas of Theoretical Physics, and more recently diversions into Algebraic Geometry and Symplectic Geometry. It is difficult to explain the ideas involved to someone who is not already mathematically literate to beyond degree level, and it's not easy to explain the point of it even to someone who is, but here goes.
Nowadays, geometry is not about triangles and circles and Euclid, who went out with the Ark. Instead, the up-to-date geometer is interested in manifolds. A manifold is a curved space of some dimension. For example, the surface of a sphere, and the torus (the surface of a doughnut), are both 2-dimensional manifolds.
Manifolds exist in any dimension. One branch of geometry, called manifold topology, aims to describe the shape of manifolds, using algebraic invariants. For example, the sphere and the torus are different manifolds because the torus has a 'hole', but the sphere does not. In higher dimensions manifolds become very complicated, both to describe topologically, and to imagine in a meaningful way.
Another branch of geometry is the study of geometrical structures on manifolds. Here the manifold itself is only the background for some mathematical object defined upon it, as a canvas is the background for an oil painting. This kind of geometry, although very abstract, is closer to the real world than you might think. Einstein's theory of General Relativity describes the Universe - the whole of space and time - as a 4-dimensional manifold.
Space itself is not flat, but curved. The curvature of space is responsible for gravity, and at a black hole space and time are so curved they get knotted up. Everything in the universe - light, subatomic particles, pizzas, yourself - is described in terms of a geometrical structure on the space-time 4-manifold. Manifolds are used to understand the large-scale structure of the Universe in cosmology, and the theory of relativity introduced the idea of matter-energy equivalence, which led to nuclear power, and the atomic bomb.
Much of my own research has been concerned with some very special geometrical structures, called special holonomy groups, which only exist in certain dimensions. I am interested in constructing examples of these structures - sometimes the first examples ever found - and in trying to understand their properties. From about 1993-2000 I did a lot of work on two unusual geometric structures: the exceptional holonomy groups G2 and Spin(7) in dimensions 7 and 8.
When I first started doing this, I thought it would be no use to anyone, ever. But then I found out about a branch of Theoretical Physics called String Theory. Basically, there are two modern theories of physics: general relativity, which describes the universe at very large scales, and quantum theory, which describes the universe at very small scales. But, embarrassingly, these theories are incompatible, and physicists have never yet succeeded in fitting them together in one consistent theoretical framework.
The best chance of unifying these two theories seems to be through String Theory, which is a bizarre branch of Theoretical Physics that models particles not as points but as 1-dimensional objects - as 'loops of string'. One weird feature of String Theory is that it prescribes the dimension of the Universe, and although physicists keep changing their minds about what the dimension should be, the answer is never four. First the dimension of the Universe was supposed to be 26, and then went down to 10.
A few years ago, though, String Theorists declared that perhaps, after all, the dimension of the Universe is 11. To account for the difference between this and the four dimensions we see, the other 7 dimensions have to be 'rolled up' into a 7-dimensional manifold with a very small radius, of about 10-33cm. It turns out that the geometrical structure on this 7-manifold must be the exceptional holonomy group G2, one of the structures of which I had found the only known examples. And so, until it went out of fashion again six months later, a number of physicists were writing research papers about 'Joyce manifolds', which was nice while it lasted.
Around about 2000, my attention shifted to calibrated geometry. In most branches of mathematics, if you're studying some class of 'things', there is usually a natural class of 'subthings' that live inside them; so you have groups and subgroups, and so on. In differential geometry, the 'things' are manifolds M, and the 'subthings' are submanifolds, which are subsets N of M which are themselves manifolds, of smaller dimension than M, and smoothly embedded in M. So, for instance, a knot in space is a 1-dimensional submanifold of a 3-dimensional manifold.
When the 'things' are manifolds M with special holonomy, then the natural class of 'subthings' are called calibrated submanifolds, which are submanifolds N compatible with the geometric structure on M in a certain way. Effectively, N must satisfy a nonlinear partial differential equation. One consequence of this equation is that closed calibrated submanifolds N have minimal volume: any other submanifold N* close to N has volume at least as large as that of N. So we can think of calibrated submanifolds as being like bubbles, because the surface tension in a bubble makes it minimize its area subject to some constraints, such as containing a fixed volume of air, or having its boundary along some fixed curve. Think of this next time you do the washing up.
Anyway, what I wanted to study was the singularities of calibrated submanifolds, which are bad points where the smooth structure of the submanifold breaks down. The simplest kinds of singularities look like the vertex of a cone (quite complicated cones, though). There are several reasons why such singularities are important. One is that singular calibrated submanifolds can occur as limits of nonsingular calibrated submanifolds, so we need to know about singularities to understand what kinds of changes can happen to nonsingular calibrated submanifolds. In washing up terms, I'm asking: how do bubbles pop?
It turns out, at least for the questions I'm interested in, the difficulty of understanding singularities increases with dimension - both the dimension of the submanifold N, and of the ambient manifold M. So I decided to focus on a class of calibrated submanifolds called special Lagrangian 3-folds, which live in Calabi-Yau 3-folds, with holonomy SU(3), since this is the case with both the smallest submanifold dimension, 3, and smallest ambient manifold dimension, 6, which is not already well understood. (Calibrated submanifolds of dimension 1 are straight lines, and of dimension 2 are complex curves, so 3 is the first interesting dimension.) I found lots of examples of singularities, and developed a general theory of a special class called isolated conical singularities.
Once again, though, the String Theorists wanted to join in. Calabi-Yau 3-folds are the manifolds-at-the-bottom-of-the-universe for the 10-dimensional version of String Theory (the universe has this dimension on Tuesdays and Thursdays). And special Lagrangian 3-folds also have a rôle in String Theory: one can consider not just closed loops of string, but also strings with ends, and when a string with ends moves in a Calabi-Yau 3-fold, the ends have to stay on a special Lagrangian 3-fold. So special Lagrangian 3-folds are boundary conditions for the string, or A-branes in physical jargon.
Using physical, quantum-theoretic reasoning, String Theorists have made some seriously way-out conjectures about bizarre relationships between pairs of Calabi-Yau 3-folds M,M*, under the general name of 'Mirror Symmetry'. Much to mathematicians' surprise, the conjectures seem to be true. One reasonably geometric form of this relationship is called the SYZ Conjecture, and has to do with families of special Lagrangian 3-folds in M,M*. In washing up terms, the SYZ Conjecture says that you should be able to fill the washing-up bowl M with bubbles (including some popping bubbles), with exactly one bubble passing through each point. And then, if you turn each bubble in the family inside-out, the new family should fill up some completely different washing-up bowl M*, with exactly one bubble passing though each point.
I don't have any hope of proving the whole SYZ Conjecture myself, but I have been thinking about its main ingredient, families ('fibrations') of special Lagrangian 3-folds in M with one passing through each point of M , and in particular about the small-scale behaviour of the fibrations near a singularity of one of the fibres. I've been able to construct many local examples of such fibrations, and show they need not be smooth, for instance.
In 2003 I changed direction again. My research on special Lagrangian 3-fold singularities led me to conjecture the existence of invariants of Calabi-Yau 3-folds M which 'count' special Lagrangian 3-folds N in M, and are unchanged (invariant) under a large class of continuous changes to the Calabi-Yau structure on M. I couldn't prove this. However, translating my conjecture through Mirror Symmetry gives a mirror conjecture about invariants counting algebro-geometric objects (semistable coherent sheaves) on the mirror M*, which I thought I would be able to prove, using all the machinery of algebraic geometry.
So, over a couple of years I retrained myself from a moderately competent differential geometer to a fully incompetent algebraic geometer. I more-or-less proved my mirror conjecture, in a series of papers on 'Configurations in Abelian categories' that is so long and complicated that only my trusty postdoc Sven can face reading more than the introductions. This opened up some interesting questions on 'Donaldson-Thomas invariants' which I pursued with postdoc Yinan Song, finally finishing a 180 page algebraic geometer paper on generalized Donaldson-Thomas invariants in 2009.
In 2006 I set up a research group of postdocs and graduate students in the general area of Homological Mirror Symmetry. This is a duality between the algebraic geometry of one Calabi-Yau 3-fold X, and the symplectic geometry of another Calabi-Yau 3-fold X*. So I needed also to retrain myself as a fully incompetent symplectic geometer. This is an ongoing project.
The area of symplectic geometry involved in Homological Mirror Symmetry is called Fukaya categories. If X is a symplectic manifold, the derived Fukaya category DbFuk(X) of X is a triangulated category whose objects are (roughly) Lagrangian submanifolds L in X, or more generally complexes of Lagrangian submanifolds in X. The morphisms Hom(L,L*) in DbFuk(X) between two Lagrangians L and L* in X is the Lagrangian Floer homology HF*(L,L*) of L and L*, which is (roughly) the homology of a complex of chains on L∩L*. The differential on this complex is defined by 'counting' J-holomorphic curves in X with boundary in L+L*.
I began with a joint project with postdoc Manabu Akaho, to extend the definition of Lagrangian Floer homology HF*(L,L*) to immersed Lagrangians L,L*. In doing so I had to learn about the current state of the subject for embedded Lagrangians, a 1400 page, unfinished book by Fukaya, Oh, Ohta and Ono, and I found that the whole area was (in my opinion) a horrendously complicated horrible mess, and also that the foundations of the area are not well worked out; the dynamics of the subject during its inception in the 1990s seem to have driven people to proceed as quickly as possible to claiming proofs of important applications in geometry, whilst passing rather hastily over the formidable analytic problems involved.
So, since 2007 I have been working on the foundations of those areas of symplectic geometry concerned with moduli spaces of J-holomorphic curves, namely Lagrangian Floer homology, and open and closed Gromov-Witten invariants. Any such theory must address four problems:
For the moment (2009 - ∞) I am working on problems (a) and (c), though I hope to move on to (d) later. For (a), I wish to find the 'correct' definition of Kuranishi space, and develop the theory of Kuranishi spaces rigorously. I believe I have found an answer to this in my theory of 'd-manifolds and d-orbifolds', which you can read about here. For (c), I am developing new homology and cohomology theories, whose (co)chains are Kuranishi spaces (or better, d-orbifolds) together with some extra data. Then the virtual chain/cycle of a moduli space is just the moduli space itself, with some choice of extra data. Thus, we can form virtual chains or cycles without perturbing the moduli spaces. This will lead to huge technical simplifications in the theory of Lagrangian Floer homology; a large part of Fukaya-Oh-Ohta-Ono is taken up by algebraic manoeuvres to get round the problems caused by perturbing moduli spaces to form virtual chains.
In the future, I hope to bring together my interests in Fukaya categories and Lagrangian Floer homology, and in special Lagrangian geometry. If X is a Calabi-Yau manifold, there is a (rather imprecise) conjecture that the derived Fukaya category DbFuk(X), as a triangulated category, should carry a Bridgeland stability condition Z such that the Z-(semi)stable objects in DbFuk(X) should be represented by special Lagrangians. Thus, there should be some deep connection between special Lagrangian geometry and Floer theory. As a Bridgeland stability condition is at some level an algebraic object, the conjecture suggests that viewed in the right way, the existence or not of special Lagrangians, and the 'boundaries' of their moduli spaces, should be controlled by rigid algebraic criteria, rather than happening in some chaotic way when nasty singularities occur.
Furthermore, a Lagrangian L gives an object in DbFuk(X) if and only if it has unobstructed Lagrangian Floer homology. So the conjecture suggests that having unobstructed Lagrangian Floer homology should be an important condition in special Lagrangian geometry. One question I would like to explore is whether the kinds of singularities of special Lagrangian submanifolds that can occur as limits of families of nonsingular special Lagrangians with unobstructed Lagrangian Floer homology, are simpler than those singularities occurring as limits of families of nonsingular special Lagrangians with obstructed Lagrangian Floer homology.
In October 2011, together with together with Kobi Kremnizer, Balázs Szendröi and Raphaël Rouquier, I started (yet) another research group on Geometry and Representation Theory, funded by an EPSRC Programme Grant for £1.8 million. You can read about our research programme here. I'll report further on how this has developed my research interests, once I have found out what Representation Theory is.