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* *G*_{2} 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^{-33}cm. It turns out that the geometrical
structure on this 7-manifold must be the exceptional holonomy group
*G*_{2}, 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 *D ^{b}*Fuk(

I began with a joint project with postdoc Manabu Akaho, to extend the definition of Lagrangian Floer homology
*HF _{*}*(

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:

- Decide what kind of geometric structure you want to put on moduli spaces of
*J*-holomorphic curves. That is, we must define some class of geometric spaces, perhaps along the lines of topological spaces, manifolds, schemes in algebraic geometry,.... Develop the basic properties of such spaces, in particular analogues of differential-geometric concepts such as smooth maps, submersions, orientations, boundaries. - Prove that the moduli spaces of
*J*-holomorphic curves in your problem carry this geometric structure (preferably uniquely). Prove that relationships between moduli spaces (for instance, writing the 'boundary' of a moduli space in terms of fibre products of other moduli spaces) lift to identities between these geometric spaces. - Make a machine which translates spaces with your geometric structure into algebraic-topological data. For instance, this
data might be a rational number (the 'number of points' in the geometric space), or a homology class (a
*virtual class*), or a chain or cycle in some chain complex (a*virtual chain*or*virtual cycle*). Ensure that this machine translates identities between geometric spaces into identities between the algebraic-topological data. - Use the algebraic-topological data and the identities upon it to prove some interesting geometrical applications, for example, define Gromov-Witten invariants or Lagrangian Floer homology, prove the Arnold Conjecture,.... This is now a problem in algebra.

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
*D ^{b}*Fuk(

Furthermore, a Lagrangian *L* gives an object in *D ^{b}*Fuk(

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.