





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, Symplectic Geometry, and Derived 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 uptodate 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 2dimensional 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 4dimensional 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 spacetime 4manifold. Manifolds are used to understand the largescale structure of the Universe in cosmology, and the theory of relativity introduced the idea of matterenergy 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 19932000 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 1dimensional 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 7dimensional manifold with a very small radius, of about 10^{33}cm. It turns out that the geometrical structure on this 7manifold 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 1dimensional submanifold of a 3dimensional 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 3folds, which live in CalabiYau 3folds, 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. CalabiYau 3folds are the manifoldsatthebottomoftheuniverse for the 10dimensional version of String Theory (the universe has this dimension on Tuesdays and Thursdays). And special Lagrangian 3folds 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 CalabiYau 3fold, the ends have to stay on a special Lagrangian 3fold. So special Lagrangian 3folds are boundary conditions for the string, or Abranes in physical jargon.
Using physical, quantumtheoretic reasoning, String Theorists have made some seriously wayout conjectures about bizarre relationships between pairs of CalabiYau 3folds 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 3folds in M,M*. In washing up terms, the SYZ Conjecture says that you should be able to fill the washingup 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 insideout, the new family should fill up some completely different washingup 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 3folds in M with one passing through each point of M , and in particular about the smallscale 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 3fold singularities led me to conjecture the existence of invariants of CalabiYau 3folds M which 'count' special Lagrangian 3folds N in M, and are unchanged (invariant) under a large class of continuous changes to the CalabiYau structure on M. I couldn't prove this. However, translating my conjecture through Mirror Symmetry gives a mirror conjecture about invariants counting algebrogeometric 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 moreorless proved my mirror conjecture, in a series of
papers on 'Configurations in Abelian categories'. This opened up some
interesting questions on 'DonaldsonThomas invariants' which I pursued
with postdoc Yinan Song, in
a 2008 algebraic geometry paper on 'generalized DonaldsonThomas
invariants', now published in Memoirs of the AMS. Almost
simultaneously, Kontsevich and Soibelman produced their own theory of
generalized DonaldsonThomas invariants. We got our paper onto the
arXiv first (just), but Kontsevich still gets the credit in every talk
on the
subject I go to.
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 CalabiYau 3fold X, and the symplectic geometry of another CalabiYau 3fold 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}F(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 D^{b}F(X) between two Lagrangians L and L* in X is the Lagrangian Floer cohomology 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' Jholomorphic 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 cohomology HF*(L,L*) to immersed Lagrangians L,L*. In doing so I had to learn about the current state (in 20067) of the subject for embedded Lagrangians, then 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. (Matters have improved since 2007, though.)
So, since 2007 I have been working on the foundations of those areas of symplectic geometry concerned with moduli spaces of Jholomorphic curves, namely open and closed GromovWitten invariants, Lagrangian Floer cohomology, Fukaya categories, Symplectic Field Theory, ... . Any such theory must address four problems:
In
October 2011, together with together with Kobi Kremnizer, Balázs
Szendrői 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.
This has taken my research in (yet) another direction. On the geometry
side of the grant, we were thinking about generalizations of DonaldsonThomas invariants DT^{ α}(τ)
of CalabiYau 3folds, which are numbers. There are two natural ways
you can generalize this: firstly, you can find vector spaces V^{ α}(τ) with dimension dim V^{ α}(τ) = DT^{ α}(τ). This is called categorification,
and there are lots of reasons why it is a good thing to do; for
instance, in String Theory DonaldsonThomas invariants are interpreted
as 'numbers of BPS states', which are dimensions of the vector space of
BPS states, so if we can define such vector spaces V^{ α}(τ), they will
probably give a mathematical definition of BPS states in String Theory.
Secondly, you can enhance DT^{
α}(τ) from a number to an element of a larger 'motivic' ring (e.g.
of polynomials or rational functions), so that the new DT^{
α}(τ) contain more information. The way we do it, the DT^{
α}(τ) are called motivic
invariants, and again there are deep reasons why this is a good
idea.
We started in October 2011 by trying to do categorification and motivic invariants using conventional techniques in DonaldsonThomas theory, and it was not going well. Then, as with Kuranishi spaces, an idea came along from Derived Algebraic Geometry which suddenly changed everything, this time provided in a seminar by Bertrand Toën, who was not visiting his girlfriend so far as I know. In November 2011, Pantev, Toën, Vaquié and Vezzosi introduced the idea of 'kshifted symplectic structure' on a derived scheme or derived stack in Derived Algebraic Geometry, and showed that derived moduli schemes and stacks of coherent sheaves on a CalabiYau mfold have (2m)shifted symplectic structures. So CalabiYau 3fold moduli spaces have 1shifted symplectic structures. This is a new geometric structure on them, which we did not know about before.
Jointly
with subsets of {Oren BenBassat, Dennis Borisov, Chris Brav, Vittoria
Bussi, Delphine Dupont, Sven Meinhardt, Pavel Safronov, Balázs
Szendrői}, I have been writing a long, ongoing series of papers using
these new PTVV ideas to make a lot of progress in the categorification
and motivic invariants projects, and do several other interesting and
unexpected things as well. This is a second example in my career in
which understanding what Derived Algebraic Geometry had to say
completely transformed a problem which initially appeared to have not
much to do with it, and also seemed intractable. Most people that go
into Derived Algebraic Geometry never come out again: they go native,
and just stay there proving theorems about derived thingies that noone
else can understand, and so the subject has acquired a certain
reputation. But in my experience, it can also be useful for ordinary
working mathematicians.