Prof. Dominic Joyce FRS

My research is mostly in Differential Geometry, with occasional forays into some more esoteric areas of Theoretical Physics. 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.

My own research is 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.

Since 2000, my attention has 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.

Recently (since 2003) I have 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, for a couple of years I have been retraining myself from a moderately competent differential geometer to a fully incompetent algebraic geometer, and I have more-or-less proved my mirror conjecture -- in a series of papers on 'Configurations in Abelian categories' that is so long and complicated that I can't get anyone to read it, though if I ever get another graduate student I might make them do it. This has opened up some interesting questions on 'Donaldson-Thomas invariants' that I hope to pursue in future.

When I've finished my current algebraic projects I'm intending to return to special Lagrangian geometry again. One angle I'd like to explore is the importance of the condition that a Lagrangian have defined Floer homology. To do this I will need to further expand my areas of ignorance, and retrain myself to become incompetent in symplectic geometry and Lagrangian Floer homology as well.

I am assembling a research group of postdocs and graduate students in the general area of Homological Mirror Symmetry, working on both sides of the mirror, with particular foci (on the Lagrangian side) on special Lagrangians, their singularities, and Lagrangian Floer homology, and (on the complex side) on Donaldson-Thomas invariants and extensions of them, and moduli of (semi)stable coherent sheaves; and on both sides, on stability conditions (of Bridgeland type) on triangulated categories (of Lagrangians or coherent sheaves), and the effect of changing stability conditions (upon invariants counting (semi)stable objects, and other structures).

At present I have EPSRC funding for a Visiting Fellow, Dr Manabu Akaho from Tokyo, who will be coming to Oxford for 1 year from October 2006 to work on a symplectic geometry project to do with Lagrangian Floer homology for immersed Lagrangian submanifolds, and a postdoc, Dr Yinan Song of UBC, Canada, who will be coming to Oxford for 3 years from October 2006 to work on an algebraic geometry project to do with Donaldson-Thomas invariants, together with a DPhil student, Martijn Kool from Utrecht University. Over 2007-9 the group will also host two EU Marie Curie Research Fellows, Spiro Karigiannis from Michigan State who will be working on a new construction of G₂-manifolds, and Tommaso Pacini from Imperial College who will be working on special Lagrangian singularities. If you're interested in joining this group, and especially if you have relevant expertise or can bring your own funding, then please contact me.