My books

Research Interests

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 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'. This opened up some interesting questions on 'Donaldson-Thomas invariants' which I pursued with postdoc Yinan Song, in a 2008 algebraic geometry paper on 'generalized Donaldson-Thomas invariants', now published in Memoirs of the AMS. Almost simultaneously, Kontsevich and Soibelman produced their own theory of generalized Donaldson-Thomas 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 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 DbF(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 DbF(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 LL*. 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 cohomology HF*(L,L*) to immersed Lagrangians L,L*. In doing so I had to learn about the current state (in 2006-7) 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 J-holomorphic curves, namely open and closed Gromov-Witten invariants, Lagrangian Floer cohomology, Fukaya categories, Symplectic Field Theory, ... . Any such theory must address four problems:

  1. 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.
  2. 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.
  3. 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 (co)homology class (a virtual class), or a chain or cycle in some chain complex (a virtual chain or virtual cycle) computing (co)homology. Ensure that this machine translates identities between geometric spaces into identities between the algebraic-topological data.
  4. 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 cohomology, prove the Arnold Conjecture, .... This is now a problem in homological algebra.
  There are two main approaches to the most general version of this machine: the Fukaya school, due to Fukaya, Oh, Ohta and Ono,  in which the answer to (a) is called a Kuranishi space, introduced by Fukaya and Ono in 1999, and the Hofer school, due to Hofer, Wysocki and Zehnder, in which the answer to (a) is called a polyfold, introduced by Hofer in 2005. The Fukaya school have answers to all of (a)-(d), and are well into applications of their machinery, but there is controversy in the symplectic geometry community over the correctness / completeness of parts of their answers to (a),(b),(c), especially (b), which is formidably difficult. The Hofer school is proceeding more slowly, and currently (mid-2015) are still mostly working on (a),(b), as far as I know. Their work is widely accepted as rigorous, I believe.
  I should also say that there is also a third approach, exemplified for instance by Paul Seidel and his collaborators Mohammed Abouzaid, Nick Sheridan and Ivan Smith, which manages to avoid the messy analytic problems in (a)-(c) almost entirely, and make the relevant moduli spaces into manifolds. The cost is that they usually study rather restricted situations, such as exact Lagrangians in a noncompact exact symplectic manifold. They are doing wonderful work, but this is not where my current interests lie.

  Over 2007 - ∞ I have been concocting my own, rival programme to Kuranishi spaces and polyfolds. For the first few years I only worked on parts (c),(a), and I saw what I was doing as providing improvements and upgrades to Fukaya's programme: finding a better definition of Kuranishi space, and a better machine for defining virtual cycles and chains for Kuranishi spaces. While I hope that my work will be adopted by the Fukaya school, I have been gradually drawn into attempting the full monty, and developing a theory that will do all of (a)-(d). I expect this will absorb the majority of my research time for a few years from 2015 onwards.
  In 2007 I tried to write down new homology and cohomology theories I called Kuranishi homology KH*(Y) and Kuranishi cohomology KH*(Y) of a manifold Y, in which the (co)chains [X,f,G] are equivalence classes of a compact, (co)oriented Kuranishi space X, a morphism (submersion) f : XY, and so extra '(co)gauge-fixing data' G. There are canonical isomorphisms KH*(Y)≅H*(Y), KH*(Y)≅H*(Y). The point is that it is really easy to form virtual cycles/chains for moduli spaces with Kuranishi structures, you just choose gauge-fixing data, and there is no perturbation. I was using a mish-mash of several FOOO definitions of Kuranishi space,  with some minor modifications of my own.
   By the end of the Kuranishi (co)homology project, I was developing serious doubts about the definition and general theory of Kuranishi spaces. I posted it on the arXiv anyway (this may have been a mistake), as I thought it was a cool idea, but I did not publish it, as I intended to go away and find the 'right' definition of Kuranishi space, and then revise the Kuranishi (co)homology paper with the new definition.
  In 2009 I found what I needed. I was getting interested in Derived Algebraic Geometry, which I suspected was connected to the Kuranishi space problem, as well as to other algebraic geometry questions I was interested in. I had a short conversation with Jacob Lurie, who is awesome, was visiting his girlfriend in Oxford, and who told me to read his student David Spivak's thesis on 'Derived smooth manifolds' (now in Duke Mathematical Journal, 2010). When I did, immediately scales fell from my eyes, and I had a revelation:
Kuranishi spaces ought to be defined to be derived smooth orbifolds.
Actually this is really obvious if you understand both theories well, but at the time few people had looked seriously at both, maybe only me. It also explains the source of some of the problems in the Fukaya theory: they were lacking some key ideas from derived geometry (especially, the necessity for working in higher categories), and doing the job without them is like banging in nails with a screwdriver.
  This changed my research direction. To do Kuranishi spaces and symplectic geometry properly, I needed a theory of Derived Differential Geometry: definitions of derived manifolds and derived orbifolds, including with boundary and corners, and good notions of boundaries, orientations, submersions, transverse fibre products, ... in the derived context. And I needed the theory to be easy enough to market to symplectic geometers, who are mostly simple-minded folk, and not very enthusiastic about scary things like ∞-categories.
  Over 2009-2015 I developed my theory of Derived Differential Geometry, which you can read about here. I first defined a 2-category dMan of 'd-manifolds', which is a 2-category truncation and simplification of Spivak's ∞-category DerManSpiof derived manifolds. Then I started to add my own bells and whistles: 2-categories of d-orbifolds dOrb, d-manifolds and d-orbifolds with boundary and corners dManb,dManc,dOrbb,dOrbc, and I proved a lot of theorems on their geometry.
  Following Spivak, all of the d-manifold and d-orbifold theory is written in terms of
'C-rings' and 'C-algebraic geometry'. However, in 2014 I found a new way to define a 2-category of Kuranishi spaces Kur by an 'atlas of charts' definition similar to (but more complicated than) the Fukaya-Oh-Ohta-Ono definition, which has an equivalence of 2-categories KurdOrb, and also versions with boundary and corners Kurb, Kurc. I regard this as the 'correct' definition of Kuranishi space that I had been searching for for 7 years, and I think it should replace other definitions of Kuranishi space in the literature.
  As they are equivalent 2-categories, (new definition) Kuranishi spaces and d-orbifolds are interchangeable; more-or-less any theorem you prove about d-orbifolds translates immediately to one about Kuranishi spaces, and vice versa. I intend to use the Kuranishi space approach in future work aimed at symplectic geometers, since it is close to things they know already.
  I am currently (mid-2015) writing a new version of the Kuranishi (co)homology project using the new definition of Kuranishi space, a few years later than I hoped in 2007. But I have learned a lot in the mean time, and the new theory is more elegant, powerful, and rigorous than the old. At that point I will have finished my answers to (a),(c) above, and will start work on (b),(d).
  For (b), I want to construct Kuranishi structures on moduli spaces by a 'representable 2-functor' approach based on Grothendieck's representable functors in algebraic geometry. For (d), the good behaviour of my virtual chain theory at the (co)chain level will lead to significant technical simplifications compared to existing theories, and in some cases the ability to prove more.

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. This has taken my research in (yet) another direction. On the geometry side of the grant, we were thinking about generalizations of Donaldson-Thomas invariants DTα(τ) of Calabi-Yau 3-folds, which are numbers. There are two natural ways you can generalize this: firstly, you can find vector spaces Vα(τ) with dimension dimVα(τ) = DTα(τ). This is called categorification, and there are lots of reasons why it is a good thing to do; for instance, in String Theory Donaldson-Thomas 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 Vα(τ), they will probably give a mathematical definition of BPS states in String Theory. Secondly, you can enhance DTα(τ) from a number to an element of a larger 'motivic' ring (e.g. of polynomials or rational functions), so that the new DTα(τ) contain more information. The way we do it, the DTα(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 Donaldson-Thomas 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 'k-shifted 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 Calabi-Yau m-fold have (2-m)-shifted symplectic structures. So Calabi-Yau 3-fold moduli spaces have -1-shifted symplectic structures. This is a new geometric structure on them, which we did not know about before.

Jointly with subsets of {Oren Ben-Bassat, 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.