UK-JAPAN MATHEMATICAL FORUM
Keio University, 24-29 March 2011



ABSTRACTS OF LECTURE COURSES AND GUEST LECTURES

Chris Budd (University of Bath): Study groups with industry
"Study Groups with Industry" began in Oxford in 1968. They are now well-established in the United Kingdom, and internationally, as a model of academic-industry cooperation. Professor Chris Budd will give an introduction to "Study Groups with Industry", followed by a Mini study group with sample problems. Three sample problems will be presented. Participants will have the opportunity to work in groups on these problems, or just to observe. Students and researchers, and anyone with an interest in academic-industry cooperation, are welcome.

David Elworthy (University of Warwick): Kusuoka's degree theorem and hypo-elliptic Laplacians
In 1987 Kusuoka showed that the McKean-Singer formula χ(M) = STr PT{*} for the Euler characteristic of a compact Riemannian manifold M in terms of the supertrace of its Hodge-Kodaira heat semigroup, is a consequence of an infinite dimensional degree theorem. The latter was a stochastic analytic result and to prove it he had to overcome several technical difficulties because of the lack of differentiabilty of Brownian paths.
These difficulties do not arise if one uses processes with C1 paths instead of Brownian motion. For a class of such processes a similar formula to McKean-Singer's for χ(M) arises in terms of a super trace. It is not clear how such formulae could be proved analytically. If one approximates the Brownian motion by paths of Ornstein-Uhlenbeck processes, generated by hypo-elliptic Laplacians in the terminology of Bismut, one gets a result for such operators to which Bismut's techniques can be applied in order to recover the usual formula. One also gets a Rice type formula for the number of fixed points of stochastic flows.
In this talk I shall describe these results assuming little background knowledge of stochastic analysis or hypo-elliptic Laplacians.

Martin Hairer (University of Warwick): Ergodicity and hypoellipticity of stochastic PDEs
In recent years, much attention has been devoted to the study of stochastic partial differential equations (SPDEs) that arise in a number of areas like turbulence, quantum field theory, material science, spatial models in epidemiology. The solutions to such SPDEs form Markov processes, but on infinite-dimensional functional spaces. While there are situations where the traditional tools of the theory of Markov processes can be applied to such systems, they do fail in many important cases. In particular, very little was known until recently about the "hypoellptic" case where noise acts only on some degrees of freedom of the system and is transmitted to the remaining degrees of freedom via the nonlinearity.
This spurred a number of recent advances in the theory. These have centered around the realization that the total variation metric, while very effective for countable state space Markov chains and for finite-dimensional diffusion processes, is much less useful in the infinite dimensional setting. The introduction of tools like the asymptotic strong Feller property have lead to progress on a number of problems which had previously been out of reach.
The aim of these lectures is to give an introduction to the main tools around which this theory is built and to illustrate its power with the example of the two-dimensional stochastically forced Navier-Stokes equations.

Nigel Hitchin (University of Oxford): Generalized geometry and Poisson geometry
Generalized geometry is a differential geometric structure which captures some of the features of supersymmetric field theories but has developed by following analogies with traditional differential geometry. The study of generalized complex structures leads quite quickly to holomorphic Poisson geometry, which is in many respects an underdeveloped field. We shall discuss the interactions between the two areas.

Dominic Joyce (University of Oxford): D-manifolds and d-orbifolds: a theory of derived differential geometry
"Kuranishi spaces" are a class of geometric spaces introduced in 1990 by Fukaya and Ono, as the geometric structure on moduli spaces of J-holomorphic curves in a symplectic manifold, and used in the work of Fukaya, Oh, Ohta and Ono on Lagrangian Floer cohomology and Fukaya categories. Although their definition was sufficient for their applications, it did not give a very satisfactory notion of geometric space -- notions of morphisms, or even of when two Kuranishi spaces are "the same", are not well behaved -- so the theory of Kuranishi spaces was never developed very far. The subject of these lectures began as a project to find the "right" definition of Kuranishi space, which I believe I have done.
In these lectures I will describe a new class of geometric objects I call "d-manifolds". D-manifolds are a kind of "derived" smooth manifold, where "derived" is in the sense of the derived algebraic geometry of Jacob Lurie, Bertrand Toen, etc. The closest thing to them in the literature is the "derived manifolds" of David Spivak (Duke Math. J. 153 (2010), 55-128). But d-manifolds are rather simpler than Spivak's derived manifolds -- d-manifolds form a 2-category which is constructed using fairly basic techniques from algebraic geometry, but derived manifolds form an infinity-category (simplicial category) which uses advanced ideas like homotopy sheaves and Bousfeld localization.
Manifolds are examples of d-manifolds -- that is, the category of manifolds embeds as a subcategory of the 2-category of d-manifolds -- but d-manifolds also include many spaces one would regard classically as singular or obstructed. A d-manifold has a virtual dimension, an integer, which may be negative. Almost all the main ideas of differential geometry have analogues for d-manifolds -- submersions, immersions, embeddings, submanifolds, orientations, transverse fibre products, and so on -- but the derived versions are often stronger. For example, the intersection of two submanifolds in a manifold exists as a manifold if the intersection is transverse, but it always exists as a d-manifold. There are also good notions of d-manifolds with boundary and d-manifolds with corners, and orbifold versions of all this, d-orbifolds. I claim that the (morally and aesthetically) "right" definition of Kuranishi space in the work of Fukaya-Oh-Ohta-Ono is that they are d-orbifolds with corners.
A useful property of d-manifolds and d-orbifolds is that they have well-behaved virtual cycles or virtual chains. So, for example, if X is a compact oriented d-manifold of virtual dimension k, and Y is a manifold, and f : X → Y is a 1-morphism, then we can define a virtual class [X] in the homology group Hk(Y;Z), which is unchanged under deformations of X,f. This will be important in applications of d-manifolds and d-orbifolds.
Many important areas of mathematics involve "counting" moduli spaces of geometric objects to define enumerative invariants or homology theories -- for instance, Donaldson and Seiberg-Witten invariants for 4-manifolds, Donaldson-Thomas invariants of Calabi-Yau 3-folds, Gromov-Witten invariants in algebraic or symplectic geometry, instanton Floer cohomology, Lagrangian Floer cohomology, contact homology, symplectic field theory, Fukaya categories, ... In all of these (at least over C for the algebraic geometry cases, and away from reducible connections, etc.) the moduli spaces concerned will be oriented d-manifolds or d-orbifolds, and the counting can be done using virtual cycles or chains. So, d-manifolds and d-orbifolds provide a unified way of looking at these counting problems. There are truncation functors from geometric structures currently used to define virtual classes to d-manifolds and d-orbifolds. For instance, any moduli space of solutions of a smooth nonlinear elliptic p.d.e. on a compact manifold is a d-manifold. In algebraic geometry, a C-scheme with a perfect obstruction theory can be made into a d-manifold. In symplectic geometry, Kuranishi spaces and polyfold structures on moduli spaces of J-holomorphic curves induce d-orbifold structures.
All this is work in progress. I am writing a book on it; you can download a preliminary version from my webpage at http://people.maths.ox.ac.uk/~joyce/

Xue-Mei Li (University of Warwick): Effective diffusions on OM
We investigate random perturbations of vertical diffusions on the orthonormal frame bundle. From the conserved quantities such as the momentum and eigenvalues of the the vertical derivative flow we obtain a system of effective diffusions.

Mark Pollicott (University of Warwick): Circle packing, reflections and ergodic theory geometry
Circle packings have been studied by Apollonius, Descartes, Soddy and, more recently Sarnak, Bourgain, Oh et al. We discuss statistical properties of some circle packings and, more general, configurations of circles in the plane arising from reflections. We use techniques from ergodic theory and Thermodynamic Formalism.

Tadashi Tokieda (University of Cambridge): What happens when we add many relativistic velocities
A neat formula is presented for addition of an arbitrary number of relativistic velocities in (1+1)-dimensional space-time in terms of elementary symmetric polynomials. Such a formula looks ungeneralizable to (1+3)-dimensions since Lorentz boosts do not commute in space dimension > 1. Nevertheless we find a generalization, by maneuvering inside the Clifford algebra of Minkowski space regarded as quaternions tensored with themselves. One or two physical applications are given.

Further abstracts of lecture courses and guest lectures will be added to this page in due course.



Return to Homepage
Last update: 14 March 2011