Keio University, 15-20 July 2012


Jürgen Berndt (King's College London): Polar actions on symmetric spaces (Lecture Notes)
An isometric action of a connected Lie group H on a Riemannian manifold M is called hyperpolar if there exists a connected closed flat submanifold Σ of M such that Σ meets each orbit of the action and intersects it orthogonally. An elementary example of a hyperpolar action comes from the standard representation of SOn on Rn. Further examples of hyperpolar actions can be constructed from Riemannian symmetric spaces. Let M = G/K be a Riemannian symmetric space and denote by o a fixed point of the K-action on M. Then the isotropy representation of K on the tangent space ToM of M at o induces a hyperpolar action. Dadok established in 1985 a remarkable, and mysterious, relation between hyperpolar actions on Euclidean spaces and Riemannian symmetric spaces. He proved that for every hyperpolar action on Rn there exists a Riemannian symmetric space M = G/K with dim M = n such that the orbits of the hyperpolar action on Rn and the orbits of the K-action on ToM are the same via a suitable isomorphism between Rn and ToM. For symmetric spaces of compact type the hyperpolar actions are reasonably well understood. In the talk I want to focus on symmetric spaces of noncompact type where the situation is much more involved because of the noncompactness of the isometry groups. With my collaborators Diaz-Ramos and Tamaru I developed an approach based on Langlands and Chevalley decompositions of parabolic subalgebras of noncompact semisimple Lie algebras. Geometrically this involves horospherical decompositions of symmetric spaces and boundary components of their maximal Satake compactifications. This approach leads to many new examples, partial classifications, and interesting open problems.

John Bolton (University of Durham): Almost complex curves in the product of two 3-spheres (Lecture Notes)
It has been shown by Butruille, that there are just four 6-dimensional manifolds which admit a homogeneous non-Kaehler, nearly-Kaehler structure. Almost complex curves in the nearly Kaehler 6-sphere have been studied by several authors, but in joint work, F Dillen, B Dioos, L Vrancken, and I have instigated the study of almost complex curves in another of these manifolds, namely the product of two 3-spheres. I will describe the relevant homogeneous nearly-Kaehler structure, give some examples of almost complex curves, and classify the almost complex 2-spheres.

Chris Budd (University of Bath): Geometry based numerical methods for weather forecasting (Lecture Notes)
This talk will describe how geometrically based methods using ideas from optimal transport theory and reliant on efficient solvers of the Monge-Ampere equation can be used to solve complex meteorological problems in several spatial dimensions. A good geometrical understanding of these methods allows useful estimates for the error in using them to calculate the solutions, especially close to singularities.

Chris Budd and Karin Mora (University of Bath): Some interesting mathematical problems arising in industry (Lecture Notes)
We will look at the general issues that arise when applying mathematics in an industrial context, and will then examine some case studies including the control of the Smart Grid, the illumination pattern of lighting sources and some problems from the food industry.

David Elworthy (University of Warwick): The expected number of fixed points of stochastic flows and Kusuoka's McKean-Singer formula (Lecture Notes)
I will describe a Rice type formula for the expected number of fixed points of a class of stochastic flows at a given time. It is closely related to the McKean-Singer formula for the Euler characteristic of a compact Riemannian manifold: this would give the algebraic number of fixed points of our flow. The original proof of this by McKean & Singer was operator theoretic. It was shown by Kusuoka in 1987, to be a consequence of an infinite dimensional degree theorem. His methods also lead to the Rice formula. For his proof he had to use stochastic analysis 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 the Rice formula can be proved and a similar formula to McKean-Singer's for the Euler-characteristic arises in terms of a supertrace. It is not clear how the latter could be proved analytically. If one approximates the Brownian motion by paths of Ornstein-Uhlenbeck processes, one gets a result for certain operators to which Bismut & Lebeau's techniques should be applicable in order to recover the usual formulae. The talk is based on joint work with AbdulRahman Al-Hussein.

Ryushi Goto (Osaka University): Logarithmic Poisson structures and generalized geometry
A logarithmic Poisson structure is a holomorphic Poisson structure which preserves the ideal sheaf of a divisor D on a complex manifold. A logarithmic Poisson structure on a compact Kähler manifold gives rise to deformations of generalized Kähler structures such that the divisor D arises as a generalized Kähler submanifold. This is a generalization of the stability theorem of ordinary Kähler manifolds coupled with complex submanifolds, due to Kodaira and Spencer. Using the unobstructedness theorem of deformations of Poisson Kähler manifolds, we obtain that every compact Poisson Kähler manifold admits bihermitian structures. We describe several examples of logarithmic Poisson structures and higher Poisson structures (i.e., Nambu structures) also. Then we discuss an approach to the extended deformations by higher Poisson structures which are beyond the ones in generalized geometry. References:
[1] Unobstructed K-deformations of generalized complex structures and bihermitian structures, to appear in Adv. Math.;
[2] Deformations of generalized Calabi-Yau and generalized SU(n)-structures, math.DG/0512211, to appear in Osaka J. Math. 49 (2012);
[3] Deformations of generalized complex and generalized Kähler structures, J. Differential Geom. 84 (2010), 525-560;
[4] Poisson structures and generalized Kähler submanifolds, J. Math. Soc. Japan 61 (2009), 107-132.

Martin Hairer (University of Warwick): Rough stochastic PDEs
In these lectures, we will provide an overview of recent developments linking the theory of (controlled) rough paths with the solutions to a class of stochastic partial differential equations. The equations we will consider are Burgers-type equations perturbed by space-time white noise. More precisely, they are of the form
(1) ∂tu = ∂x2u + G(u)∂xu + ξ(x,t) ;
where ξ is space-time white noise, u(x,t) ∈ Rn and G is a smooth n x n-matrix valued function. The problem arising in the study of such equations is that it is not even clear what the product G(u)∂xu actually means: for fixed time, u(.,t) is typically Cα only for α < 1/2, while such expressions are well-posed only for α > 1/2 ! The usual trick of considering weak solutions and formally integrating by parts also fails if G is not of gradient type...
The outline of the lectures is as follows:
1. In the first lecture, we will study the linear equation ∂tu = ∂x2u + ξ(x,t) and we will give an overview of its properties, as well as of the standard methods used to perturb such equations by a sufficiently �nice� nonlinearity.
2. In the second lecture, we will give an introduction to the theory of controlled rough paths and we will see how this can be used in general to break through the α = 1/2 barrier mentioned above. Originally, this theory was developed by Lyons to give a pathwise notion of solution to ordinary stochastic differential equations.
3. In the last lecture, we will see how it is possible to combine the tools developed in the first two lectures to provide a good notion of solution to (1) and we will expand on the notion in which sense these solutions are �unique�.

Mark Haskins (Imperial College London): The geometry and topology of compact G2-manifolds
Riemannian 7-manifolds with holonomy group the exceptional Lie group G2, or G2-manifolds for short, are very special geometric objects of interest to both geometers and theoretical physicists (in M-theory). I will give an introduction to the subject and review some of the main developments in the area before describing my recent joint work with Corti, Nordstrom and Pacini. We construct many new compact G2-manifolds, a large fraction of which also contain interesting 3-dimensional submanifolds, so-called associative submanifolds. In many cases we understand the diffeomorphism type of the 7-manifolds on which these G2-metrics exist. Along the way we will run into various other parts of mathematics: non-compact Calabi-Yau 3-folds, weak Fano 3-folds, (lattice polarised) K3-surfaces, nonlinear elliptic PDE, gluing methods, lattice arithmetic and 7-dimensional topology. I will try to explain how all these fit together in G2-geometry.

Dominic Joyce (University of Oxford): D-manifolds and d-orbifolds: a theory of derived differential geometry (Lecture Notes: Talk 1, Talk 2, Talk 3)
"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

Kazumasa Kuwada (Ochanomizu University): Optimal transportation costs of heat distributions in stochastic geometric analysis (Lecture Notes)
On a complete Riemannian manifold, a contraction property of the Wasserstein distance between heat distribution is known to be equivalent to the presence of a lower Ricci curvature bound. The Lp-Wasserstein distance is the optimal transportation cost of two probability measures whose cost function is given by p-th power of the distance function. In this talk, we consider three extensions of this relation in different directions mainly based on a coupling method of Brownian motions. The first one is the monotonicity of a transportation cost associated with the Perelman�s L-functional on a backward Ricci flow. This is an extension of Topping�s result by stochastic analysis. The second one deals with a modified transportation cost associated with the total variation between heat distributions on spaceforms. It yields a comparison theorem for the total variation. The third one is a new contraction property for the Lp-Wasserstein distance which is equivalent to the curvature-dimension condition, that is, the combination of an upper dimension bound and a lower Ricci curvature bound.

Xue-Mei Li (University of Warwick): Convergence of stochastic processes and collapsing of manifolds (Lecture Notes)
Our talk is concerned with the convergence of stochastic processes on Berger spheres, associated to the Hopf fibration, considered together with the problem of collapsing manifolds. A family of manifolds equipped with a family of Riemannian metrics converge to a Riemannian manifold of lower dimension while keeping the sectional curvatures bounded.

Mario Micallef (University of Warwick):
1. Minimal hypersurfaces in manifolds with a lower bound on scalar curvature
I shall first review well-known results of Simons and Schoen-Yau on stable minimal hypersurfaces in manifolds with lower curvature bounds. Then I shall describe some joint work with Vlad Moraru on an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. An application of this result and counterexamples to some natural conjectured generalisations to higher dimensions will be described by Vlad Moraru in his talk.
2. Minimal surfaces of high codimension
After reviewing some long established results, I shall present a new technique for studying the stability of covers of two-dimensional minimal surfaces of high codimension. I shall also pose some open problems on the local behaviour of two-dimensional area minimising surfaces in Euclidean space of any dimension.
3. Approximation of harmonic maps by alpha-harmonic maps
In a famous paper, Sacks and Uhlenbeck introduced alpha-harmonic maps to construct non-trivial harmonic maps of the two-sphere in manifolds with a non-contractible universal cover. I shall describe a joint work with Tobias Lamm on the question of which harmonic maps of the two-sphere to itself can be constructed in this way.

Reiko Miyaoka (Tohoku University): Hypersurface geometry and moment map (Lecture Notes)
Recently, isoparametric hypersurfaces in the sphere have been classified except for one case, and it turns out that they are all homogeneous unless g=4. When g=4, there exist infinitely many non-homogeneous examples which are given by level sets of the Cartan-Munzner polynomials associated with Clifford systems. Using the spin action induced from the Clifford system, we describe the polynomial in terms of the moment map. Although this spin action is rather small, it is interesting that we can describe our hypersurfaces in both homogeneous and non-homogeneous cases uniformly.

Tadashi Tokieda (University of Cambridge): A different way of perturbing dynamical systems
Traditionally `perturbation' in a dynamical system means perturbing the initial conditions or perturbing the vector field. The unusual idea I wish to convey is that perturbing the geometry of the phase space is sometimes interesting. As an illustration, we solve the stability problem of the `Thomson heptagon' of point vortices on the plane (zero curvature). This problem is difficult because of various degeneracies, which the traditional method overcomes by recourse to the Birkhoff normal form. We instead perturb the dynamics away from degeneracies by embedding it in a parametric family of such dynamics on surfaces of nonzero curvatures. By looking at bifurcations that occur when the curvature crosses 0, we can deduce that the heptagon is stable. In this illustration, we are trading the difficulty of the Birkhoff normal form for a different difficulty of equivariant bifurcation theory, so we end up handling degeneracies elsewhere (conservation law of mathematical difficulties). But it is a different kind of degeneracy handled by a different point of view. [joint work with J. Montaldi]

Alexander Veselov (University of Loughborough): Topology and integrability
A fruitful interaction between topology and theory of integrable systems was known for a long time although it was not always clearly emphasized. I will discuss three different examples of such interaction:
I. Integrable gradient flows in elementary Morse theory.
II. Topology of Toda lattice and Steenrod's cycle realisation problem.
III. Kohno-Drinfeld Lie algebra, Gaudin model and stable rational curves.


Kotaro Kawai (Tohoku University): G2-manifolds and calibrated submanifolds
The SYZ conjecture explains mirror symmetry of compact Calabi-Yau 3-folds in terms of dual fibrations by special Lagrangian 3-tori. In M-theory, fibrations of coassociative 4-folds in G2-manifolds are expected to play the same role as special Lagrangian fibrations. In this talk, we focus on (co)associative submanifolds in a G2-manifold Y with an S1- or T2-action. We characterize (co)associative submanifolds by submanifolds in Y/S1 and Y/T2 and then we construct examples of (co)associative submanifolds.

Graham Kemp (University of Loughborough): Dirac magnetic monopoles, coadjoint orbits and geometric quantization
I will describe a natural generalization of Dirac's magnetic monopole from the sphere to a coadjoint orbit of any compact Lie group G. The classical phase space may be identified with an orbit of the tangent bundle TG of G. Classically, the 'magnetic geodesic flow' has been shown to be integrable only in special cases. Quantizing this system using geometric quantization leads to representation theory and the quantum problem is then integrable in terms of Kostant's branching formula.

Karin Mora (University of Bath): Dynamics of a spinning disk impacting with friction (Lecture Notes)
Rotating machinery such as spindles, pumps or turbines have magnetic bearing systems integrated within. Such a system is comprised of a rigid shaft spinning within a magnetic ring which in turn is protected by a bearing. A nonsmooth, three degrees of freedom model for collisions with friction between the shaft (assumed to be a disk) and bearing (circular wall) is presented. Numerical simulations and analytical nonsmooth methods confirm periodic impacts observed in experiments but also reveal chattering phenomena, finite or infinite number of impacts accumulating at a point; the latter leads to more damaging dynamics such as sliding, rolling or a combination of both. Further, it is investigated how these dynamics depend on initial conditions and system parameters such as imbalance eccentricity and rotational speed. This is joint work with Chris Budd (Bath) and Patrick Keogh (Bath).

Vlad Moraru (University of Warwick): Area minimising surfaces and splitting of 3-manifolds with lower bounds on scalar curvature
I will prove a splitting theorem for 3-manifolds with scalar curvature bounded below by R0 and which contain an area minimising surface of genus g and, if g is not equal to 1, whose area is equal to 2πR0(1-g). This talk will be a continuation of Micallef's talk on "Minimal hypersurfaces in manifolds with a lower bound on scalar curvature". I will also exhibit counterexamples to some natural conjectured generalisations of the area comparison result in Micallef's talk.

Takayuki Moriyama (Kyoto University): Special Legendrian submanifolds in toric Sasaki-Einstein manifolds (Lecture Notes)
A Sasaki-Einstein manifold is a Riemannian manifold S whose cone C(S) is a Calabi-Yau manifold. A submanifold L in a Sasaki-Einstein manifold S is a special Legendrian submanifold if the cone C(L) is a special Lagrangian submanifold in C(S). In this talk, we will see that every toric Sasaki-Einstein manifold admits a special Legendrian submanifold which arises as a real form.

Ryoichi Suzuki (Keio University): A Clark-Ocone type formula under change of measure for canonical Lévy processes (Lecture Notes)
In this talk we introduce a Clark-Ocone type formula under change of measure for canonical Lévy processes by using Malliavin calculus for canonical Lévy processes, based on Solé et al. (2007, SPA) and Delong-Imkeller (2010, SPA). This result is a generalization of the Karatzas-Ocone theorem, that is, Clark-Ocone type formula under change of measure for Brownian motions.

Satoshi Ueki (Tohoku University): Integral geometry and its applications
Integral geometry treats integration of geometric invariants like as intersection number. There are relationships between the integral of intersection numbers and volumes of submanifolds. These integral formulae are applied to the Hamiltonian volume minimizing problems. I will talk about the integral formula and the known result for the Hamiltonian volume minimizing problem.

Benjamin Volk (University of Oxford): Blowup functor for d-orbifolds
Building upon Joyce's theory of d-orbifolds, we will sketch how one can define stable almost complex structures on d-orbifolds and almost complex d-orbifold bordism. We will then discuss how one could define a "blow up functor" from (almost complex) d-orbifold bordism to (almost complex) effective orbifold bordism and explain how this could lead to a general understanding of integrality properties of virtual cycles for almost complex d-orbifolds.

Jack Waldron (University of Oxford): Formality of Fukaya categories
Fukaya Categories are notoriously difficult to calculate compared to their somewhat simpler cohomology. For complex Lagrangian submanifolds in hyperkaehler manifolds, both invariants are the same. I will discuss non-trivial applications of this result and the tools used in its proof.

Yang Yuxin (University of Bristol): The vanishing of L2 harmonic one-forms on based path spaces
We prove the triviality of the first L2 cohomology class of based path spaces of Riemannian manifolds, and the consequent vanishing of L2 harmonic one-forms. We give explicit formulae for closed and co-closed one-forms; these are considered as extended Clark-Ocone formulae. A feature of the proof is the use of the temporal structure of these path spaces to relate a rough exterior derivative operator on one-forms to the exterior differentiation operator defined earlier by Elworthy and Li, the one used to construct the de Rham complex and the self-adjoint Laplacian on L2 one-forms. This Laplacian is shown to have a spectral gap. This is based on joint work with K. D. Elworthy.

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Last update: 8 August 2012