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Online Talks

Durham Cathedral

London Mathematical Society Durham Symposium
Methods of Integrable Systems in Geometry
Friday 11th August - Monday 21st August 2006

List of abstracts

Uwe Abresch (Ruhr-Universität Bochum)Saturday 12th August 09:20
On holomorphic quadratic differentials
(Joint work with Harold Rosenberg, Paris 7)

A basic tool in the theory of constant mean curvature (cmc) surfaces in space forms is the holomorphic quadratic differential discovered by Heinz Hopf. However, for more general target spaces the (2,0)-part of the second fundamental form of a cmc surface fails to be holomorphic.

The basic new result is that for cmc surfaces in the product spaces  S²×R  and H ²×R  holomorphicity can be restored with the help of explicit, geometrically defined correction terms.

Our generalized holomorphic quadratic differential is good enough to proceed along the lines of Hopf and prove that an immersed cmc sphere  S²  in such a product space must in fact be one of the embedded, rotationally-invariant surfaces described in the work of W.-Y. Hsiang and R. Pedrosa, which are the simplest cmc surfaces in these product spaces. The distance spheres do not have constant mean curvature any more.

The next step is to investigate the scope of the new construction. More precisely, we ask for which class of (oriented) Riemannian 3-manifolds  (M³,g)  there exists a correction field  L  that induces a holomorphic quadratic differential on any immersed cmc surface. There is an amazingly simple necessary and sufficient condition, namely,  L  must satisfy a certain explicit inhomogeneous ODE system.

Integrability for this ODE system is by no means automatic; it rather imposes serious restrictions on the geometry of the 3-manifold. A tedious classification reveals that solutions exist if and only if  (M³,g)  is a homogeneous bundle over a surface with totally-geodesic fibers. Again an analogue to Hopf's theorem can be established.

The preceding results suggest that homogeneous 3-manifolds with at least 4-dimensional isometry groups are an appropriate setting for global results about minimal surfaces and cmc surfaces. In order to test this thesis, we have started studying minimal surfaces in the Heisenberg group. If time permits, the talk will end with an outlook on these results.

Alexander Bobenko (TU Berlin)Tuesday 15th August 16:30
On organising principles of discrete differential geometry. Geometry of spheres.

Christoph Bohle ( Technische Universität Berlin)Wednesday 16th August 17:30
Constrained Willmore Tori in the 4-Sphere
Constrained Willmore surfaces are the critical points of the Willmore functional with respect to conformal variations. Examples include all CMC-surfaces in space forms. In my talk I explain how a combination of ideas from Hitchin's paper 'Harmonic 2-Tori in the 3-Sphere' with methods of quaternionic holomorphic geometry yields a geometric proof of the fact that all constrained Willmore tori in the 4-sphere are either obtained by some twistorial construction or are of finite type, i.e., have a spectral curve of finite genus.

David Calderbank (University of York)Friday 18th August 17:30
Submanifolds of celestial spheres: geometry and integrability
Many integrable systems arise naturally in the classical differential geometry of surfaces in ordinary Euclidean space: for example, CMC surfaces or K-surfaces are described by integrable systems.

There is a similar story for submanifold geometry in the celestial n-spheres which arise as projective light cones in Minkowski (n+1,1)-space. The integrable theories of conformally-flat submanifolds, isothermic surfaces and Willmore surfaces fit naturally into this framework. In this talk I plan to outline a novel and systematic approach to this geometry developed with Fran Burstall.

Peter Clarkson (University of Kent)Tuesday 15th August 11:00
The Painlevé Equations - Nonlinear Special Functions

Boris Dubrovin (SISSA, Trieste)Wednesday 16th August 16:30
Do we already know enough integrable systems?

Jenya Ferapontov (Loughborough University)Sunday 13th August 17:30
Integrable 3-dimensional equations of the dispersionless Hirota type.
I will discuss equations of the form

F(uxx, uxy, uxt, uyy, uyt, utt)=0,

a single relation among the six second order partial derivatives of a function u(x, y, t). Particular examples thereof include the Boyer-Finley equation uxy=e(utt), the dispersionless Hirota equation (a - b) e(uxy)+(b - c) e(uyt)+(c - a) e(utx)=0, etc. The integrability is defined as the existence of infinitely many hydrodynamic reductions. This leads to an involutive system of third order PDEs for the function F, which is invariant under the natural action of the symplectic group Sp(6). The geometry of the integrability conditions will be analysed.

Kenji Fukaya (University of Kyoto)Monday 14th August 09:20
Smooth correspondence and Topological Field theory
I would like to overview the general procedure to find algebraic system by using smooth correspondence by moduli spaces. The story includes : Finding appropriate algebraic structure and its `homotopy theory' to have correct statement of well-definedness. Working out appropriate transversality theory of moduli space. Working out orientation problem of moduli space to obtain correct sign convention. Such procedure works in several situations and also applicable for several purposes.

Alexander Givental (UC Berkeley)Wednesday 16th August 09:20
Picard-Lefschetz periods and integrable hierarchies
The way how integrable hierarchies (such as KdV, Extended Toda, 2-Toda, Gelfand-Dickey, and Kac-Wakimoto) arise in the Gromov-Witten theory and its axiomatic extension turns out to be better described in mirror-symmetric terms of singularity theory and Picard-Lefschetz periods. This relationship can be conveniently framed into a loop-space setting which at a first glance seems a manifestation of the standard boson-fermion formalism, but in fact is remarkably different from (and in a sense Laplace-dual to) it. In the talk, we will try to describe the ingredients of this theory and outline their relationships.

Mark Haskins (Imperial College London)Sunday 20th August 11:00
Special Lagrangian cones, integrable systems and spectral geometry
We show how integrable systems machinery (more specifically spectral curves) can be combined with other techniques from geometry and analysis to give geometric information about special Lagrangian cones in dimension three and (time permitting) also in higher dimensions.

Udo Hertrich-Jeromin (University of Bath)Saturday 12th August 11:00
Conformally flat hypersurfaces with cyclic Guichard net
I plan to discuss a recent classification result on 3-dimensional generic conformally flat hypersurfaces. Conformally flat hypersurfaces carry special curvature line coordinates, a "Guichard net". The hypersurface can be reconstructed from a conformal image in Euclidean 3-space of this triply orthogonal system. The considered hypersurfaces correspond to Guichard nets in Euclidean space with one family of circular arcs, that is, they form a cyclic system.

Frédéric Hélein (Paris 7)Saturday 12th August 17:30
Integrable systems associated with 4-symmetric spaces
Hamiltonian stationary Lagrangian surfaces in C2, CP2 or more generally in a complex Hermitian homogeneous manifold of complex dimension 2 are instances of completely integrable systems associated with 4-symmetric spaces (results of F. Hélein and P. Romon). In contrast with the theory of harmonic maps into a homogeneous manifold, where a Cartan involution plays a central role, the key of the structure of such integrable systems is an automorphism of order 4 of the Lie algebra of symmetries. Recently new examples of such systems have been constructed by I. Khemar for surfaces in octonions. A possible supersymetric interpretation will be discussed.

Hiroaki Kanno (Nagoya University)Saturday 19th August 17:30
Topological strings on local Calabi-Yau manifolds and instantons in gauge theories
One of the most important ideas in the recent developments of topological string theory is the topological vertex, which allows us to obtain all genus amplitudes of topological strings on (local) toric Calabi-Yau manifolds. We show how the problem of instanton (BPS state) counting in gauge theories is related to the amplitudes on local Hirzebruch surfaces and local curves. For non-nef cases there are subtleties on what topological vertex computes and we hope to make some comments on this issue.

Lionel Mason (University of Oxford)Monday 14th August 11:00
Holomorphic discs and global problems in integrable systems or Integrability and Twistor Strings
Holomorphic discs: This lecture will review some global integrable problems in gometry whose solution can be encoded in the problem of finding holomorphic discs in a complex manifold with boundary constriained to lie on a totally real submanifold. These problems include the construction of Zoll projective structures on surfaces (i.e., whose geodesics are all closed), anti-self-dual conformal structures of split signature on S2×S2 and Einstein-Weyl structures that are asymptotically de sitter.

The twistor-strings lecture: recent progress in twistor-string theory has led to new twistor construction for non-self dual fields for both Yang-Mills and anti-self-dual conformal structures at least in Euclidean signature. Although these equations are not integrable, these constructions lead to new ways of looking at these equations. Thus far the main applications have been in perturbative quantum field theory, but there is the possiblity of applications elsewhere also.

Ian McIntosh (University of York)Tuesday 15th August 17:30
Hamiltonian stationary surfaces in R4 and CP2
(Includes joint work with Pascal Romon and work of my research student Richard Hunter.)
In this talk I will decsribe progress made in understanding the construction of Hamiltonian stationary tori in R4 and CP2 using the spectral curve methods and show how this fits in with the previous understanding of minimal Lagrangian tori.

Mario Micallef (University of Warwick)Saturday 19th August 09:20
Isotropic minimal surfaces and holomorphic curves in flat tori
Isotropicity of a minimal surface is characterised by the vanishing of certain holomorphic differentials. Holomorphic curves in a complex torus with a flat metric are precisely the minimal surfaces which are maximally isotropic. I will discuss the deformation of a holomorphic curve in a complex torus with a flat metric to a minimal surface with a prescribed degree of isotropicity. This is based on joint work with Elisabeta Nedita.

Todor Milanov (Stanford University)Friday 18th August 11:00
Gromov-Witten invariants of CP1 and integrable hierarchies
Let X be a compact Kahler manifold. The total ancestor potential of X is a certain generating function of Gromov-Witten invariants of X. In this talk, using the mirror model of CP1, I will explain how to construct Hirota Bilinear Equations for the total ancestor potential of CP1. Our hope is that the construction can be generalized for other manifolds as well.

Motohico Mulase (UC Davis)Sunday 13th August 11:00
Integrable Systems in Gromov-Witten Theory
(Based on joint work with B. Safnuk and A. Hodge)
The talk will survey several new proofs of the Witten-Kontsevich theorem due to Okounkov-Pandharipande, Mirzakhani, Kim-Liu and Kazarian-Lando. Some ideas for understanding the strucutre of the Gromov-Witten invariants of a higher dimentional target will be examined.

Atsushi Nakayashiki (Kyushu University)Saturday 19th August 11:00
Abelian functions as a D-module
All elliptic functions which are singular at a point on an elliptic curve are obtained by differentiating one fundamental elliptic function like Weierstrass' pe-function. We discuss what happens in cases of higher dimensions.

Ulrich Pinkall (TU Berlin)Sunday 13th August 09:20
Integrable systems for real time simulation of fluid flow
In the limit of an infinitely thin vortex filaments one obtains from the Euler equations of fluid flow a purely local evolution equation for space curves, the so called "smoke ring flow". The smoke ring flow turns out to be a completely integrable Hamiltonian system related to the non-linear Schroedinger equation.

Here we study a discrete integrable version of the smoke ring flow, modified to take into account a finite thickness of the vorticity tubes and therefore to include non-local interactions between the different parts of the curves.

We apply this modified smoke ring flow to the design and visualization of fluid flow. It turns out that this approach allows the simulation of fluids like smoke in real time, which is of great importance for many applications in computer graphics.

Emma Previato (Boston University)Wednesday 16th August 11:00
Commuting partial differential operators
Beginning in the 1970s solutions of integrable evolution equations were produced using theta functions of spectral curves. In the theory under development, theta functions of spectral varieties of higher dimension are identified with partial differential operators and the resulting nonlinear equations are investigated.

Masa-Hiko Saito (Kobe University)Friday 18th August 09:20
Geometry of equations of Painlevé type
In the first half of this talk, I would like to explain the meaning of the Painlevé property for algebraic differential equations from the view point of geometry of spaces of initial conditions (or phase spaces) and their compactifications.
I will introduce the following two conditions on differential equations and the compactifications of their phase spaces;
  1. (n-log)-condition,
  2. Okamoto-Painlevé condition (introduced by Sakai and Saito-Takebe-Terajima).
I will explain the reason why these conditions are necessary for the Painlevé property.

In the second half of the talk, I will discuss about the moduli space of stable parabolic connections over a Riemann surface and their natural compactifications. I will explain about the results of Inaba-Iwasaki-Saito and Inaba on a property of Riemann-Hilbert correspondences which gives a complete proof of the Painlevé property for the non-linear differential equations arising from the isomonodromic deformations of stable parabolic connections.

Wolfgang Schief (New South Wales)Saturday 19th August 16:30
The geometry of shell membranes: The Lamé equation
We show that shell membranes in equilibrium subjected to constant normal loading are governed by an integrable system provided that the lines of curvature on the membranes coincide with the lines of principal stress. We prove that the membrane geometry uniquely determines the stress distribution unless the Gauss map is conformally flat in terms of (appropriately scaled) curvature coordinates. This condition is equivalent to demanding that there exist a Combescure transform of the membrane which is minimal and leads to a reduction of the Gauss-Mainardi-Codazzi equations to the classical Lamé equation uxx = n(n+1)P(x)u for the distinct value of n = -1/2.

Martin Schmidt (Universität Mannheim)Saturday 12th August 16:30
On immersions of tori in 3-space.

Ken Stephenson (Tennessee)Sunday 20th August 09:20
Discretizing conformal geometry
The talk will propose "circle packing" as a model for faithfully discretizing an area of classical geometry; in this instance, the geometry of conformal surfaces and analytic functions. This talk is related to integral systems only indirectly through the work of Alexander Bobenko and his collaborators, wherein circle patterns provide the basis of discrete integrable systems. My goal is to distinguish traditional numerical approximation from true discretization, in which the fundamental geometric behaviors are manifest directly by the discrete entities. In the case of circle packing, the result is a "quantum" conformal geometry which is classical in the limit. The talk will illustrate several well known classical results in discrete and visually accessible form.

Iskander Taimanov (Novosibirsk)Sunday 13th August 16:30
Surfaces in three-dimensional Lie groups

Chuu-Lian Terng (UC Irvine)Tuesday 15th August 09:20
The n×n KdV flows
(Joint work with K. Uhlenbeck)
We construct a new hierarchy of integrable systems from an unusual splitting of the algebra of loops in sl(n,C). We call it the n×n KdV hierarchy. It is known that the 2×2 hierarchy is the KdV hierarchy. We prove that the n×n KdV hierarchy is gauge equivalent to the Gel'fand-Dikii hierarchy. Since our flows come from a splitting of loop algebra, symplectic structures, Bäcklund transformations, inverse scattering, and tau functions for this hierarchy can be constructed via the splitting.

Alexander Veselov (Loughborough University)Friday 18th August 16:30
Logarithmic Frobenius structures and Coxeter root systems
I will discuss the geometry of special solutions to the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation related to the hyperplane arrangements and the role of Coxeter configurations in this theory. The talk is based on a recent joint work with M.V. Feigin.

Richard Ward (Durham University)Sunday 20th August 16:30
Periodic and doubly-periodic monopoles
This talk deals with BPS monopoles in three dimensions, which are periodic either in either one or two of the dimensions. In each case, I review what is known about the corresponding Nahm transform, and about exact and approximate solutions.

Computer Visualization Evening Sunday 13th August 20:00
This will take place in Holgate House, Grey College. Several groups using mathematical visualization tools for experimentation and testing conjectures will give demonstrations.
  1. Ulrich Pinkall (TU-Berlin): "Mathematical Visualization and Virtual Reality"
  2. Richard Palais (UC-Davies): "The Virtual Mathematical Museum"
  3. Nicolas Schmitt (Tuebingen): "Constant mean curvature Platonic n-noids "
  4. Alexander Bobenko(TU-Berlin): "Construction of Alexandrov and Koebe polyhedra"

Willmore EveningWednesday 16th August 20:00
This will take place in Holgate House, Grey College. After a short anecdotal introduction by John Bolton, there will be a talk by Franz Pedit describing some aspects of the Willmore conjecture.