Fran Burstall: "Polynomial conserved quantities for isothermic surfaces"
Isothermic surfaces constitute a geometric integrable system of both classical and modern interest. They include CMC
surfaces in 3-dimensional space forms and the class of "special" isothermic surfaces discussed by Darboux and Bianchi in relation to surfaces isometric to a quadric.
I shall describe recent work with my student, Susana Santos, which provides a simple approach to and substantial extension of these classes.
Oleg Chalykh: "KP solitons, Weyl algebra and Calogero-Moser spaces"
As is well-known, the rational solutions of the KP hierarchy are parameterized by certain algebro-geometric data (consisting, roughly, of a rational cuspidal curve together with a line bundle on it). Berest and Wilson observed around 1998 that the same data appeared implicitly in the work of Cannings and Holland (1994) on classification of ideals of the Weyl algebra.
I will start by briefly recalling this relation. Next, following our recent work with Yu. Berest (mathQA/0410194), I will explain how to relate the ideals of the Weyl algebra to the points of the so-called Calogero-Moser spaces. In
particular, this leads to a much more transparent proof of the well-known result of G.Wilson, giving a parameterization of the rational KP solitons by pairs of nxn matrices X,Y satisfying the `rank one' condition: rk([X,Y]+id)=1.
Further generalizations will be mentioned.
Adam Doliwa: "Generalized isothermic lattices"
The generalized isothermic lattice is the two dimensional quadrilateral lattice satisfying simultaneously a quadratic and the projective Moutard constraints. A posteriori, the lattice is characterized by "Steiner's version" of the cross-ratio condition of Bobenko and Pinkall. Indeed, when the quadric under consideration is the Moebius sphere one recovers their definition of discrete isothermic surfaces.
In my talk I will concentrate on presenting basic geometric constructions which encode integrability of the generalized isothermic lattice.
Eugene Ferapontov: "Differential-geometric aspects of integrability of multi-dimensional quasilinear systems"
A multi-dimensional quasilinear system is said to be integrable if it
possesses `sufficiently many' hydrodynamic reductions parametrized by
arbitrary functions of a single variable. It has been observed that this condition provides an effective classification criterion and partial classification results were obtained.
The integrability conditions lead to complicated over-determined involutive PDEs which possess interesting differential-geometric properties. For instance, the necessary condition for integrability of a multi-dimensional system of hydrodynamic type can be formulated in terms of the Haantjes tensor of the corresponding pencil of matrices. Some further geometric aspects of multi-dimensional integrability will be discussed.
Frederic Helein: "Hamiltonian stationary Lagrangian surfaces in symmetric spaces"
Hamiltonian stationary Lagrangian surfaces are critical points of the restriction of the area functional to the set of Lagrangian surfaces, the Euler-Lagrange equation being computed by using uniquely Hamiltonian vector fields. Beside the fact that they satisfy the Lagrangian constraint they are characterized by a third order elliptic PDE. Particular solutions are the special Lagrangian surfaces in a Calabi-Yau manifold.
In a joint work with Pascal Romon we have shown that this problem is completely integrable for surfaces in symmetric Hermitian spaces of complex dimension two. The most important examples are the complex plane (or the quaternion line)
and the complex projective plane.
Recently these results has been extended by Idrisse Khemar for surfaces in the octonion line.
Claus Hertling: "What Painleve III and polarized mixed Hodge structures
have in common"
Both are related to certain vector bundles on P1 with real structure and meromorphic connection with irregular poles at zero and infinity. More precisely, to 1-parameter orbits of them. Painleve III is the semisimple case of rank 2, polarized mixed Hodge structures are the nilpotent case of any rank.
We will discuss them and the general case. This is related to work of Its/Novokshenov, Schmid, Simpson, Sabbah,
Dubrovin, and the tt^* equations of Cecotti and Vafa.
Tim Hoffmann: "Discrete S-isothermic and S-cmc surfaces"
Discrete s-isothermic surfaces are a discretization of isothermic surfaces build from spheres. In analogy to the smooth case they can be charachterized by a Moutard equation in a Minkowski space.
Knowing the Darboux transforation for this class of discrete surfaces allows to define discrete surfaces of constant mean curvature (cmc) and to derive some of their geometric properties.
Andrew Hone: "Bilinear recurrences, integer sequences and elliptic curves"
The explicit solution of the initial value problem for a family of fourth order bilinear recurrences is presented in terms of the Weierstrass sigma function of an associated elliptic curve. For particular initial conditions these recurrences generate sequences of integers, including elliptic divisibility sequences and the Somos sequence.
Recent joint work with Everest and Ward concerning the appearance of prime divisors are briefly discussed. Some analogous results on the Kleinian sigma function in genus two with Braden and Enolskii, related to the Backlund transformation for an integrable Hamiltonian mechanical system, will also be mentioned.
Spyridon Kamvissis: "On the nonlinear steepest descent method"
We will review the history of the nonlinear steepest descent method for the asymptotic evaluation of the solutions of Riemann-Hilbert factorization problems, and then focus on some recent results on the "non-self-adjoint" extension of the theory.
In particular we will consider the case of the semiclassical focusing NLS problem. We will explain how the nonlinear steepest descent method gives rise to a maximin variational problem for Green potentials with external field in two dimensions and we will announce results on existence and regularity of solutions to this variational problem.
Igor Krichever: "Trisecant conjecture and integrable disrete equations"
We present recent results on the characterization of the Jacobians and Primmians via integrable linear differential equations and related infinite-dimensional analogs of Calogero-Moser type systems.
We will disscus also the same connections of the famous Walters trisecant conjecture integrable difference equations.
Christian Mercat: "Discrete Holomorphies and integrability"
Holomorphy can be discretized in many ways. We will present a quadratic version of it, the so called cross-ratio preserving maps, associated to the celebrated circle packings, and explain its integrability through Baecklund transformations.
Though very simple, its linearized version allows for a tighter similarity with the theory of complex analysis and we will present the main features linking the three theories.
the spectral data of a varying periodic curve or surface,
soliton surfaces and curve flows from varying boundary or initial data.
Experiments with the discrete systems often leads to interesting new questions about the dynamics of their smooth counterparts.
Nikolai Reshetikhin: "The 6-vertex model with fixed boundary conditions"
The 6-vertex model with fixed boundary conditions on an N × N region of a square lattice is studied in the thermodynamical limit.
I will present arguments indicating that the system develops a macroscopical configuration (the limit shape) as N → ∞. This limit shape is a solution to certain variational problem.
I also will discuss this variational problem and the structure of fluctuations around the limit shape.
Ulrike Scheerer: "Discrete minimal surfaces from circle patterns"
Discrete minimal surfaces can be considered as a subclass of s-isothermic surfaces built from spheres. We focus on the construction of discrete analogues to continous minimal surfaces (for example some Plateau problems) from a combinatorial picture of the curvature lines (and their image under the Gauss map). As in the continous case, there is a Weierstrass-type representation relating discrete minimal surfaces to planar orthogonal circle patterns. We present some convergence results of such circle patterns to holomorphic functions.
Martin Schmidt: "Towards a proof of the Willmore conjecture"
I present a shortened version of a proof of the Willmore conjecture,
due to new observations one can circumvent some of the previous technical difficulties.
This talk will focus on Fermi curves of local minimizers of the Willmore functional.
Boris Springborn: "A discrete Laplace-Beltrami operator for simplicial surfaces"
We define a discrete Laplace-Beltrami operator for simplicial surfaces.
It depends only on the intrinsicgeometry ofthe surface and its edge
weights are positive. Our Laplace operator is similar to the one defined
by Pinkall and Polthier (the so called "cotan formula") except hat it is
based on the intrinsic Delaunay triangulation of the simplicial surface.
This leads to new definitions of discrete harmonic and holomorphic
functions, discrete mean curvature, and discrete minimal surfaces.
Alexander Veselov: "Coincident root loci and Calogero-Moser systems"
In 1857 Arthur Cayley addressed the question (which he prescribed to Sylvester) of how to determine when a polynomial has a multiple root of a given multiplicity or, more generally, several roots with prescribed multiplicities. The corresponding varieties are known as coincident root loci and the question is what are the algebraic equations defining them.
We will discuss some interesting relations of this problem with the theory of quantum Calogero-Moser systems and related theory of Jack polynomials, which have been recently discovered by B. Feigin, Jimbo, Miwa, Mukhin,
Kasatani, A.N. Sergeev and the speaker.
Steffen Weissmann: PORTAL presentation
The PORTAL is a virtual reality theater aimed at the visualization of complicated mathematical structures.
For visualizations of such structures in three dimensions, immersive virtual reality as in the portal, which gives the user the feeling of moving around the objects being viewed, is vastly superior to the two-dimensional view presented on a workstation screen.
We show selected applications related to integrable systems:
Applications of the discrete Sinus-Gordon equation
- Pendulum chain
- helicoids with handles
- discrete minimal surfaces
shallow water waves as solutions of the KP2 equation.