skip to content

Timetable (GMRW09)

Inverse scattering and integrability

Monday 21st November 2005 to Sunday 27th November 2005

Monday 21st November 2005
10:00 to 11:00 LJ Mason ([Oxford])
The twistor theory of the Ernst Equation
11:30 to 12:30 Integrable reductions of Einstein's field equations: monodromy transform and the linear integral equation methods

For each of the known today integrable reductions of Einstein's field equations for space-times with two commuting isometries, the monodromy transform (similarly to the well known Inverse Scattering Transform applied successfully for many other completely integrable equations) provides us with a unified and convenient mapping of the complete space of local solutions of the symmetry reduced field equations in terms of a finite set of unconstrained coordinate-independent functions of the spectral parameter (analogous to the scattering data). These set of functions arises as the monodromy data for the fundamental solution of associated linear systems (``spectral problems'') and they can serve as free independent ``coordinates'' in the infinite dimensional space of the local solutions. The direct and inverse problems of such ``coordinate transformation'', (monodromy transform), i.e. the problems of calculation of the monodromy data for given solution of the field equations and of calculation of the solution, corresponding to given monodromy data, possess unique solutions. In principle, the monodromy data functions can be calcul ated also from some boundary, or initial, or characteristic initial data for the fields, and many physical properties of solutions are simply ``encoded'' in the analytical structures of these functions. However, to find the solutions of the mentioned above direct and inverse problems, we have to solve explicitly the systems of ordinary differential and linear singular integral equations respectively, that can occur a difficult problems in many cases.

In the introduction we give a short survey of various integrable symmetry reductions of Einstein's field equations and mention some interrelationships between various developed linear integral equation methods. We describe also in a unified manner the common structure of various integrable reductions of Einstein's field equations -- the (generalized) hyperbolic and elliptic Ernst equations for vacuum and electrovacuum space-times, for Einstein - Maxwell - Weyl fields, for stiff matter fluids as well as their matrix generalizations for some string gravity models with coupled gravity and dilaton, axion and Abelian vector fields. The structure of the direct problem of the monodromy transform and general construction of the linear singular integral equation solving the inverse problem will be considered and some applications of this approach for construction of infinite hierarchies of exact solutions will be presented. In this context we present also another linear integral equation forms of integrable hyperbolic symmetry reductions of Einstein's field equations which provides a solution (viz. linearization) of the characteristic initial value problems for colliding waves and for evolution of inhomogeneous cosmological models.

14:30 to 15:30 Quasi-stationary routes to the Kerr black hole

In this talk I shall discuss quasi-stationary transitions from rotating equilibrium configurations of normal matter to rotating black holes via the extreme Kerr metric. For the idealized model of a rotating disc of dust, rigorous results derived by means of the 'inverse scattering method' are available. They are supplemented by numerical results for rotating fluid rings with various equations of state.

References: gr-qc/0205127, gr-qc/0405074, gr-qc/0506130

Tuesday 22nd November 2005
11:30 to 12:30 Isomonodromic tau-functions on Hurwitz spaces and their applications

We discuss Jimbo-Miwa tau-functions corresponding to Riemann-Hilbert problems with quasi-permutation monodromy groups; these tau-functions are sections of certain line bundles on Hurwitz spaces. We show how to compute these tau-functons explicitly in terms of theta-functions and discuss their applications in several areas: large N expansion in Hermitian matrix models, Frobenius manifolds, determinants of laplacians over Riemann surfaces and conformal factor of Ernst equation.

14:30 to 15:30 Periodic instantons \& monopoles in gauge theory (and gravity) INI 1
16:00 to 17:00 Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability

A (d+1)-dimensional dispersionless PDE is said to be integrable if it possesses infinitely many n-component hydrodynamic reductions parametrized by (d-1)n arbitrary functions of one variable. Among the most important examples one should primarily mention the three-dimensional dKP and the Boyer-Finley equations, as well as the four-dimensional heavenly equation descriptive of self-dual Ricci-flat metrics. It was observed that the integrability in the sense of hydrodynamic reductions is equivalent to the existence of a scalar pseudopotential playing the role of dispersionless Lax pair. Lax pairs of this type constitute a basis of the dispersionless d-bar and twistor approaches to multi-dimensional equations.

Wednesday 23rd November 2005
16:00 to 17:00 Anti-self-dual conformal structures with null Killing vectors INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons