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Isomonodromic tau functions, the constructive approach to conformal maps, and black holes.

Presented by: 
Bruno Carneiro da Cunha
Wednesday 11th September 2019 - 10:00 to 11:00
INI Seminar Room 1
Recent developments on the relation between the Riemann-Hilbert problem and the representation theory of Virasoro algebras allowed for explicit expansions of the isomonodromic tau functions in terms of conformal blocks. In this talk I will describe how these expansions can be used to constructively solve the connection problem of ordinary differential equations of the Fuchsian type. The simplest non-trivial case of 4 regular singular points (the Heun equation) -- as well as a particular confluent limit -- are solved by generic Painlevé transcendents of the sixth and fifth type. On the formal side, these relations allow us to conjecture an interpretation of the zeros of the tau functions in the general case. On the application side, the explicit expansions are useful for high precision numerical calculations of the accessory parameters of conformal maps, as well as the determination of (quasi)-normal modes of metric vibrations for a variety of black hole backgrounds in general relativity.

Co-authors include: T. Anselmo, J.-J. Barragán-Amado, J. P. Cavalcante, R. Nelson, D. Crowdy and E. Pallante.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons