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

57 mins 31 secs, 1.26 GB, MPEG-4 Video 1280x720, 30.0 fps, 44100 Hz, 2.98 Mbits/sec

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Description:	Carneiro da Cunha, B Wednesday 11th September 2019 - 10:00 to 11:00


Created:	2019-09-11 11:09
Collection:	The complex analysis toolbox: new techniques and perspectives
Publisher:	Isaac Newton Institute
Copyright:	Carneiro da Cunha, B
Language:	eng (English)


Abstract:	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.

Abstract:

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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