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International Conference "Analytic Theory of Differential and Difference Equations" Dedicated to the Memory of Andrey Bolibrukh
February 2, 2021 18:30, online, Moscow
 


Dynamics of extremal polynomials, isomonodromic and isoharmonic deformations

Vladimir Dragović

The University of Texas at Dallas, Richardson, TX
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Abstract: We establish a synergy of integrable billiards, extremal polynomials, Riemann surfaces, combinatorics, potential theory, and isomonodromic deformations. The cross-fertilization between ideas coming from these distinct fields lead to new results in each of them. We introduce a new notion of iso-harmonic deformations. We study their isomonodromic properties in the first nontrivial examples and indicate the genesis of a new class of isomonodromic deformations. The talk is based on the current research with Vasilisa Shramchenko and:

[1] Dragović V., Radnović M., Periodic ellipsoidal billiard trajectories and extremal polynomials, Comm. Math. Phys. 372 (2019), 183–211.
[2] Dragović V., Shramchenko V., Algebro-geometric solutions of the Schlesinger systems and the Poncelet-type polygons in higher dimensions, IMRN 2018:13, 4229–4259.
[3] Dragović V., Shramchenko V., Algebro-geometric approach to an Okamoto transformation, the Painlevé VI and Schlesinger equations, Ann. Henri Poincare 20:4 (2019), 1121–1148.
[4] Dragović V., Radnović M., Caustics of Poncelet polygons and classical extremal polynomials, Regul. Chaotic Dyn. 24:1 (2019), 1–35.
[5] Andrews G., Dragović V., Radnović M., Combinatorics of the periodic billiards within quadrics, Ramanujan J., DOI: 10.1007/s11139-020-00346-y (arXiv:1908.01026).

Language: English

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