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Created as a celebration of mathematical pioneer Emma Previato, this comprehensive book highlights the connections between algebraic geometry and integrable systems, differential equations, mathematical physics, and many other areas. The authors, many of whom have been at the forefront of research into these topics for the last decades, have all been influenced by Previato's research, as her collaborators, students, or colleagues. The diverse articles in the book demonstrate the wide scope of Previato's work and the inclusion of several survey and introductory articles makes the text accessible to graduate students and non-experts, as well as researchers. This first volume covers a wide range of areas related to integrable systems, often emphasizing the deep connections with algebraic geometry. Common themes include theta functions and Abelian varieties, Lax equations, integrable hierarchies, Hamiltonian flows and difference operators. These powerful tools are applied to spinning top, Hitchin, Painleve and many other notable special equations.
Integrable Systems and Algebraic Geometry: Volume 1
Integrable systems: a celebration of Emma Previator's 65th birthday Ron Donagi and Tony Shaska; 1. Trace ideal properties of a class of integral operators Fritz Gesztesy and Roger Nichols; 2. Explicit symmetries of the Kepler Hamiltonian Horst Knoerrer; 3. A note on the commutator of Hamiltonian vector fields Henryk Zoladek; 4. Nodal curves and a class of solutions of the Lax equation for shock clustering and Burgers turbulence Luen-Chau Li; 5. Solvable dynamical systems in the plane with polynomial interactions Francesco Calogero and Farrin Payandeh; 6. The projection method in classical mechanics A. M. Perelomov; 7. Pencils of quadrics, billiard double-reflection and confocal incircular nets Vladimir Dragovic, Milena Radnovic and Roger Fidele Ranomenjanahary; 8. Bi-flat F-manifolds: a survey Alessandro Arsie and Paolo Lorenzoni; 9. The periodic 6-particle Kac-Van Moerbeke system Pol Vanhaecke; 10. Integrable mappings from a unified perspective Tova Brown and Nicholas M. Ercolani; 11. On an Arnold-Liouville type theorem for the focusing NLS and the focusing mKdV equations T. Kappeler and P. Topalov; 12. Commuting Hamiltonian flows of curves in real space forms Albert Chern, Felix Knoeppel, Franz Pedit and Ulrich Pinkall; 13. The Kowalewski top revisited F. Magri; 14. The Calogero-Francoise integrable system: algebraic geometry, Higgs fields, and the inverse problem Steven Rayan, Thomas Stanley and Jacek Szmigielski; 15. Tropical Markov dynamics and Cayley cubic K. Spalding and A. P. Veselov; 16. Positive one-point commuting difference operators Gulnara S. Mauleshova and Andrey E. Mironov.