Ideas and methods of integrable systems played a key role in many celebrated as well as recent advances in algebraic geometry, mathematical physics, probability, and many other areas of pure and applied mathematics. As a particularly striking example one may mention Krichever's recent proof of the famous Welters' conjecture, characterizing Jacobians of algebraic curves among all principally polarized abelian varieties. The conference will bring together experts and young researchers, mathematicians and physicists, people with different backgrounds and different takes on integrable systems, to discuss the latest achievements in this very dynamic field. This conference will be held at at Columbia University in New York, NY, on May 4-7, 2011.

A differential equation is called integrable if its solutions may be given by a closed formula, without a recourse to numerical or other approximations. Many highly nontrivial integrable equations were discovered and analyzed both classically and recently. They describe important phenomena in both classical (e.g. certain water waves) and modern theoretical physics and connect to some of the deepest mathematical structures known in pure mathematics. In applied mathematics, they form a basis of perturbative understanding of nearby problems and give a powerful calibration tool for numerical investigations. This conference will bring together the experts working on various aspects of integrability, and will aim to advance our understanding of integrable systems and their applications in algebra, geometry, and physics.

Project Report

, on integrable systems in algebra, geometry, and physics, took place at Columbia University, May 4-7, 2011. The conference brought together the leading world experts on integrable systems, from some of the founders of the theory, to younger mathematicians and physicists currently working in areas as diverse as representation theory, moduli, and probability, where integrability plays a major role. We had 82 registered participants, including 32 graduate students and 11 postdoctoral researchers. The talks at the conference presented the breadth of recent and ongoing research related to integrable systems, and allowed the younger mathematicians to learn about the recent developments and ongoing research, and to be exposed to the wealth of ideas in integrability, and its interrelations with algebra and geometry. The conference schedule allowed for ample time for conversations among the speakers and gave the graduate students and postdocs the opportunity to talk to the experts and learn from them directly in a collegial atmosphere. Integrable systems has been a very active area of current research, yielding many new insights, approaches, and results in diverse areas of mathematics and physics. In particular, Novikov's conjecture, proven by Shiota (both speakers at the conference), provided a solution to the classical Schottky problem of characterizing Jacobians among all principally polarized abelian varieties. Their approach was largely initiated by Krichever, in his construction of algebraic solutions of integrable systems, and recently yielded a proof by Krichever of the famous longstanding Welters trisecant conjecture, giving a much stronger characterization of Jacobians. From the algebraic viewpoint, Etingof, Veselov, and others recently studied the elliptic Calogero-Moser system for complex crystallographic groups. The quantum elliptic Calogero-Moser system was studied by Nekrasov and Shatashvili; Wiegmann studied application to quantum liquids, Its, Deift and others studied applications to asymptotic problems. These, and many other recent developments in the theory were presented by the above experts and other speakers at the conference. The Versatility of Integrability was unique in bringing together the full spectrum of researchers studying and applying the methods of integrable systems. It provided an invaluable venue for the experts to bring their expertise to bear on adjoining fields, and gave the students an opportunity to see the field in all its variety and intricate beauty, learning more outside of their primary Ph.D. research topic. More details are available at the conference website

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Joanna Kania-Bartoszynsk
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Columbia University
New York
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