University of Colorado

Forty-Fourth Annual DeLong Lecture Series

Department of Mathematics

Professor Maxim Kontsevich

Classical integrable systems and applications

Classical integrable systems are mechanical systems with quasi-periodic dynamics. In the hamiltonian formulation, they have a maximal commuting family of first integrals. The basic examples are the Euler top and geodesics on an ellipsoid. I will describe two applications which show that integrable systems are very fundamental mathematical objects by themselves. First, one can generalize (hypothetically) the Langlands correspondence in the functional field case to higher dimensions. Automorphic forms are replaced by an appropriate version of an algebraic integrable system. Secondly, in Mirror Symmetry, the basic underlying geometric structure is of a manifold with an integral affine structure. Such a structure arises in real integrable systems.

Date	Time	Room	Title
Monday, February 25, 2008	4:00-5:00 pm	BESC 180	Definitions, examples and a general construction Following this lecture, there will be a reception in honor of Professor Kontsevich at the Koenig Alumni Center, 1202 University Avenue (the SE corner of Broadway and University).
Wednesday, February 27, 2008	4:00-5:00 pm	ECCR 265	Integrable systems and a higher-dimensional generalization of the Langlands correspondence
Friday, February 29, 2008	4:00-5:00 pm	ECCR 265	Integrable systems and collapsing Calabi-Yau varieties in mirror symmetry

Maxim Kontsevich

Maxim Kontsevich was educated at Moscow State University between 1980 and 1985, where he was a student of I.M. Gelfand. Between 1985 and 1990, he was a junior researcher at the Institute for Problems of Information Transmission in Moscow. In 1991, he moved to the West, where he was based at the Max-Planck-Institut für Mathematik in Bonn until 1994. He also visited the Institute for Advanced Study and Harvard University. He received his Ph.D. from Bonn University in 1992. Between 1994 and 1995 he was a professor at the University of California, Berkeley, and since then he has been a professor at the IHES (Institut des Hautes Études Scientifiques) in France. He has also held visiting professorships at Rutgers University (1997-2002) and the University of Miami (2005-present).

Professor Kontsevich has received several major prizes and distinctions, including the Otto-Hahn-Medaille from the Max Planck Society (1992); the Mayor of Paris's Prize (first European Congress of Mathematicians, 1992), the Prize of the International Congress of Mathematical Physics (Brisbane, Australia, 1997) and the Fields Medal (International Congress of Mathematicians, 1998). In January 2008, he was awarded the Crafoord Prize, which he shared with Edward Witten. He has also been elected to the Academiae Europaeae (2000) and the Académie des Sciences (2002). He has given many invited addresses, including to the European Congress of Mathematicians (1992), the International Congress of Mathematicians (1994), and the International Congress of Mathematical Physics (1994).

Professor Kontsevich's research involves many areas of mathematics and mathematical physics. In the early 1990s, he mainly worked on topological quantum field theories, their foundations, and their applications to pure mathematics, including homological mirror symmetry, deformation quantization, and new constructions of varieties over non-archimedean fields. Recently, he has been concentrating on the foundations of noncommutative algebraic geometry and the interplay between quantization and prime characteristic. He has formulated several new conjectures on motives, some of which will be explained in these lectures.

DeLong Lecture Series
This Lecture Series is funded by an endowment given by Professor Ira M. DeLong, who came to the University of Colorado in 1888 at the age of 33. Professor DeLong essentially became the mathematics department by teaching not only the college subjects but also the preparatory mathematics courses. Professor DeLong was a prominent citizen of the community of Boulder as well as president of the Mercantile Bank and Trust Company, organizer of the Colorado Education Association, and president of the charter convention that gave Boulder the city manager form of government in 1917. After his death in 1942, it was decided that the bequest he made to the mathematics department would accumulate interest until income became available to fund DeLong prizes for undergraduates and DeLong Lectureships to bring outstanding mathematicians to campus each year.

If you have any questions concerning this lecture series, please contact Richard Green.

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