The Shaw Prize in Mathematical Sciences 2012 is awarded to Maxim Kontsevich for his pioneering works in algebra, geometry and mathematical physics and in particular deformation quantization, motivic integration and mirror symmetry.
2012 年的邵逸夫奖, 数学奖颁予法国高等科学研究所的教授马克西姆·康采维奇, 以表彰他在代数, 几何和数学物理上的开创性工作.
Maxim Kontsevich 也是 1998 年的菲尔兹奖(Fields Medal)得主.
颁奖典礼已经于 9 月 17 日进行.
邵逸夫奖的官网是 shawprize, 在这里可以找到获奖得主及其工作的简单介绍, 此外, 还有颁奖视频.
on the Prize in Mathematical Sciences 2012
Traditionally the interaction between mathematics and theoretical physics has been concerned with topics ranging from dynamical systems and partial differential equations to differential geometry to probability theory. For the last two decades, modern algebra and algebraic geometry (which is the study of the solutions of systems of polynomial equations in several variables via algebraic methods) have taken a central position in this interaction. Physical insights and intuition, especially from string theory, have led to a number of unexpected and striking predictions in both classical and modern algebraic geometry. Thanks to the efforts of many mathematicians new techniques and theories have been developed and some of these conjectures have been proven.
Maxim Kontsevich has led the way in a number of these developments. Among his many achievements is his early work on Witten’s conjecture concerning the topology and geometry of the moduli (that is parameter) spaces of all algebraic curves of a given genus, his solution of the problem of deformation quantization, his work in mirror symmetry and in a different direction the theory of motivic integration.
Quantization is the process of passing from classical to quantum mechanics and it has been realized by different mathematical theories. One of these is the algebraic theory of deformation quantization. This takes place on a Poisson manifold (that is a manifold with a Poisson bracket on functions) for which there are two natural algebras, the classical observables which are the functions under point-wise multiplication and the Poisson algebra where the multiplication comes from the Poisson structure. The problem is to give a formal deformation in powers of a parameter h, in which the zeroth order term is the classical algebra of observables and the next order term is the given Poisson algebra. The construction of such a deformation was carried out in special cases (Weyl, Moyal, Fedosov…) but the general case proved formidable. It was resolved brilliantly by Kontsevich using ideas from quantum field theory.
The discovery by physicists of mirror pairs of Calabi–Yau manifolds has led to a rich and evolving mathematical theory of mirror symmetry. The physics predicts that there is a relation between the symplectic geometry (that is a geometry coming from classical mechanics) on such a manifold and the algebraic/complex geometry of the mirror manifold. When carried out in certain examples for which explicit computations can be made, this led to some remarkable predictions in classical enumerative geometry, concerning the counting of curves in higher dimensional spaces. Some of these predictions have since been proven. Kontsevich introduced homological mirror symmetry which predicts that further refined objects associated with the symplectic geometry of the manifold are related to ones associated with the complex geometry of its mirror. These conjectures and their generalizations have been proven in significant special cases. From the beginning Kontsevich has played a leading role in the development of the mathematical theory of mirror symmetry. He continues to revisit the original formulation and to provide clearer conceptual answers to the mathematical question: “What is mirror symmetry?”
Motivic integration is another invention of Kontsevich. It is an integration theory which applies in the setting of algebraic geometry. Unlike the usual integral from calculus whose value is a number, the motivic integral has its values in a large ring which is built out of the collection of all varieties (the zero sets of polynomial equations). It satisfies many properties similar to the usual integral and while appearing to be quite abstract, when computed and compared in different settings it yields some far reaching information about algebraic varieties as well as their singularities. It has been used to resolve some basic questions about invariants of Calabi–Yau varieties and it is also central to many recent developments concerning the uniform structure of counting points on varieties over finite fields and rings.
Through his technical brilliance in resolving central problems, his conceptual insights and very original ideas, Kontsevich has played a substantial role in shaping modern algebra, algebraic geometry and mathematical physics and especially the connections between them.
Mathematical Sciences Selection Committee
The Shaw Prize
17 September 2012, Hong Kong
Autobiography of Maxim Kontsevich
I was born in 1964 in a suburb of Moscow, close to a big forest. My father is a well-known specialist in Korean language and history, my mоther was an engineer (she is retired now), and my elder brother is a specialist in computer vision. The apartment where I grew up was very small and full of books – about half of them in Korean or Chinese.
I became interested in mathematics at age 10 – 11, mainly because of the influence of my brother. Several books at popular level were very inspiring. Also, my brother was subscribed to the famous monthly “Kvant” magazine containing many wonderful articles on mathematics and physics addressed to high-school kids, sometimes explaining even new results or unresolved problems. Also, I used to take part in Olympiads at various levels and was very successful.
In the Soviet Union, some schools had special classes for gifted children, with an additional four hours per week devoted to extra-curricular education (usually in mathematics or physics) taught by university students who had passed through the same system themselves. At age 13 – 15 I was attending such a school in Moscow, and from 1980 till 1985 was studying mathematics at Moscow State University. Because of my previous training in High School, I never attended regular courses, but instead went to several graduate and research-level seminars where I learned a huge amount of material. My tutor was Israel Gelfand, one of the greatest mathematicians of the 20th Century. His weekly seminar, on Mondays, was completely unpredictable, and covered the whole spectrum of mathematics. Outstanding mathematicians, both Soviet and visitors from abroad, gave lectures. In a sense, I grew up in these seminars, and also had the great luck to witness the birth of conformal field theory and string theory in the mid-80s. The interaction with theoretical physics remains vitally important for me even now. After graduating from university, I became a researcher at the Institute for Information Transmission Problems. Simultaneously, I began to learn to play the cello and for several years enjoyed the good company of my musician friends with whom I played some obscure pieces of baroque and renaissance music.
In 1988, I went abroad for the first time, to Poland and France. Also in 1988, I wrote a short article concerning two different approaches to string theory, and maybe because of this result, was invited to visit the Max Planck Institute for Mathematics in Bonn for three months in 1990. At the end of my stay there was an annual informal meeting of mostly European mathematicians, called Arbeitstatgung, where the latest hot results were presented. The opening lecture by Michael Atiyah was about a new surprising conjecture of Witten concerning matrix models and the topology of moduli spaces of algebraic curves. In two days I came up with an idea of how to relate moduli spaces but with a completely new type of matrix model, and explained it to Atiyah. People at MPIM were very impressed and invited me to come back the following year. During the next 3 – 4 years I was visiting mostly Bonn, and also IAS in Princeton and Harvard. My then future wife Ekaterina, whom I met in Moscow, accompanied me, and in 1993 we were married. In Bonn I finished several works which became very well-known: one on Vassiliev invariants, and another on quantum cohomology (with Yu Manin, whose seminar I had attended back in Moscow). Scientifically, a very important moment for me was Spring 1993 when I came to the idea of homological mirror symmetry, which was an opening of a grand new perspective. In 1994, I accepted an offer from Berkeley, but one year later I moved to IHES in France, where I continue to work. In 1999 my wife and I were granted French citizenship (keeping our Russian citizenship as well), and in 2001 our son was born.
For a few years I visited simultaneously Rutgers University, where my teacher Gelfand moved to after the perestroika, and IAS in Princeton. During the last six years I have regularly visited the University of Miami.
In my work I often change subjects, moving from Feynman graphs to abstract algebra, differential geometry, dynamical systems, finite fields. Still, mirror symmetry remains the major line. The interaction during the last two decades between mathematics and theoretical physics has been an amazing chain of breakthroughs. I am very happy to be a participant in this dialogue, not only absorbing mathematical ideas from string theory, but also giving something back, like a recent wall-crossing formula which I discovered with my long-term collaborator Yan Soibelman, and which became a very important tool in the hands of physicists, simultaneously answering questions concerning supersymmetric particles, and solving the classical problem about asymptotics for equations depending on small parameter.
17 September 2012