Recently, some of the most profound advances in algebra and geometry have been inspired by ideas from physics. Maxim Kontsevich has led the way in a number of these developments.
Beginning with Heisenberg’s introduction of quantum mechanics, the mathematical process of quantization – that is, of passing from classical to quantum mechanics – has been a central theme. One version, known as deformation quantization, has for its natural setting classical spaces known as Poisson manifolds. Their exact quantization had been carried out in special cases, but in general this proved to be a formidable problem. It was resolved brilliantly by Kontsevich, who used ideas from quantum field theory.
Kontsevich’s invention of motivic integration, a striking new conceptual tool, allowed him and others to resolve some problems in algebraic geometry (the study of solutions to polynomial equations to several variables) that had previously seemed way out of reach.
The discovery by string theorists of “mirror symmetry” led to a series of unexpected mathematical predictions which assert that two apparently different geometries appearing in string theory: symplectic geometry, which is connected with classical mechanics, and algebraic geometry, are “mirror” to each other. Thanks to the contributions of many mathematicians these assertions have gradually been proven. The modern understanding of mirror symmetry is framed by fundamental insights and advances. Many of these are due to Kontsevich, who, beginning with his 1994 “homological mirror symmetry conjecture”, keeps revisiting the original formulation to provide clearer conceptual answers to the mathematical question “What is mirror symmetry?”.
Mathematical Sciences Selection Committee
The Shaw Prize
29 May 2012 Hong Kong