# The 2010 Inoue Science Prize to Toshiyuki Kobayashi

December 21, 2010

Institute for the Physics and Mathematics of the Universe (IPMU)

The 2010 Inoue Science Prize was awarded to Toshiyuki Kobayashi, professor of The University of Tokyo Graduate School of Mathematics Sciences and Senior Scientist at IPMU. The Inoue Science Prize is awarded to scientists under the age of fifty who have made significant contributions in the area of basic natural sciences. The award ceremony will take place on February 4, 2011.

Title of the prize: Analysis of symmetry of infinite dimensions

Statement of the Inoue Foundation:

Kobayashi’s contributions present a beautiful harmony of all areas of basic concepts that constitute mathematics, namely algebra, geometry and analysis. He has developed a large scale theory starting from the symmetry principle and extending over a wide area of pure mathematics. One of them is to create a discontinuous group theory that goes beyond the framework of Riemannian geometry. Kobayashi studied a problem of rigidity in discontinuous groups for semisimple symmetric spaces with indefinite metrics, formulated a concept of local rigidity and global rigidity, and for the first time constructed an example in which the rigidity principle does not apply for arbitrarily high dimensions. Furthermore, he invented a method for judging the presence of nontrivial deformation in the deformation problem of fundamental groups, significantly adding to the world of geometry. These pioneering contributions influenced a wide research areas such as symplectic geometry, harmonic maps and graph theory, and have lead to development beyond pure mathematics.

In the theory of branching rule, which is a mathematical description of symmetry breaking of the representation theory in infinite dimensions, various difficulties in analysis arising from infinite dimensions have been obstructing its advancement. On this front, Kobayashi advocated the concept of “purely discrete decomposition” for the restriction of infinite dimensional representations by fully exploiting complex analysis, algebraic analysis and pure algebraic methods, and invented a fundamental theory of discrete branching rule. This theory is now widely used as a new method for attacking a variety of difficult problems in other areas such as number theory or global analysis.

Kobayashi has also made original contributions in singular unitary representations and non-commutative harmonic analysis. For example, he has shown that a representation of infinite dimensions of the conformal transformation groups can be constructed in the space of global solutions to the Yamabe operator on any psuedo-Riemannian manifold. As a result his approach has become widely used in geometry, automorphic functions, special functions and partial differential equations.

Kobayashi’s contributions are fully justify the Inoue Science Prize because they have impacted greatly many areas of mathematics and have pioneered new areas of research.