## Prof. Toshiyuki Kobayashi at the Department of Mathematics Won Humboldt Research Award on July 23

Prof. Toshiyuki Kobayashi, a professor of the Graduate School of Mathematical Sciences and of the Department of Mathematics, Faculty of Science, recently won a Humboldt Research Award in Mathematics. The award ceremony was held at the German presidential residence, the Bellevue Palace in June, 2008.

The Alexander von Humboldt Foundation, a non-profit organization established by the Federal Republic of Germany, selected Prof. Kobayashi as a recipient of its Humboldt Research Award in Mathematics for 2008 for his pioneering contributions to geometric analysis, in particular to the theory of lattices for homogeneous spaces and representation theory. The prize is awarded to outstanding scientists and scholars whose fundamental discoveries have had a significant impact on their discipline and who are expected to continue to produce cutting-edge achievements. At the School of Science, Dr. Masatoshi Koshiba, an honorary professor emeritus of the University of Tokyo and Dr. Akito Arima,former President of the University of Tokyo had previously won the Award in Physics. This is the first time the University of Tokyo has been awarded in Mathematics.

Prof. T. Kobayashi is a mathematician known for his innovative work on representation theory. This is the mathematics of symmetries, ranging from numbers to infinite dimensions, and from geometry to physics. Prof. Kobayashi's pioneering contributions are a harmonious blend of analysis, geometry and algebra, and demonstrate striking new perspectives. He has thus opened up several new fields, including the theory of discontinuous groups beyond the classical Riemannian setting, the theory of discrete breaking of symmetry, and the theory of visible action on complex manifolds towards a unified theory of multiplicity-free representations.

The details of these three achievements are as follows.

### Achievement 1:

In the late 1980s, Prof. Kobayashi pioneered the theory of discontinuous groups beyond the classical Riemannian setting, the first systematic study in this area. He discovered and analyzed mysterious phenomena concerning the global geometry of locally homogeneous spaces of high dimension, and single-handedly established the foundation of the theory. His achievements opened up a new research field that ranges from geometry to Lie group theory, ergodic theory, and representation theory.### Achievement 2:

His second major achievement concerns his highly original discoveries in branching laws of unitary representations. Branching laws are a mathematical description of broken symmetries. Due to the various analytic difficulties caused by infinite dimensions even for building blocks cases, a deeper understanding of branching laws of unitary representations was considered to be hopeless in the past. However, Prof. Kobayashi developed an epoch-making prototype for discretely decomposable branching laws, elucidated this discovery in full generality by using micro-local analysis and algebraic representation theory, and established the theory of discrete breaking symmetries. Being without continuous spectrum, discrete branching laws can be described using algebraic methods. Thanks to Prof. Kobayashi's theory of discrete breaking symmetries, new vistas in representations were opened from a different viewpoint, and moreover, applications to different fields such as the topology of modular varieties and non-commutative harmonic analysis are beginning to burgeon.### Achievement 3:

His third major achievement concerns multiplicity-free representations. The algebraic structure of multiplicity-freeness (each building block is used no more than once) is sometimes hidden in powerful mathematical methods such as the Taylor series and the Fourier transform. He discovered the geometric trick that this algebraic property propagates from fibers to sections. This is the machinery that makes new out of old, and big out of small. Prof. Kobayashi thus developed systematic and synthetic applications of his original theory of 'visible actions' on complex manifolds to multiplicity-free theorems on both finite and infinite dimensional representations.The essential contribution of Prof. Kobayashi's research is the fact that he not only has opened up new research areas but has at the same time developed the fundamental theory by putting forward extraordinary insights. With his unflagging passion for research, he has recently held a number of invited lectures and keynote lectures abroad.

Being a graduate of the University of Tokyo, Prof. Kobayashi has devoted himself to educational activities for the students since he was first appointed at the University of Tokyo at the age of 24. In addition to research and education, over the years he has served the academic community in several capacities at home and abroad, including as a member of key councils, as well as working on the editorial boards and as managing editor of several academic journals.

We wish Prof. Kobayashi health, happiness and continued success.