# Toshiyuki Kobayashi Awarded the 2015 JMSJ Outstanding Paper Prize

Toshiyuki Kobayashi, professor at the Graduate School of Mathematical Sciences, The University of Tokyo, and principal investigator at Kavli IPMU, received the 2015 JMSJ Outstanding Paper Prize.

Since 2010, the JMSJ Outstanding Paper Prize has been awarded to the authors of most outstanding articles published in the Journal of the Mathematical Society of Japan (JMSJ) in the previous year. Kobayashi and his coauthors, J. Hilgert and J. Möllers, received this honor for their paper, “Minimal representations via Bessel operators”, which was published in JMSJ vol. 66 (2), (2014), pp.349–414.

An “infinite-dimensional representation” is a mathematical concept that provides an algebraic understanding of “symmetries” in the broad sense. Although this concept is closely related with quantum theory, it is mathematically difficult to analyze. Recent progress in algebraic representation theory has revealed that among unitary representations there are only a few building blocks, and that they are realized in relatively “small infinite-dimensional spaces". The “minimal representation" is the most distinguished representation of this property.

Kobayashi proposed considering a “minimal representation” from a different perspective where “small representations of a group” viewed from algebraic representation theory are regarded as “large symmetries in a representation space” viewed from global analysis. Based on this viewpoint, Kobayashi initiated geometric analysis with minimal representations as a motif. Since 2003, he has published many articles in this area, which total over 1,000 pages, with his collaborators in Germany, France, the United States, Denmark, and Japan.

As predicted by Kobayashi, this new analysis not only has been fruitful in representation theory but also has realized significant interactions in various areas of mathematics, such as conformal geometry, symplectic geometry, deformation theory of the Fourier transform, conservative quantities for partial differential equations, and special functions associated with fourth order differential equations.

Existing methods of “induction” cannot construct minimal representations directly. However, the JMSJ article provides an explicit construction for the minimal representation using a Lagrangian submanifold of the minimal nilpotent coadjoint orbit, which generalizes the classical Schrödinger model of the oscillator representation.

“Geometric analysis of minimal representations has opened the door to new avenues of research. This infinite-dimensional space with a large symmetry has been a spring of new discoveries for me. It is delightful because the topic interacts with various areas of mathematics, and the more I understand, the more discoveries are born from the spring” says Kobayashi.

The award ceremony was held at the annual meeting of the Mathematical Society of Japan, which occurred at Meiji University from March 21 to 24, 2015.

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