Kyushu University [Takaaki Nomura (Professor) Faculty of Mathematics, Department of Mathematics]

My research area is analysis on domains, spaces or manifolds which admit Lie group actions. Recent interests are in Siegel domains in complex Euclidean spaces. These domains are multi-variable or matrix-variable generalizations of the usual upper half-plane in the complex plane. Current emphasis of the research is put on analytic and geometric characterizations of symmetric domains in the category of homogeneous Siegel domains. However, in the future, I plan to investigate representation-theoretic decomposition (e.g. irreducible decomposition) of various function spaces on the domain, and also to construct Fourier analysis on Siegel domains that includes well-developed Fourier analysis on symmetric spaces. Making good use of non-associative algebra structure which is geometrically introduced in the tangent space as well as of the Lie algebras of the Lie groups, I would like to weave the harmony produced by the interplay of geometry and analysis.

Research

Research Interests

Geometric harmonic analysis on homogeneous Siegel domains
keyword : Siegel domains, harmonic analysis, normal j algebra, Jordan algebra
1998.04～2018.03.

Current and Past Project

Under the idea of thorough investigation by Lie groups, we gather forefront specialists from the two research areas in Japan and in Germany, one is from Lie group representations and harmonic analysis on homogeneous spaces, and the other is from geometry of Lie groups, at Paderborn University. We cover the latest topics and also hope the new research direction in the above two research areas.
Groups and their homogeneous spaces are basic objects in science and to unravel their structure and that of certain classes of functions on them is of fundamental importance.
A classical example in that respect form Fourier series and integrals. Harmonic analysis on locally compact groups is the search for a non-commutative generalization of this theory and can be applied in other branches of mathematics such as number theory, singularity theory, probability and integrable systems.
At the conference special emphasis will be put on symmetric spaces, special functions and q-deformations and their representations.
The plan is to organize a joint seminar focused upon analysis with the keyword harmonic analysis as a core and to gather together forefront researchers in both countries within the framework of academic program under the cooperation of Ministry of Higher Education, Scientific Research and Technology in Tunisia (MHESRT) and Japan Society for the Promotion of Science (JSPS).
Lie group representation theory and harmonic analysis on Lie groups and　on their homogeneous spaces have huge amounts of substances as a　significant and indispensable area of mathematics and are closely　interrelated with various other mathematical fields.　We carry out the research in this very active area of mathematics　based on the concept of quantization proposed by Berezin.

Academic Activities

Reports

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1.	Takaaki Nomura,Focusing on symmetry characterization theorems for homogeneous Siegel domains,Vol. 23 (2010), 47-67,Sugaku Expositions,2010.06.
2.	Hideyuki Ishi, Takaaki Nomura,Spherical Fourier transforms of the Berezin kernels on symmetric Siegel domains,vol. 1487, 69--78.,数理解析研究所講究録,2006.05.
3.	Around symmetry conditions on homogeneous Siegel domains.

Papers

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1.	Takaaki Nomura, Hideto Nakashima,Clans defined by representations of Euclidean Jordan algebras and the associated basic relative invariants,Kyushu J. Math.,Vol.67,No.1,2013.03.
2.	Hideyuki Ishi, Takaaki Nomura,An Irreducible homogeneous non-selfdual cone of arbitrary rank linearly isomorphic to the dual cone,Infinite dimensional harmonic analysis,129--134,2009.01.
3.	Hideyuki Ishi, Takaaki Nomura,Irreducible homogeneous non-symmetric cones linearly isomorphic to the dual cones,Comtemporary Geometry and Topology and Related Topics,167--171,2008.11.
4.	Hideyuki Ishi and Takaaki Nomura,Tube domain and an orbit of a complex triangular group,Math. Z.,259巻、697--711,2008.06.
5.	Chifune Kai, Takaaki Nomura,A characterization of symmetric tube domains by convexity of Cayley transform images,Differential Geom. Appl.,vol. 23, 38--54,2005.01.
6.	Chifune Kai, Takaaki Nomura,A characterization of symmetric cones through pseudoinverse maps,J. Math. Soc. Japan,vol. 57, 195--215,2005.01.
7.	Takaaki Nomura,Geometric norm equality related to the harmonicity of the Poisson kernel for homogeneous Siegel domains,J. Funct. Anal.,vol. 198, 229--267,2003.01.
8.	Takaaki Nomura,Family of Cayley transforms of a homogeneous Siegel domain parametrized by admissible linear forms,Differential Geom. Appl.,vol. 18, 55--78,2003.01.
9.	Takaaki Nomura,Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains,Differential Geom. Appl.,vol. 15 , 91--106,2001.01.
10.	Takaaki Nomura,A characterization of symmetric Siegel domains through a Cayley transform,Transform. Groups,vol. 6, 227--260,2001.01.
11.	Takaaki Nomura,On Penney's Cayley transform of a homogeneous Siegel domain,J. Lie Theory,vol. 11, 185--206,2001.01.

Presentations

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1.	On Poguntke's commuting subalgebra of the distinguished Laplacian.
2.	Homogeneous convex cones, clans, and basic relative invariants.
3.	NASPA New Otani.
4.	RIMS, Kyoto University.

Educational

Educational Activities

Analysis is my principal teaching area of mathematics. Thus I teach calculus in the freshman classes. For the undergraduate courses I teach Lebesgue's integration theory, elementary functional analysis and function theory of one complex variable. In the seminar or courses for the final year undergraduate students or for graduate students, the theme is non-commutative harmonic analysis which is related to my research area. I give introductory lectures on that subject, and in the reading course, books are chosen from the area of harmonic analysis (commutative or non-commutative).

Social

Professional and Outreach Activities

Lecturer in extension courses (July, 2006)
Excellent Students in Science Program (2009--2011)

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