Organization
School of Computing Visiting Professor
Title
Visiting Professor
External link
WAKAYAMA MASATO
Degree
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Dr．Science
Research Interests
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Automorphic forms and zeta functions
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Representation theory
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Harmonic analysis
Research Areas
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Natural Science / Mathematical analysis
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Natural Science / Algebra
Professional Memberships
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American Mathematical Society
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日本数学会
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American Mathematical Society
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Mathematical Society of Japan
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Australian Mathematical Society
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THE JAPAN SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS
Committee Memberships
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Mathematical Society of Japan Councilor
1998
Papers
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Partition functions for non-commutative harmonic oscillators and related divergent series
Kazufumi Kimoto, Masato Wakayama
Indagationes Mathematicae 2024.6
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Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals
Kazufumi Kimoto, Masato Wakayama
Annales de l’Institut Henri Poincaré D 2022.12
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Remarks on the hidden symmetry of the asymmetric quantum Rabi model
Cid Reyes-Bustos, Daniel Braak, Masato Wakayama
Journal of Physics A: Mathematical and Theoretical 54 ( 28 ) 2021.7
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Heat kernel for the quantum Rabi model: II. Propagators and spectral determinants
Cid Reyes-Bustos, Masato Wakayama
Journal of Physics A: Mathematical and Theoretical 54 ( 11 ) 2021.3
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Determinant Expressions of Constraint Polynomials and the Spectrum of the Asymmetric Quantum Rabi Model Reviewed
Kazufumi Kimoto, Cid Reyes-Bustos, Masato Wakayama
International Mathematics Research Notices 2020.4
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Spectral Degeneracies in the Asymmetric Quantum Rabi Model
Cid Reyes-Bustos, Masato Wakayama
Mathematical Modelling for Next-Generation Cryptography 117 - 137 2017.7
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HERMITIAN SYMMETRIC SPACES OF TUBE TYPE AND MULTIVARIATE MEIXNER-POLLACZEK POLYNOMIALS Reviewed
Jacques Faraut, Masato Wakayama
MATHEMATICA SCANDINAVICA 120 ( 1 ) 87 - 114 2017
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Equivalence Between the Eigenvalue Problem of Non-Commutative Harmonic Oscillators and Existence of Holomorphic Solutions of Heun Differential Equations, Eigenstates Degeneration, and the Rabi Model Reviewed
Masato Wakayama
INTERNATIONAL MATHEMATICS RESEARCH NOTICES ( 3 ) 759 - 794 2016
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Wreath determinants for group-subgroup pairs Reviewed
Kei Hamamoto, Kazufumi Kimoto, Kazutoshi Tachibana, Masato Wakayama
Journal of Combinatorial Theory, Series A 133 76 - 96 2015.7
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The quantum Rabi model and Lie algebra representations of sI(2) Reviewed
Masato Wakayama, Taishi Yamasaki
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 47 ( 33 ) 2014.8
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Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun's differential equation Reviewed
Masato Wakayama
PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 89 ( 6 ) 69 - 73 2013.6
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Invariant differential operators on the Heisenberg group and Meixner-Pollaczek polynomials Reviewed
Jacques Faraut, Masato Wakayama
Advances in Pure and Applied Mathematics 4 ( 1 ) 41 - 66 2013.4
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Non-linear algebraic differential equations satisfied by a certain family of elliptic functions Reviewed
Masato Wakayama, Keitaro Yamamoto
RAMANUJAN JOURNAL 30 ( 2 ) 173 - 186 2013.2
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Milnor-Selberg zeta functions and zeta regularizations Reviewed
Nobushige Kurokawa, Masato Wakayama, Yoshinori Yamasaki
J. Geom. Phys. 64 ( 1 ) 120 - 145 2013
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Spectrum of non-commutative harmonic oscillators and residual modular forms Invited Reviewed
Kazufumi Kimoto, Masato Wakayama
Noncommutative geometry and physics 3 237 - 267 2013
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Quantum α-determinants and q-deformations of hypergeometric polynomials Invited Reviewed
Kazufumi Kimoto, Masato Wakayama
RIMS Kokyuroku Bessatsu B36 97 - 111 2012
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Multiple finite Riemann zeta functions Reviewed
K Kimoto, N Kurokawa, S Matsumoto, M Wakayama
ACTA ARITHMETICA 116 ( 2 ) 173 - 187 2005
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Erratum: Non-commutative harmonic oscillators-I (Forum Mathematicun (2002) 14:4 (539-604))
Alberto Parmeggiani, Masato Wakayama
Forum Mathematicum 15 ( 6 ) 955 - 963 2003
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Zeta extensions Reviewed
N Kurokawa, M Wakayama
PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 78 ( 7 ) 126 - 130 2002.9
Books
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多変数超幾何函数 - ゲルファント講義1989 -（共著）
若山 正人
2016.6
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Mathematics for Modern Industrial Technology (6 volumes), Editor
Kodansha 2008
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現代技術への数学入門（6巻）（編著）
講談社 2008
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技術に生きる現代数学(編著)
岩波書店 2008
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Contemporary Mathematics in Technology, Editor,（Coauthers: Y. Ninomiya, S. Taniguchi, M. Kaneko, Y. Fukumoto, O. Saeki）
Iwanami Publishers 2008
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オイラー入門, William Dunham 著(共訳)
シュプリンガ－東京 2004
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Absolute Casimir Elements,（Coauther: S. Kurokawa）
Iwanami Publishers 2002
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数学の最先端 21世紀への挑戦 第２巻, どこでも熱核, J. Jorgensen and S. Lang著（分担訳）
シュプリンガ－東京 2002
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絶対カシミール元（共著）
岩波書店 2002
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Zeta's World,（Coauthers: T. Umeda, S. Kurokawa, S. Nakajima)
Nippon Hyoronsha Publishers 1999
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ゼータの世界(共著)
日本評論社 1999
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エンカルタ電子大百科辞典, 数学部門(執筆・監修)
マイクロソフト社
MISC
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Symmetry of asymmetric quantum Rabi models Reviewed
Masato Wakayama
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 50 ( 17 ) 2017.4
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Wakayama Masato
RIMS Kokyuroku 1977 132 - 143 2015.12
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Wakayama Masato, Inoue Hiroto
RIMS Kokyuroku 1961 74 - 80 2015.8
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Wakayama Masato, Inoue Hiroto
RIMS Kokyuroku 1925 26 - 30 2014.11
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WAKAYAMA Masato, INOUE Hiroto
RIMS Kokyuroku 1825 197 - 204 2013.2
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Higher regularizations of zeros of cuspidal automorphic L-functions of GL(d)
Masato Wakayama, Yoshinori Yamasaki
JOURNAL DE THEORIE DES NOMBRES DE BORDEAUX 23 ( 3 ) 751 - 767 2011
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Alpha-determinant cyclic modules and jacobi polynomials
Kazufumi Kimoto, Sho Matsumoto, Masato Wakayama, Kazufumi Kimoto
Transactions of the American Mathematical Society 361 ( 12 ) 6447 - 6473 2009.12
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Hermitian Symmetric Spaces of Tube Type and Multivariate Meixner-Pollaczek Polynomials
Jacques Faraut, Masato Wakayama
2008.12
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RUELLE TYPE L-FUNCTIONS VERSUS DETERMINANTS OF LAPLACIANS FOR TORSION FREE ABELIAN GROUPS
Nobushige Kurokawa, Masato Wakayama, Yoshinori Yamasaki
INTERNATIONAL JOURNAL OF MATHEMATICS 19 ( 8 ) 957 - 979 2008.9
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Period deformations and Raabe's formulas for generalized gamma and sine functions
Nobushige Kurokawa, Masato Wakayama
KYUSHU JOURNAL OF MATHEMATICS 62 ( 1 ) 171 - 187 2008.3
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Invariant theory for singular alpha-determinants
Kazufumi Kimoto, Masato Wakayama
JOURNAL OF COMBINATORIAL THEORY SERIES A 115 ( 1 ) 1 - 31 2008.1
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Invariant theory for singular $\alpha$-determinants, (with Kazufumi Kimoto)
Journal of Combinatorial Theory, Series A 115 1 - 31 2008
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q-analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function
Kenichi Kawagoe, Masato Wakayama, Yoshinori Yamasaki
FORUM MATHEMATICUM 20 ( 1 ) 1 - 26 2008
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Invariant theory for singular $\alpha$-determinants, (with Kazufumi Kimoto)
Journal of Combinatorial Theory, Series A 115 1 - 31 2008
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$q$-Analogues of the Riemann zeta, the Dirichlet $L$-functions and a crystal zeta function, (with Kenichi Kawagoe and Yoshinori Yamasaki)
Forum Mathematicum 20 1 - 26 2008
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Splitting density for lifting about discrete groups
Yasufumi Hashimoto, Masato Wakayama
TOHOKU MATHEMATICAL JOURNAL 59 ( 4 ) 527 - 545 2007.12
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Splitting density for lifting about discrete groups
Yasufumi Hashimoto, Masato Wakayama
Tohoku Mathematical Journal 59 ( 4 ) 527 - 545 2007.12
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Jackson-Mellin's transform of modular forms and q-zeta functions
Shai Haran, Nobushige Kurokawa, Masato Wakayama
KYUSHU JOURNAL OF MATHEMATICS 61 ( 2 ) 551 - 563 2007.9
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Quantum alpha-determinant cyclic modules of U-q (gl(n))
Kazufumi Kimoto, Masato Wakayama
JOURNAL OF ALGEBRA 313 ( 2 ) 922 - 956 2007.7
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On The Spectral Zeta Function For The Noncommutative Harmonic Oscillator
Takashi Ichinose, Masato Wakayama
Reports on Mathematical Physics 59 ( 3 ) 421 - 432 2007.6
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On the spectral zeta function for the noncommutative harmonic oscillator
Takashi Ichinose, Masato Wakayama
REPORTS ON MATHEMATICAL PHYSICS 59 ( 3 ) 421 - 432 2007.6
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Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators
Kazufumi Kimoto, Masato Wakayama
Kyushu Journal of Mathematics 60 ( 2 ) 383 - 404 2007.5
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Jackson's integral of the Hurwitz zeta function
Nobushige Kurokawa, Katsuhisa Mimachi, Masato Wakayama
Rendiconti del Circolo Matematico di Palermo 56 ( 1 ) 43 - 56 2007.2
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Quantum $\alpha$-determinant cyclic modules of ${\cal U}_q(\frak{gl}_n)$, (with Kazufumi Kimoto)
Journal of Algebra 313 922--956 2007
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Splitting density for lifting about discrete groups, (with Yasufumi Hashimoto)
Tohoku Mathematical Journal 59 527 - 545 2007
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Elliptic curves arising from the spectral zeta function for non-commutative harmonic oscillators and Gamma(0)(4)-modular forms
K. Kimoto, M. Wakayama
CONFERENCE ON L-FUNCTIONS 201 - + 2007
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Elliptic curves arising from the spectral zeta functions for non-commutative harmonic oscillators and $\Gamma_0(4)$-modular forms, (with Kazufumi Kimoto)
Proceedings of the conference on $L$-functions, World Scientific 201 - 218 2007
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Integral representations of q-analogues of the Hurwitz zeta function
Masato Wakayama, Yoshinori Yamasaki
MONATSHEFTE FUR MATHEMATIK 149 ( 2 ) 141 - 154 2006.10
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Apery-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators
Kazufumi Kimoto, Masato Wakayama
KYUSHU JOURNAL OF MATHEMATICS 60 ( 2 ) 383 - 404 2006.9
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Algebraicity and transcendency of basic special values of Shintani's double sine functions
N Kurokawa, M Wakayama
PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY 49 361 - 366 2006.6
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若山 正人
数学通信 11 ( 1 ) 174 - 176 2006.5
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非可換調和振動子のスペクトルゼータ関数(スペクトル・散乱理論とその周辺)
若山 正人
数理解析研究所講究録 1479 26 - 39 2006.4
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Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities
Masao Ishikawa, Masato Wakayama
Journal of Combinatorial Theory. Series A 113 ( 1 ) 113 - 155 2006.1
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Milnor's multiple gamma functions,(with Nobushige Kurokawa and Hiroyuki Ochiai)
Journal of the Ramanujan Mathematical Society 21 153--167 2006
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Integral representations of $q$-analogues of the Hurwitz zeta function, Monatshefte f\"ur Mathematik,(with Yoshinori Yamasaki)
Monatshefte f\"ur Mathematik 149 141 - 154 2006
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Applications of minor summation formulas III, Pl\"ucker relations, Lattice paths and pfaffinas, (with Masao Ishikawa)
Journal of Combinatorial Theory, Series A 113 113--155 2006
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Milnor's multiple gamma functions, (with Nobushige Kurokawa and Hiroyuki Ochiai)
Journal of the Ramanujan Mathematical Society 21 2006 2006
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Alpha-determinant cyclic modules of $\frak{gl}_n(\Bbb C)$, (with Sho Matsumoto)
Journal of Lie Theory 16 393 - 405 2006
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Alpha-determinant cyclic modules of gl(n)(C)
S Matsumoto, M Wakayama
JOURNAL OF LIE THEORY 16 ( 2 ) 393 - 405 2006
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Alpha-determinant cyclic modules of gl(n)(C)
S Matsumoto, M Wakayama
JOURNAL OF LIE THEORY 16 ( 2 ) 393 - 405 2006
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Generalized zeta regularizations, quantum class number formulas, and Appell's O-functions
N Kurokawa, M Wakayama
RAMANUJAN JOURNAL 10 ( 3 ) 291 - 303 2005.12
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Gamma and sine functions for Lie groups and period integrals
N Kurokawa, M Wakayama
INDAGATIONES MATHEMATICAE-NEW SERIES 16 ( 3-4 ) 585 - 607 2005.12
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Regularizations and finite ladders in multiple trigonometry
N Kurokawa, M Wakayama
JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 57 ( 4 ) 1197 - 1216 2005.10
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Certain families of elliptic functions defined by q-series
N Kurokawa, M Wakayama
RAMANUJAN JOURNAL 10 ( 1 ) 23 - 41 2005.8
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Hierarchy of the Selberg Zeta functions
Y Hashimoto, M Wakayama
LETTERS IN MATHEMATICAL PHYSICS 73 ( 1 ) 59 - 70 2005.7
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Hierarchy of the Selberg zeta functions
Yasufumi Hashimoto, Masato Wakayama
Letters in Mathematical Physics 73 ( 1 ) 59 - 70 2005.7
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Integral representations of q-analogues of the Hurwitz zeta function
Masato Wakayama, Yoshinori Yamasaki
Monatsh. Math., 149 (2006) No. 2, 141-154 2005.3
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Extensions of zeta functions - Examples and a study of the double sine functions
N Kurokawa, M Wakayama
ACTA APPLICANDAE MATHEMATICAE 86 ( 1-2 ) 179 - 201 2005.3
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Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations
T Ichinose, M Wakayama
KYUSHU JOURNAL OF MATHEMATICS 59 ( 1 ) 39 - 100 2005.3
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Hierarchy of the Selberg zeta functions,(with Yasufumi Hashimoto)
Letters in Mathematical Physics 73 59--70 2005
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Certain families of elliptic functions defined by $q$-series, (with Nobushige Kurokawa)
The Ramanujan Journal 10 23--41 2005
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Remarks on Shintani's zeta function
Journal of the Mathematical Sciences, University of Tokyo 12 289--317 2005
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Multiple finite Riemann zeta functions, (with Kazufumi Kimoto, Nobushige Kurokawa and Sho Matsumoto)
Acta Arithmetica 116 173--187 2005
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Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent heun equations
Takashi Ichinose, Masato Wakayama
Kyushu Journal of Mathematics 59 ( 1 ) 39 - 100 2005
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Zeta functions for the spectrum of the non-commutative harmonic oscillators, (with Takashi Ichinose)
Communications in Mathematical Physics 258 697--739 2005
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A q-logarithmic analogue of Euler's sine integral (*)
N Kurokawa, M Wakayama
RENDICONTI DEL SEMINARIO MATEMATICO DELLA UNIVERSITA DI PADOVA 114 51 - 62 2005
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Gamma and sine functions for Lie groups and period integrals, (with Nobushige Kurokawa)
Indagationes Mathematicae N.S. 16 585 - 607 2005
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Differential algebraicity of multiple sine functions,(with Nobushige Kurokawa)
Letters in Mathematical Physics 71 75--82 2005
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Differential algebraicity of multiple sine functions,(with Nobushige Kurokawa)
Letters in Mathematical Physics 71 75--82 2005
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Remarks on Shintani's Zeta Function
Wakayama Masato
Journal of mathematical sciences, the University of Tokyo 12 ( 2 ) 289--317 - 317 2005
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Multiple finite Riemann zeta functions,(with Kazufumi Kimoto, Nobushige Kurokawa and Sho Matsumoto)
Acta Arithmetica 116 173--187 2005
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Zeta functions for the spectrum of the non-commutative harmonic oscillators,(with Takashi Ichinose)
Communications in Mathematical Physics 258 697--739 2005
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Euler's constants for the Selberg and the Dedekind zeta functions
Y Hashimoto, Y Iijima, N Kurokawa, M Wakayama
BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN 11 ( 4 ) 493 - 516 2004.10
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Zeta functions of q-perfect numbers
Nobushige Kurokawa, Masato Wakayama
Rendiconti del Circolo Matematico di Palermo 53 ( 3 ) 381 - 389 2004.10
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Ruelle type zeta functions for tori and some arithmetics
N Kurokawa, M Wakayama
INTERNATIONAL JOURNAL OF MATHEMATICS 15 ( 7 ) 691 - 715 2004.9
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Higher Selberg zeta functions
N Kurokawa, M Wakayama
COMMUNICATIONS IN MATHEMATICAL PHYSICS 247 ( 2 ) 447 - 466 2004.5
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A note on spectral zeta functions of quantum groups
N Kurokawa, M Wakayama
INTERNATIONAL JOURNAL OF MATHEMATICS 15 ( 2 ) 125 - 133 2004.3
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Zeta regularizations
N Kurokawa, M Wakayama
ACTA APPLICANDAE MATHEMATICAE 81 ( 1-3 ) 147 - 166 2004.3
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q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function
Kenichi Kawagoe, Masato Wakayama, Yoshinori Yamasaki
Forum Math., 20 (2008) No. 1, 1-26 2004.2
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A generalization of Lerch's formula, (with Nobushige Kurokawa)
Czechoslovak Mathematical Journal 54 949--954 2004
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A generalization of Lerch's formula, (with Nobushige Kurokawa)
Czechoslovak Mathematical Journal 54 949--954 2004
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Extremal values of double and triple trigonometric functions
Nobushige Kurokawa, Masato Wakayama
Kyushu Journal of Mathematics 58 ( 1 ) 141 - 166 2004
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Remarks on zeta regularized products, (with Kazufumi Kimoto)
International Mathematics Research Notices 2004:17 855--875 2004
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Some Examples of Generalized Zeta Regularized Products
Kazufumi Kimoto, Nobushige Kurokawa, Chie Sonoki, Masato Wakayama
Kodai Mathematical Journal 27 ( 3 ) 321 - 335 2004
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On q-analogues of the Euler constant and Lerch's limit formula
N Kurokawa, M Wakayama
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 132 ( 4 ) 935 - 943 2004
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Absolute tensor products, (with Nobushige Kurokawa)
International Mathematics Research Notices 2004:5 249--260 2004
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Zeta regularized product expressions for multiple trigonometric functions
Nobushige Kurokawa, Masato Wakayama
Tokyo Journal of Mathematics 27 ( 2 ) 469 - 480 2004
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Zeta regularizations, (with Nobushige Kurokawa)
Acta Applicandae Mathematicae 81 147--166 2004
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Ruelle type zeta functions for tori and arithmetics, (with Nobushige Kurokawa)
International Journal of Mathematics 15 691--715 2004
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Absolute tensor products
N Kurokawa, M Wakayama
INTERNATIONAL MATHEMATICS RESEARCH NOTICES 2004:5 ( 5 ) 249 - 260 2004
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Remarks on zeta regularized products
K Kimoto, M Wakayama
INTERNATIONAL MATHEMATICS RESEARCH NOTICES 2004:17 ( 17 ) 855 - 875 2004
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Masao Ishikawa, Masato Wakayama
J. Combin. Theory Ser. A. 113, (2006) 113-155 2003.12
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Duplication formulas in triple trigonometry
N Kurokawa, M Wakayama
PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 79 ( 8 ) 123 - 127 2003.10
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Analyticity of polylogarithmic euler products
Nobushige Kurokawa, Masato Wakayama
Rendiconti del Circolo Matematico di Palermo 52 ( 3 ) 382 - 388 2003.10
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On q-basic multiple gamma functions
N Kurokawa, M Wakayama
INTERNATIONAL JOURNAL OF MATHEMATICS 14 ( 8 ) 885 - 902 2003.10
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Gamma factors and functional equations of higher Riemann zeta functions
KUROKAWA N.
Kyushu University Preprint Series in Math. 2003-10 2003
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Zeta regularizations and q-analogue of ring sine functions
Kazufumi Kimoto, Chie Sonoki, Nobushige Kurokawa, Masato Wakayama
Kyushu Journal of Mathematics 57 ( 1 ) 197 - 215 2003
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Absolute derivations and zeta functions, (with Nobushige Kurokawa and Hiroyuki Ochiai)
Documenta Mathematica (Extra Volume Kato) 563--582 2003
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A variation of Euler's approach to values of the Riemann zeta function, (with Masanobu Kaneko and Nobushige Kurokawa)
Kyushu Journal of Mathematics 57 ( 175--192 ) 2003
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A comparison between the sum over Selberg's zeroes and Riemann's zeroes, Journal of the Ramanujan Mathematical Society, 18, (2003), 221-236 (Corrections, ibid. 18, (2003), 415-416), (with Nobushige Kurokawa)
Journal of the Ramanujan Mathematical Society 18 221 - 236 2003
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Gamma factors and functional equations of higher Riemann zeta functions, (with Nobushige Kurokawa, Sachiko Matsuda)
Kyushu University Preprint Series in Math 2003-10 2003
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Half zeta functions, (with Miki Hirano, Nobushige Kurokawa)
Journal of the Ramanujan Mathematical Society 18 ( 195--209 ) 2003
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Cauchy-schwarz type inequalities for categories
Nobushige Kurokawa, Masato Wakayama
Kyushu Journal of Mathematics 57 ( 2 ) 325 - 331 2003
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Half zeta functions, (with Miki Hirano, Nobushige Kurokawa)
Journal of the Ramanujan Mathematical Society 18 195--209 2003
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Absolute derivations and zeta functions, (with Nobushige Kurokawa and Hiroyuki Ochiai)
Documenta Mathematica Extra Volume Kato 565--584 2003
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A variation of euler's approach to values of the riemann zeta function
M. Kaneko, Nobushige Kurokawa, Masato Wakayama
Kyushu Journal of Mathematics 57 ( 1 ) 175 - 192 2003
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A comparison between the sum over Selberg's zeroes and Riemann's zeroes, Journal of the Ramanujan Mathematical Society, 18, (2003), 221-236 (Corrections, ibid. 18, (2003), 415-416), (with Nobushige Kurokawa)
Journal of the Ramanujan Mathematical Society 18 221 - 236 2003
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A variation of euler's approach to values of the riemann zeta function
M. Kaneko, Nobushige Kurokawa, Masato Wakayama
Kyushu Journal of Mathematics 57 ( 1 ) 175 - 192 2003
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Explicit Capelli identities for skew symmetric matrices
K Kinoshita, M Wakayama
PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY 45 449 - 465 2002.6
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Casimir effects on Riemann surfaces
N Kurokawa, M Wakayama
INDAGATIONES MATHEMATICAE-NEW SERIES 13 ( 1 ) 63 - 75 2002.3
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Non-commutative harmonic oscillators-I
A Parmeggiani, M Wakayama
FORUM MATHEMATICUM 14 ( 4 ) 539 - 604 2002
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Multiple trigonometry and zeta functions, (with Nobushige Kurokawa and Hiroyuki Ochiai)
Journal of the Ramanujan Mathematical Society 17 101--113 2002
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Zeta extensions
Nobushige Kurokawa, Masato Wakayama
Proceedings of the Japan Academy Series A: Mathematical Sciences 78 ( 7 ) 126 - 130 2002
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Corrigenda and remarks to ``Non-commutative harmonic oscillators I, Forum Mathematicum, 14, (2002), 539--604 " ibid. 15, (2003), 955--963 (with Alberto Parmeggiani)
Forum Mathematicum 14 539--604 2002
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Kronecker's Jugendtraum and ring sine functions, (with Nobushige Kurokawa, Eva-Marie M\"uller-St\"uler and Hiroyuki Ochiai)
Journal of the Ramanujan Mathematical Society 17 211 - 220 2002
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Non-commutative harmonic oscillators-II
A Parmeggiani, M Wakayama
FORUM MATHEMATICUM 14 ( 5 ) 669 - 690 2002
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Zeta extensions, (with Nobushige Kurokawa)
Proceedings of the Japan Academy 78A ( 126--130 ) 2002
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A remark on systems of differential equations associated with representations of si2(r) and their perturbations
Alberto Parmeggiani, Masato Wakayama
Kodai Mathematical Journal 25 ( 3 ) 254 - 277 2002
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Casimir effects on Riemann surfaces
Nobushige Kurokawa, Masato Wakayama
Indagationes Mathematicae 13 ( 1 ) 63 - 75 2002
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Verified numerical computations of eigenvalue problems for non-commutative harmonic oscillators,(with Kaori Nagatou, Mitsuhiro T.Nakao)
Numerical Functional Analysis and Optimization 23 ( 633--650 ) 2002
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Equidistribution of holonomy restricted to a homology class about closed geodesics
Kazufumi Kimoto, Masato Wakayama
Forum Mathematicum 14 ( 3 ) 383 - 403 2002
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Corrigenda and remarks to ``Non-commutative harmonic oscillators I, Forum Mathematicum, 14, (2002), 539--604 " ibid. 15, (2003), 955--963 (with Alberto Parmeggiani).
Forum Mathematicum 14 539--604 2002
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Kronecker's Jugendtraum and ring sine functions, (with Nobushige Kurokawa, Eva-Marie M\"uller-St\"uler and Hiroyuki Ochiai)
Journal of the Ramanujan Mathematical Society 17 ( 211-220 ) 2002
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Multiple trigonometry and zeta functions, (with Nobushige Kurokawa and Hiroyuki Ochiai)
Journal of the Ramanujan Mathematical Society 17 ( 101--113 ) 2002
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Verified numerical computations of eigenvalue problems for non-commutative harmonic oscillators, (with Kaori Nagatou, Mitsuhiro T. Nakao)
Numerical Functional Analysis and Optimization 23 633--650 2002
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Equidistribution of holonomy restricted to a homology class about closed geodesics
K Kimoto, M Wakayama
FORUM MATHEMATICUM 14 ( 3 ) 383 - 403 2002
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Oscillator representations and systems of ordinary differential equations
A Parmeggiani, M Wakayama
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA 98 ( 1 ) 26 - 30 2001.1
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On ζ(3), (with Nobushige Kurokawa)
J. Ramanujan Math. Soc. 16, 205--214 2001
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On $\zeta(3)$, (with Nobushige Kurokawa)
Journal of the Ramanujan Mathematical Society 16 205--214 2001
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Roe Goodman & Nolan R.Wallach:Representations and Invariants of the Classical Groups
52 ( 4 ) 435 - 439 2000.10
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A formula for the multiplicity of the principal series in L-2(Gamma backslash G)
N Kurokawa, H Kuroyama, M Wakayama
FORUM MATHEMATICUM 12 ( 6 ) 757 - 766 2000
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Minor summation formulas of pfaffians, survey and a new identity
M Ishikawa, M Wakayama
COMBINATORIAL METHODS IN REPRESENTATION THEORY 28 133 - 142 2000
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A formula for the multiplicity of the principal series in L2(HG), (with Nobushige Kurokawa and Hitodakari Kuroyama)
Forum Math. 12 ( 757-766 ) 2000
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Minor summation formulas of Pfattians, Survey and a new identity, (with Masao Ishikawa)
Advanad Studies in Pure Mathematics 28 ( 133-142 ) 2000
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Applications of minor-summation formula II. Pfaffians and Schur polynomials
M Ishikawa, M Wakayama
JOURNAL OF COMBINATORIAL THEORY SERIES A 88 ( 1 ) 136 - 157 1999.10
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Equidistribution of holonomy about closed geodesics
P Sarnak, M Wakayama
DUKE MATHEMATICAL JOURNAL 100 ( 1 ) 1 - 57 1999.10
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Applications of minor summation formulas II, Pfaffians and Schur Polynomials, (with Masao Ishikawa)
Journal of Combinatorial Theory, Series A 88 136--157 1999
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New Schur function series
M Ishikawa, M Wakayama
JOURNAL OF ALGEBRA 208 ( 2 ) 480 - 525 1998.10
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An inequality among infinitesimal characters related to the lowest K-types of discrete series
M Wakayama
PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 74 ( 4 ) 57 - 60 1998.4
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An inequality among infinitesimal characters related to the lowest K-types of discrete series
Masato Wakayama
Proceedings of the Japan Academy Series A: Mathematical Sciences 74 ( 4 ) 57 - 60 1998
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New Schur function series, (with Masao Ishikawa)
Journal of Alg. 208 480 - 525 1998
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Another look at the differential aperators on the quantum matrix spaces and its applications, (with T\^oru Umeda)
Commnt. Math. Univ. Sancti Pauli 47 53 - 80 1998
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Another Look at the Differential Operators on Quantum Matrix Spaces and its Applications
Umeda Toru, Wakayama Masato
Commentarii mathematici Universitatis Sancti Pauli 47 ( 1 ) 53--80 - 80 1998
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Dual pairs, spherical harmonics and a Capelli identity in quantum group
M Noumi, T Umeda, M Wakayama
COMPOSITIO MATHEMATICA 104 ( 3 ) 227 - 277 1996.12
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Applications of minor-summation formula .1. Littlewood's formulas
M Ishikawa, S Okada, M Wakayama
JOURNAL OF ALGEBRA 183 ( 1 ) 193 - 216 1996.7
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Applications of minor summation formulas I, Littlewood's formulas,(with Masao Ishikawa and Soichi Okada)
Journal of Algebra 183 193--216 1996
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Minor summation formula and applications, discrete Fourier transforms
Masao Ishikawa, Masato Wakayama
RIMS Kôkyûroku 933 85 - 104 1995.12
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A Q-ANALOG OF CAPELLIS IDENTITY FOR GL(2)
T HIBI, M WAKAYAMA
ADVANCES IN MATHEMATICS 115 ( 1 ) 49 - 53 1995.9
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A quantum analogue of the Capelli identity and an elementary differential calculus on GLq (n), (with Masatoshi Noumi and T\^oru Umeda)
Duke Math. J. 76 567 - 594 1995
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A quantum dual pair $(\frak{sl}_2, \frak{o}_n)$ and the associated Capelli Identity,(with Masatoshi Noumi and T\^oru Umeda)
Letters in Mathematical Physics 34/1--8 1995
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A q-anologue of Capelli's idenfity for GL(2),(with Takayuki Hibi)
Advances in Mathematics 115 ( 49--53 ) 1995
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Minor summation formula of Pfaffians (with Masao Ishikawa)
Linear and Multilinear Algebra 39 285--305 1995
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A quantum dual pair $(\frak{sl}_2, \frak{o}_n)$ and the associated Capelli Identity, (with Masatoshi Noumi and T\^oru Umeda)
Letters in Mathematical Physics 34 1--8 1995
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Minor summation formula of Pfattians and Schur functions' identities, (with Masao Ishikawa)
Proc. Japan Acad. 71 54 - 57 1995
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Minor summation formula of Pfattians, (with Masao Ishikawa)
Linear and Multilinear Alg. 39 ( 285--305 ) 1995
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Minor summation formula of Pfaffians and Schur functions' identities, (with Masao Ishikawa)
Proceedings of the Japan Academy 71A 54--57 1995
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A QUANTUM ANALOG OF THE CAPELLI IDENTITY AND AN ELEMENTARY DIFFERENTIAL-CALCULUS ON GL(Q)(N)
M NOUMI, T UMEDA, M WAKAYAMA
DUKE MATHEMATICAL JOURNAL 76 ( 2 ) 567 - 594 1994.11
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On Radon transforms for Minkowski spaces. (with Keisaku Kumahara)
J. Fac. Gen. Edu. Tottori Univ. 27 139 - 157 1993
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Powers of 2×2 quantum matrices, (with T\^oru Umeda)
Communications in Algebra 21 21 4461 - 4465 1993
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On Radon transforms for Minkowski spaces, (with Keisaku Kumahara)
Journal of the Faculty of General Educations, Tottori Univiversity 27 139--157 1993
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POWERS OF 2X2 QUANTUM MATRICES
T UMEDA, M WAKAYAMA
COMMUNICATIONS IN ALGEBRA 21 ( 12 ) 4461 - 4465 1993
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The relation between the (]G0648[)-invariant and the spin representation in terms of the Selberg zeta function
Advanced Studies in Pure Mathematics 21 71--86 1992
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量子群上の微分学とその応用
若山正人
『第37回代数学シンポジウム報告集』 154 - 171 1992
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The relation between the $\eta$-invariant and the spin representation in terms of the Selberg zeta function
Advanced Studies in Pure Mathematics 21 71--86 1992
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A note on the Selberg zeta function for compact quotients of hyperbolic spaces
Hiroshima Mathematical Journal 21 539--555 1991
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A note on the Selberg zeta function for compact quotients of hyperbolic spaces
Hiroshima Mathematical Journal 21 539--555 1991
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Paley-wiener theorems on a symmetric space and their application
Toshio Oshima, Yutaka Saburi, Masato Wakayama
Differential Geometry and its Applications 1 ( 3 ) 247 - 278 1991
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量子群上の"定数閨秀微分作用素"
若山正人
『第30回実函数論 第29回函数解析学合同シンポジウム講演集録』 113 - 125 1991
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Review of "Remarks on Classical Invarinat Theory" by Roger Howe
Math. Rev. 90 4463 - 4463 1990
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A FORMULA FOR THE LOGARITHMIC DERIVATIVE OF SELBERG ZETA FUNCTION
M WAKAYAMA
JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 41 ( 3 ) 463 - 471 1989.7
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A formula for the logarithmic derivative of Selberg's zeta function
Masato Wakayama
Journal of the Mathematical Society of Japan 41 ( 3 ) 463 - 471 1989
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A note on Ehrenpreis' fundamental principle on a symmetric space, (with Toshio Oshima and Yutaka Saburi)
Algebraic Analysis II(Academic Press) 681--697 1988
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A note on Ehrenpreis' fundamental principle on a symmetric space, Algebraic Analysis, (with Toshio Oshima and Yutaka Saburi)
Algebraic Analysis Volume II(Academic Press) 681--697 1988
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Laplacian の spectre と Selberg zeta 関数(調和解析と数論)
若山 正人
数理解析研究所講究録 631 138 - 149 1987.10
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An elementary proof of the Trombi theorem for the Fourier Transform of (G : F), (with Masaaki Eguchi)
Hiroshima Mathematical Journal 17 471--487 1987
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An elementary proof of the Trombi theorem for the Fourier transform of $C^p(G:F)$, (with Masaaki Eguchi)
Hiroshima Mathematical Journal 17 471--487 1987
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Zeta functions of Selberg's type associated with homogeneous vector bundles
Hiroshima Mathematical Journal 15 235--295 1985
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Zeta functions of Selberg's type associated with homogeneous vector bundles
D.~Sci. Thesis, Hiroshima Universit
Hiroshima Mathematical Journal 15 235--295 1985
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Zeta functions of selberg's type for compact quotient of SU(n, 1)(n〉2)
Hiroshima Mathematical Journal 14 597--618 1984
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Zeta functions of Selberg's type for compact quotient of $SU(n, 1)\, (n\geq2)$
Hiroshima Mathematical Journal 14 597--618 1984
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SU(n,1) (n$\ge$2)のcompact商空間に於るSelberg型zeta関数について(群の表現と非可換調和解析)
若山 正人
数理解析研究所講究録 481 44 - 63 1983.2
Awards
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ナイスステップな研究者2007（文部科学省）
2007
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Nice-Step-Researcher Award 2007（by MEXT） * MEXT: Ministry of Education, Culture, Sports, Science and Technology
2007
Research Projects
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Quantum Interaction and number, representation theory, discrete dynamics
Grant number：20K03560 2020.4 - 2023.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
Grant amount：\4290000 （ Direct Cost: \3300000 、 Indirect Cost：\990000 ）
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量子相互作用の数学解析を通した表現論・数論の展開と深化
2016 - 2018
日本学術振興会 学術研究助成基金助成金(基盤研究(C))
若山 正人
Authorship：Principal investigator Grant type：Competitive
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Grant number：25610006 2013 - 2015
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research
Wakayama Masato
Authorship：Principal investigator Grant type：Competitive
Grant amount：\3770000 （ Direct Cost: \2900000 、 Indirect Cost：\870000 ）
The first theme of the project was to understand, by employing a suitable group action, the mysterious derivation/determination of geodesics on multivariate normal models given by Erisken in 1987. This was completely achieved by Hiroto Inoue, a PhD course student under my supervision. Actually, he proved that the normal model can be realized as a submanifold of a certain (pre-conjectured) Riemannian symmetric space (cone) by Riemann submersion and the mysterious derivation of geodesic follows from this fact. This result may provide a new understanding for the elliptical models including normal models. The second is expanding the representation theoretic study of α-determinant, which was originally defined in the framework of statistical and probability theory, we explored the group-subgroup determinant theory. Moreover, as the third theme, we obtain the fundamental results of multivariate Meixner-Pollaczek polynomials in the framework of harmonic analysis on Hermite symmetric spaces.
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Mathematics for Non-commutative Harmonic Oscillators and Representation Theory of alpha-determinants
Grant number：21340011 2009 - 2013
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
WAKAYAMA Masato
Authorship：Principal investigator Grant type：Competitive
Grant amount：\13130000 （ Direct Cost: \10100000 、 Indirect Cost：\3030000 ）
1.We described the special value at s=4 of the spectral zeta function of the non-commutative harmonic oscillator (NcHO) by introducing the notion of residual modular forms, which is a generalization of Eichler integrals. With this investigation, we introduced the periodic Eichler cohomology groups and determined the structure of certain congruence subgroups of SL_2(Z).
2.We showed a complete description of the eigenvalue problem for NcHO, which is defined by a matrix-valued self-adjoint ordinary differential operator, in terms of Heun's ordinary differential equations (Only the odd case was given by Ochiai in 2001). As a by-product, using the monodromy representation of Heun's ODEs, we proved that the multiplicity of the eigenvalue of NcHO is at most two.
3.We described a connection between the quantum Rabi model and NcHO, employing principal series of sl2 and confluent procedure of Heun ODE. -
Integrals and special functions in representation theory
Grant number：19204011 2007 - 2010
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
OCHIAI Hiroyuki, ISHI Hideyuki, OSHIMA Toshio, KOBAYASHI Toshiyuki, WAKAYAMA Masato
Grant amount：\29770000 （ Direct Cost: \22900000 、 Indirect Cost：\6870000 ）
The subject of this research is an integral and a sum arising in representation theory. We have obtained the differential equations, functional equations among these integrals and sums, and their expressions by a known special function in several cases. This frame is also efficient to work on the spectral problems of a systems of linear differential equation, a spectral zeta function as generating functions, and a multiple zeta function as a sum. This work is a step to a post-hypergeometric systems.
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Geometric Harmonic Analysis on Homogeneous Cones and Homogeneous Siegel Domains
Grant number：18340039 2006 - 2009
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
NOMURA Takaaki, WAKAYAMA Masato, FUJIWARA Hidenori, UMEDA Toru, ITOH Minoru, ISHI Hideyuki, KAI Chifune
Grant amount：\10660000 （ Direct Cost: \8800000 、 Indirect Cost：\1860000 ）
There has been a conjecture concerning the gaps between the degrees of the basic relative invariants associated to homogeneous cones and the symmetry of the corresponding homogeneous tube domains (Siegel domains of the first kind).However, we found a homogeneous cone that denies the conjecture. Moreover such a cone is found in any rank greater than or equal to 3. We also discovered a homogeneous cone of arbitrary rank greater than or equal to 3 that is linearly isomorphic to its dual cone. Further, we proved that the basic relative invariants are exactly the irreducible factors of the determinant of the right multiplication operators in the complexification of the clan corresponding to the cone. In the case of symmetric cone, we computed an actual decomposition of that determinant into irreducible factors.
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Grant number：18654005 2006 - 2008
文部科学省 科学研究費補助金(萌芽研究) 萌芽研究
若山 正人
Authorship：Principal investigator Grant type：Competitive
Grant amount：\1800000 （ Direct Cost: \1800000 ）
α-行列式の表現論を不変式論,特殊関数の観点から展開した.α=-1のときは,α-行列式は通常の行列式に他ならず,そのGLn-巡回加群は1次元(既約)表現である.α=1のときは,α-行列式はパーマネントであり,そのGLn-巡回加群は対称テンソルが定める既約表現となる.したがって,一般のαに対して,α-行列式が定めるGLn-巡回加群は,これら2つの既約表現を補間するものである.本年度は昨年に続き,とくにα-行列式の整数べきが定めるGLn-巡回加群の研究を行った.(1)αがgenericのとき,当該加群は全テンソル代数のべき次対称テンソルのn次対称テンソル空間と同値になり可約である.従ってその既約分解の各既約表現の重複度はKostka数で与えられる.(松本詔・若山)(2)αがgenericでないときには,当該加群は退加するが,その退加の様子を記述するために,ある多項式係数の行列(遷移行列)を研究した.(n=2のとき,遷移行列は1次となり,Jacobi多項式で与えられる.さらに,その根が退化パラメータαを与える.)また,n〓3の場合の例の計算を行った.(3)木本-史とともに,べきが1のときには,一般のnに対し,退化パラメータαは最高ウェイト(ヤング図形)に対するコンテント多項式で与えられることを示した.また,この退化パラメータαに対して,(リース行列式なる概念を定義し)新しい不変式論を展開した.その他、特殊リース行列の研究を開始した.
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Zeta functions and multiple sine functions
Grant number：18540018 2006 - 2007
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
KUROKAWA Nobushige, MIMACHI Katsuhisa, MIZUMOTO Shin-ichiro, SATOH Takakazu, WAKAYAMA Masato
Grant amount：\4010000 （ Direct Cost: \3500000 、 Indirect Cost：\510000 ）
We studied zeta functions and multiple sine functions extending the theory of usual siine function. Then, we established their fundamental properties. Especially we studied special values of zeta functions via multiple sine functions. This includes the study of quantum deformations of the Catalan constant Moreover, we studied the following themes: Kinkelin formula for multiple sine functions, Casimir energies on the infinite symmetric group, Jackson integral analogue of Euler integral, generalized Mahler measures, and Jackson-Mellin transforms as quantizations of zeta functions.
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Representation Theory and Duality associated with Non-commutative Special Functions
Grant number：16340039 2004 - 2007
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
UMEDA Toru, NOUMI Masatoshi, WAKAYAMA Masato, OCHIAI Hiroyuki, MATSUZAWA Jun-ichi, ITOH Minoru
Grant amount：\8770000 （ Direct Cost: \8200000 、 Indirect Cost：\570000 ）
Theory of special functions of non-commutative variables is a framework for us to understand deeply the dual pairs, which is a sort of revival of the classical invariant theory. The aim of the research is to link each other the three theories on special functions, invariants, and representations, to throw a new light to the mathematical world under the non-commutativity. In the center of our study, we have the Capelli identities, the equalities of the invariant differential operators, which arise from the representations of the centers of the universal enveloping algebras. Around the Capelli identities, we found many interesting phenomena caused from the transition from commutative theories to non-commutativeones. And even behind the usual commutative theories, we often found the dominating non-commutative variables.
For the goal to obtain the ultimate Capelli type identities, we made a big progress through the new ideas which combine the non-commutative matrix elements, the generalization of the notion of transfer, symbolic method, and the method of generating functions. Though the program has not been accomplished, the results we had will be very useful for the understanding of the ultimate Capelli identities.
An important example is in the treatment of Euler's pentagonal number theorem, which we understand a trace identity of matrices of infinite size. In there, we discover the fact that some identities generalizing the pentagonal number theorem are indeed sort of summation formula for q-hypergeometric series. This point of view will lead us to a new link between the representation theory and invariant theory through the infinite-dimensional spaces.
Another important discovery is the algebra, which is a very useful toolfor the higher Capelli identities as the non-commutative formal variables. This is done by M. Itoh, an investigator of this research. -
Grant number：15340012 2003 - 2006
Ministry of Education, Culture, Sports, Science and Technology Grants-in-Aid for Scientific Research(基盤研究(B)) 基盤研究(B)
Masato WAKAYAMA, Nobushige KUROKAWA, Kenji KAJIWARA, Masanobu KANEKO, Toru UMEDA
Authorship：Principal investigator Grant type：Competitive
Grant amount：\13400000 （ Direct Cost: \13400000 ）
The purpose of this research project was to take a detailed study of zeta functions and duality-"infinite sum = infinite product" based on the trace formulas viewpoint. During the period we obtained the following results :Study on the spectral zeta function for the non-commutative harmonic oscillators :1)We show that the differential equation satisfied by the generating function w_2(t) of the Ap'ery like numbers arising from the evaluation of the special values at 2 of the spectral zeta function is the Picard-Fuchs equation for the universal family of elliptic curves equipped with rational 4 torsion. The parameter t of this family can be interpreted as a modular function for a certain congruent subgroup of level 4. (+. K.Kimoto)2)A higher value analogue is formulated and is shown to be related to some quisi-modular form (+. K.Kimoto).3)Some estimation for the nth eigen-values of the non-commutative harmonic oscillators are obtained (+ T.Ichinose)Study on some q-analogue of zeta functions :1)Contour integral representation of the q-analogue of the Riemann and Hurwitz zeta functions are obtained. The representation is an q-analogue of the one Riemann discovered. As an application, special values of the q-analogue of the zeta function can be easily obtained. (+ Y.Yamasaki).2)Based on the Jackson integral, some integral representation which is different from the above is obtained (+ K.Mimachi, N.Kurokawa).Milnor's type multiple gamma and sine functions are formulated in terms of the zeta regularization and their analytic properties are derived. Moreover, the special values are studied. (+ N.Kurokawa, H.Ochiai). Umeda studies some intrinsic relation between special functions and non-commutative invariants, Kaneko develops the study of quisi-modular form and special values, and Kajiwara studies the q-Painleve IV equation from symmetry.
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絶対数論の基礎研究
Grant number：15654003 2003 - 2004
日本学術振興会 科学研究費助成事業 萌芽研究
黒川 信重, 若山 正人
Grant amount：\3000000 （ Direct Cost: \3000000 ）
絶対数論の基礎研究として、もっとも基本的かつ重要と思われる多重三角関数論とその周辺の研究を行った。多重三角関数は三角関数の無限積表示を一般化して構成し、その基本性質と応用を様々に研究した。その結果、周期性、フーリエ級数展開、微分方程式等の基本的諸性質を確立した。通常の三角関数は線形の2階微分方程式を満たすが、多重版は非線形の2階微分方程式を満たす、という違いがある。また、多重三角関数の周期の等分点における値を詳しく調べ、クロネッカーの青春の夢との関連を考察した。さらに、多重三角関数の第一の応用として、リーマンゼータ関数の正の奇数における値やディリクレL関数の値など、不明だった数論的に興味深い特殊値に対し多重三角関数を用いた表示を与えた。次に、多重三角関数の第二の応用として、セルバーグゼータ関数の関数等式に現れるガンマ因子に多重ガンマ関数による簡明な表示を与た。この結果は、多重三角関数という新たな特殊関数が多様な面白い性質を持っている事を示している。また、この研究の発展として、多重三角関数のq-類似、オイラー定数のq-類似、ゼータ関数の絶対テンソル積、多重ゼータ関数論等を研究した。特に、絶対テンソル積の研究においては、多重化されたゼータ関数が多重版のオイラー積表示を持つという顕著な結果を証明した。ここにあらわれるオイラー因子は高次の数論的対象を指し示している。さらに、関連する研究として、絶対微分や圏のスペクトル、リーマン面のカシミール・エネルギー等の研究も行った。
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Grant number：13874004 2001 - 2003
文部科学省 科学研究費補助金(萌芽的研究, 萌芽研究) 萌芽的研究, 萌芽研究
若山 正人, 黒川 信重
Authorship：Principal investigator Grant type：Competitive
Grant amount：\2000000 （ Direct Cost: \2000000 ）
当該研究期間中に代表者が研究分担者と行った研究概要は以下のとおりである。量子論における零点エネルギーの典型例であるカシミール効果の、リーマン面においての定式化を行った。その上で、跡公式を用いてカシミールエネルギーを計算し、それのセルバーグゼータ関数による積分表示を得た。さらに、系として、計量に関する新しい不変量であるカシミールエネルギーの負値性を示した。このセルバーグゼータ関数による積分表示を用いると、与えられたリーマン面の型のタイヒミュラー空間の中での第一変分として、カシミール力(引力となる)が定式化できる。その後、本研究期間中に、高次セルバーグゼータ関数を定義し詳しく研究したが、これを用いると上記の第一変分が計算されることを権氏が証明している。一方で、カシミール効果を具体的に計算するときのポイントとなるゼータ正規化(無限大の繰り込み)についての研究を行った。高次のセルバーグゼータ関数や、高次のリーマンゼータ関数(これも新たに定義した)のガンマ因子の記述も正規化の研究を通して統一的に行った。また、正規化の概念を拡張することにより、量子類数公式を得たほか、与えられたゼータ関数の零点から構成するゼータ関数の絶対テンソル積の計算に、有限体の場合に成功した。素数定理をはじめとする素元の漸近分布とカシミール元との直接的なつながりを明らかにすることも目指したが、期間中にはできなかった。ただし、そのために必要と考えられる新しいタイプのオイラー積を定義しその解析的な性質を調べた。今後の研究につなげてゆきたい。
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Grant number：11440010 1999 - 2002
Ministry of Education, Culture, Sports, Science and Technology Grants-in-Aid for Scientific Research(基盤研究(B)) 基盤研究(B)
Masato WAKAYAMA, 三鳥川 寿一, 梅田 亨, 今野 拓也, 三町 勝久, Nobushige KUROKAWA, 落合 啓之, 吉田 正章, 金子 昌信, Hiroyuki OCHIAI
Authorship：Principal investigator Grant type：Competitive
Grant amount：\14200000 （ Direct Cost: \14200000 ）
The purpose of this research project was to make a detailed study of dualities -- " infinite sum=infinite product" type identities and representations from the trace formulas point of views.During the period we obtained the following results :1) The higher and ordinary analogue of the Euler constants for the Dedekind and Selberg zeta functions (+ Kurokawa, Iijima, Hashimoto)2) Zeta regularized products and the determinant expressions of several zeta functions : a) a generalization of Lerch's formula to higher degree polynomials, b) Introducing a new notion called Donburi product and weestablished a q-analogue of Lerch' s formulas. C) Calculating several sine functions for rings and also their q-analogue (Kurokawa, Ochiai, Kimoto, Muller-Stuler, Sonoki)3) We found the nice q-analogue of the Riemann zeta function and calculated the special values (+Kurokawa, Kaneko)4) We made a description of the specter of the non-commutative harmonic oscillators and study the spectral zeta function (+A.Parmeggiani, Nagatou, Nakao, Ichinose)5) We introduced and studied about the zeta extensions, especially, investigated the higher Selberg and Riemann zeta functions (+Kurokawa, Matsuda)6) We studied the absolute derivations and gave some conjecture of the determinant expression (+Kurokawa, Ochiai)7) Multiple sine functions theory was developed (+Kurokawa, Ochiai)8) We established the explicit formula of the Capelli identity for the skew symmetric matrices (+Kinoshita)9) A density theorem for the holonomy groups was established (+ Kimoto)
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Study of special functions based on representation and invariant theories
Grant number：11440043 1999 - 2001
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
UMEDA Toru, NOUMI Masatoshi, MATSUZAWA Junichi, NOMURA Takaaki, OEHIAI Hiroyuki, WAKAYAMA Masato
Grant amount：\9600000 （ Direct Cost: \9600000 ）
The main object of the research is to find the group theoretical background behind the world of special functions and to utilize the symonetious for the special functions. Among them the theory of "dual pairs" is the key to our study, which explains many phenomina from the view-point of representation theory and the theory of invariants. We have Capilli type identities, now commtative harnomic oscillatws as the typical investizations where and pains work very well as the griding principle. On the other hand, for the hyprogeinctic from Rons and Pain lene transcendents, we have claified the grop gymmetric behind them. The helps a lot for deeper investigations of these fnctions.
As for the Capelli type identities, we got many interesting formlas including permanets and Pfuffians, not only for the determinants, Furthermore we found some Capelli type identity corresponding to the "group determinant". The invariant theoretic backgroud conneits these identities to some sphenicel functions. There are sort of unification of various objects. -
Comprehensive Study of Real Analysis and Functional Analysis
Grant number：11304009 1999 - 2000
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A).
MIYACHI Akihiko, WAKAYAMA Masato, CHO Muneo, KOBAYASI Kazuo, OHYA Masanori, OKAJI Takashi
Grant amount：\26700000 （ Direct Cost: \26700000 ）
The research is extended to the following 5 areas : (1) Harmonic analysis by real variable methods ; (2) Partial differential equations by functional- analysis method ; (3) Operator algebras and function algebras ; (4) Represen- tation theory ; (5) Quantum logics and function spaces.
Our main results are the following. In the area (1) : we obtained real variable characterization for the weighted Hardy spaces on a domain and for the Herz-type Hardy spaces ; proved boundedness of several singular integral operators with nonregular kernels ; and got Hardy space estimate for the mul- tilinear operator which is defined as a sum of products of singular integrals. In (2) : we got strong unique continuation theorem for the Maxwell equation, the Dirac equation, and for the Schrodinger equations ; studied propagation of singularities for the Schrodinger equations ; used Hardy space estimate to study the Stokes operator ; proved time-global existence of solutions to the Cauchy problem for the Navier-Stokes equation with nondecaying initial data ; studied the properties of the attractor of the nonlinear dynamical systoms related to the phase-transition phenomena ; obtained several algorithms for computing the Grothendieck residues, de Rham cohomology of algebraic varieties, and D-modules. In (3) : we gave a generalization of Putnam's inequality ; got the classification of actions of factors to operator algebras ; constructed the index theory for C^* algebras ; and solved the single generator problem on the Bohr group. In (4) : we got Hardy space theory on rank 1 semisimple Lie group ; and got summation formula for Pfaffians and its applications. In (5) : we gave a solution to an NP complete problem as an application of the quantum algorithm and the chaos theory ; studied several properties of p-hyponormal operators. -
Grant number：09440022 1997 - 1998
Ministry of Education, Culture, Sports, Science and Technology Grants-in-Aid for Scientific Research(基盤研究(B)) 基盤研究(B)
Masato WAKAYAMA, 吉田 英治, Masao ISHIKAWA, Michihiko HASHIZUME, Hisaichi MIDORIKAWA, Takuya KON-NO, Minoru WAKIMOTO
Authorship：Principal investigator Grant type：Competitive
Grant amount：\7700000 （ Direct Cost: \7700000 ）
The aim of this research project was to make a detailed study of dualities and of "infinite surn = infinite product" identities from the trace formula viewpoint, During the period we obtained good results as follows :1) The head investigator Wakayama and the investigator Ishikawa obtained quite a number of generalizations of the so-called Littlewood formulas concerning the generating functions of Schur ploynomials. They also established various formulas relating the enumerative combinatorics and represenattions of the classical groups.2) The head investigator and the collaborator Umeda (Kyoto) developed the theory of the Dual Pair, especially the theory of spherical harmonics under the quantum group symmetries. Moreover, he gave an answer of the open problem concerning the explicit construction of skew Capelli elements with his student Kinoshita.3) The head investigator and the collaborator Sarnak (Princeton) proved some density theorem for the distribution of the conjugacy class of the holonomy group on negatively-curved Riemannian locally symmmetric spaces. In order to develop this result to more precised way, the investigator Midorikawa studied and obtained an explicit relation among characters of certain representations of semi simple Lie groups. Also the investigator Kon-no constructed certain important automorphic representations of the unipotent type. Related to the trace formula, the head investigator and the collaborator Kurokawa (TIT) proved the integral represenattion of the Selberg zeta function.4) The head investigator and the collaborator Parmeggiani (Bologna) introduced the new system of differntial equation called by the non-commutative harmonic oscillator and gave a complete description of the spectrum by means of continued fractions and the oscillator represenations.5) The investigator Wakimoto made a very precised study of the represenattion of the affine Lie algebra, the super Lie algebra and vertex operators by means of modular transforms.Finally, on behalf of the investigators the head investigator would like to express their special thanks to this support.
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A study on idenities of the form "infinite sum=infinite product" by an evaluation of determinants.
Grant number：08454010 1996
Ministry of Education, Culture, Sports, Science and Technology Grants-in-Aid for Scientific Research(基盤研究(B)) 基盤研究(B)
Masato WAKAYAMA, Takuya KONNO, Eiji YOSHIDA, Junko INOUE
Authorship：Principal investigator Grant type：Competitive
Grant amount：\3800000 （ Direct Cost: \3800000 ）
The purpose of the present research project is to clarify a meaninmg of several important identities of the form "infinite product=infinite sum" by an evaluation of various determinants via the ideas : 1) the minor summation formula of Pfaffians, 2) dual pairs, 3) a precise study of (Selberg) trace formulas.(1) We developed the theory of dual pairs in a context of quantum groups (Noumi-Umeda-W., Comp.Math., 104,1996) and extended it to general rank cases (Umeda-W. ; Another look at the differential operators on the quantum matrix spaces and its applications/96). Further, we gave a new proof of Littlewood's formulas for characters by enumerative combinatorics using Pfaffinas and established several identities which involve Littilewood's formulas (Ishikawa-Okada-W., J.Alg., 183,1996). We obtained also generating functions including various symmetric functions and, in particular, gave a representation theoretic interpretation of the product representation of elliptic theta (Ishikawa-W. ; Applications of minor summation formulas II,Pfaffians and Schur Polynomials/96, New Schur function series/97).(2) We studied the trace formula on negatively curved locally symmetric Riemannian spaces in an explicit way and solved a problem concerning an equidistribution property of holonomy as an application of it (Sarnak-W. ; Equidistribution of holonomy about closed geodesics/96). As a biproduct, I got a ceratin remarkable estimate concerning an infinitesimal character of the (restricted) holonomy group (An in-equality of infinitesimal characters related to the lowest K-types of discrete series/97). Parallely, the investigator Yoshida studied the Kloosterman zeta for obtaining a good estimate of a remainder term with respect to a distribution of closed geodesic on a Riemann surface (Remarks on the Kuznetov trace formula, to appaer). Also Kon-no studied a Langlands' functoriality for automorphic representations on some classical groups of rank 2 (The residual spectrum of U (2,2) & Sp (2), to appear) and the precise study of trace formula for an explicit description of cuspidal automorphic representations (The Arthur trace formula for GSp (2) I/97). To help these studies, Inoue established a realization and an irreducible decomposition of holomorhically induced representation of some solvable group of affine transformations which act simply transitively on a homogeneous Siegel domain.
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Geometric study of linear differential equations in complex variables
Grant number：06452012 1994 - 1995
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for General Scientific Research (B)
YOSHIDA Masaaki, NAKAO Mitsuhiro, KAJIWARA Jouji, CHOU Kanchi, MIMACHI Katsuhisa
Grant amount：\7200000 （ Direct Cost: \7200000 ）
Though the Gauus's hypergeometric differential equation has a fruitful relationship with the theory of automorphic forms and differential geometry, the theory of ordinary differential equations in complex variables has been restricted to analytic and local studies. The present research aims generalizations and specializations of the Gauss's theory regarding the hypergeometric differential equation as the uniformizing equation of the obifolds with three singular points on the projective line. In 1994 and 1995, we got the following results.
(1) Equivalence problem for projective submanifolds. Let D be a symmetric space equivariantly embedded in the minimal dimensional projective space P^N. We consider the problem : For a submanifold V * P^N, state a differential geometrical condition for V so that it is locally projectively equivalent to D.This problem is crucial for studing uniformizing equations. When D is of type VI (i.e.a hyperquadric), the problem is solved.
(2) Geometry of configuration spaces. Let X (k, n) be the configuration space of n points in general postion in P^<k-1>. This space is one of the main objects in algebraic geometry in the beginning of this century. Since this space turned out to be the natural domain of definition of the hypergeometric differential equation of type (k, n), this became a hot topic again. We clarified the combinatorial topological properties of the configuration spaces of type (2, n) and (3,6).
(3) Modular interpretation of configuration spaces. Problem : Is a configuration space isomorphic to the quotient space D/GAMMA for a symmetric domain D and a discontinuous group GAMMA? There are many studies about the space X (2, n). We succeeded to solve positively the problem for X (3,6),
(4) Intersection theory for twisted (co) homolgies. We established the intersection thery for twisted homolgies. We are about to establish it for twisted cohomolgies. Comparison of (co) homolgies of a variety and those of a branched covering is also studied.
(5) Studies of Selberg-type integrals. A kind of Selberg-type integrals is studied. The theory stated in (4) is useful.
(6) The image of the hypergeometric maps. We proved that if (k, n) * (2, n), (3,6), then the image of the hypergeometric map is not locally projective equivalent to the Grassmannian embedded in the projective space by the Plucker map. This negative result destroyed the dream of many mathematicians. -
ファノ多様体と超幾何函数の代数幾何的研究
Grant number：06640157 1994
日本学術振興会 科学研究費助成事業 一般研究(C)
趙 康治, 若山 正人, 山田 泰彦, 吉田 正章, 塩浜 勝博, 梶原 壌二
Grant amount：\2000000 （ Direct Cost: \2000000 ）
(1)mixed Hodge structureと高次元におけるglobal residueを応用して、高次元の複素射影空間上のlocal systemに対する交点理論の構成の研究。
(2)層の複体のhypercohomologiesとLie環の表現、Verma moduleとの対応を調べることにより、超幾何微分方程式の量子化の導出のための指導原理を与える研究。
(3)Selberg型積分の組み合わせ論的側面を注目し、これらを統一的にあつかう研究。
(4)代数幾何で既知の各種の消滅定理とlocal systemsの(co)homology群の消滅定理間の対応関係を明確にすることにより、より一般的かつ有効な消滅定理の研究。
(5)量子群GL_q(n)上の定数係数微分作用素を導入し、それが持つ良い性質を示し、古典的な不変式論で重要な役割を果たしたCapelli恒等式の量子群版を得た。
(6)R.Howeによって提唱され、群の表現論、不変式論において重要な概念であるdual pairの理論を、最も簡単ではあるが非自明で重要な組(sl_2,on)に対し量子群類似を行った。あわせてこの組に付随したCapelli恒等式も得た。
(7)極小曲面のGauss Mapに関する藤本坦孝の結果を極小に近い曲面に量的に精密化する、笹倉の問題を解決し、その擬等角性を量的に表現し、平均曲率に関する評価式より擬等角性の評価式を導いた。
(8)自乗可積分な正則関数の作る関数空間の再生核としての核関数を、抽象Wiener空間の任意の領域に対して与え、Bergmanの核関数の無限次元化に成功した。 -
ホロノミック系からの量子群・可解格子模型・共形場
Grant number：06640248 1994
日本学術振興会 科学研究費助成事業 一般研究(C)
三町 勝久, 趙 康治, 若山 正人, R Sczech, 吉田 正章
Grant amount：\2100000 （ Direct Cost: \2100000 ）
可解格子模型の一つにXXZ模型が知られているが、近年Jimbo‐MiwaらはXXZ模型を表現論的に再構成し、その相関関数の積分表示(Contour積分による)を与えた。しかしこの表示は物理量を計算する際にはあまり都合か良くなく検討の余地が十分ある。我々はこれをq-解析の立場から再検討した。その結果、1点函数はラマヌジャンの等式と呼ばれるq-解析の基本等式により直ちに積分できることがわかった。
ヤコビ多項式の多変数化のq-アナログとして導入されたのがマクドナルド多項式であり、これはあるq-差分方程式系の解で適当な条件を満たすものとして定義される。このときに用いられるq-差分作用素(マクドナルド作用素)はqが1のときは展開環のセンターから来ることからqの場合もそのように期待される。そして実際にA型の場合ではあるが量子展開環Uq(gl(n))のセンターの動径部分を計算すると問題のマクドナルド作用素が得られることを示した。
mixed Hodge structureと高次元におけるglobal residueを応用して、高次元の複素射影空間上のlocal systemに対する交点理論の構成の研究。
層の複体のhypercohomologiesとLie環の表現、Verma moduleとの対応を調べることにより、超幾何微分方程式の量子化の導出のための指導原理を与える研究。
Selberg型積分の組み合わせ論的側面を注目し、これらを統一的にあつかう研究。
代数幾何で既知の各種の消滅定理とlocal systemの(co)homology群の消滅定理間の対応関係を明確にすることにより、より一般的かつ有効な消滅定理の研究。 -
Grant number：05640190 1993
文部科学省 科学研究費補助金(一般研究(C)) 一般研究(C)
若山 正人, 原瀬 巍, 栗林 幸男, 石川 雅雄, 小島 政利, 熊原 啓作
Authorship：Principal investigator Grant type：Competitive
Grant amount：\2000000 （ Direct Cost: \2000000 ）
本研究の主題は,量子群理論,とりわけ量子等質空間や量子群の様々な軌道空間における微分概念の定式化とそれに伴った幾何学,さらには量子群対称性に基づく調和解析を展開し,応用を図ることであった.若山は野海氏,梅田氏と共同で不変式論的見地から微分作用素環の定式化を試みるとともにDual Pairの量子群理論の重要例を構成した.またそれに関連して量子群版球面調和函数の理論を作った.一方では,量子座標環における座標に関する微分の自然な定義を与えることにも成功し,これら定数係数微分作用素がq差分作用素の編み上げで得られる様子も解明した.これによって矩系行列に対応する量子空間上でも座標に関する微分が自然に理解できるようになり,重要な応用としてグラスマン多様体上に定式化されるGelfandの一般化された超幾何函数(微分方程式)の理論の量子群版が議論できる基盤を作った.そのほか,これに関係して,解の積分表示と相俟ったラドン変換の理論を量子群の枠組みでも考察する必要が出てきた.熊原,若山はそういった見地から,古典論でのラドン変換の研究を行った.また量子化されたパフィアンについての重みつき和公式を石川,若山で開発した.これも表現の指標の様々な母函数公式や,先のDual Piarの理論,球函数の研究と深く関連しており今後解明すべき我々の視点からの問題に役立つものと考えている.
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Grant number：05230045 1993
文部科学省 科学研究費補助金(重点領域研究) 重点領域研究
若山 正人
Authorship：Principal investigator Grant type：Competitive
Grant amount：\1200000 （ Direct Cost: \1200000 ）
研究の主題は,古典的不変式論で枢要を占めてきたCapelli恒等式の量子群対応物とDual Pairの量子群理論を構築することである.そのために成さねばならない最初のことは,量子座標環における座標に関する微分の自然な定義を与えることであった.手始めにそれをGL_q(2)という特別な場合日比氏との共同で試みたこれは実験的な結果ではあったが,そのアイデアは野海,梅田氏との共同研究で生かされた.すなわちそこでは,量子行列環Mat_q(n)上の定数係数微分作用素を量子群GL_q(n)の両側からの自然な作用のみならず,それ自身の左右からのかけ算作用素とも整合的であるように偏極作用素を通して定義し,然るべき性質を持つことを示した.さらにそこでは量子R行列でコントロールされたYang-Baxter方程式による定式化と枠組みを生かし,不変微分作用素の量子包絡環の中心による記述という観点から量子群版Capelli恒等式を得た.更なる研究により量子群版球面調和函数の理論を作り,それを通じてもっとも基本的ではあるが典型的なDual Pair(sl_2,o_n)の量子群対応物と付随するCapelli恒等式を証明することに成功した.より最近では梅田氏と共に,先の定数係数微分作用素がq差分作用素の編み上げで得られる様子を解明した.これによって矩系行列に対応する量子空間上でも座標に関する微分が自然に理解できるようになり,応用としてシンプレクティクリー環のq類似であるU_q(sp_<2m>)のq振動表現を定式化する事もできた.これは他にもGelfandの一般化されたグラスマン多様対上の超幾何函数の理論の量子群理論の構成にも応用が見込まれるものである.また(量子化された)パフィアンについての重みつき和公式を石川氏と共に得たが,これは表現の指標の様々な母函数公式や,先のCapelli恒等式,Dual Pairの理論とも深く関連している.
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リーマン空間の2重微分形式の研究
Grant number：04640068 1992
日本学術振興会 科学研究費助成事業 一般研究(C)
小島 政利, 下村 克己, 栗林 幸雄, 石川 雅雄, 若山 正人, 熊原 啓作
Grant amount：\1800000 （ Direct Cost: \1800000 ）
n+1次元ユークリッド空間におけるn次元2次超曲面(quadric hypersurface)は座標変換によって次の標準形に持ち来される:
(I)Σ^^<n+1>__<i=1>λ_i(x^i)^2=0,
(II)Σ^^<n+1>__<i=1>λ_i(x^i)^2+1=0,
(III)Σ^^<n+1>__<i=1>λ_i(x^i)^2+2x^<n+1>=0
【numerical formula】
定理1 n(] SY.gtorsim.[)2qのとき,固有n次元2次超曲面がq-flat(R^q≡0)であるための必要十分条件は,n=2q,でかつ[I](2次錐)の場合に限る。
定理2 n(] SY.gtoreq. [)2qのとき,固有n次元2次超曲面がq-constant(R^q=kg^<2q>)あるいはq-Einstein(cR^q=kg^<2q-1>)であるための必要十分条件は,n次元超球面に限る。
定理3 n(] SY.gtoreq. [)4qのとき,固有n次元2次超曲面がq-conformally flat(conR^q=0)であるための必要十分条件は,n-2q+2個の係数λiが等しいことである。
系 定理3より,n(] SY.gtoreq. [)4q,q(] SY.gtoreq. [)2のとき,(q-1)-conformally flatではないがq-conformally flatな例が与えられる。 -
関数解析の超準解析的研究
Grant number：03640158 1991
日本学術振興会 科学研究費助成事業 一般研究(C)
栗林 幸男, 下村 克己, 若山 正人, 小島 政利, 熊原 啓作, 赤井 逸
Grant amount：\1700000 （ Direct Cost: \1700000 ）
(1)n個の実変数の複素数値関数の集合をMap(R^2,C)と表す。R^+={YER;Y<O},F={(O,Y):YER^+}とし,Fを含む超フィルタ-(ultrafilter)の一つをFとする。KをRまたはCまたはMap(R^n,C)とし ^<n*>Rを次のように定義する。 ^<n*R>=Π(y_1,…,y_n)K/F^n
^<n*>Rの元[x(y_1,…,y_m)]の元を超実数(hypetreal numbex)という。このとき,RC^*R…C^<(nー1)*>RC^<n+>Rc…,が成立する。しかも超実数を用いた距離を測ることにすれば, ^<n+>R(n=1,2,…)は距離空間となるのみならず各 ^<(nー1)*>Rは ^<n*>R(n=1,2,…)の中でdiscreteになることを証明することができた。これは重要な成果といってよいであろう。
(2)超関数論(theory of hyperfunctions)の研究者が楔と呼ぶ領域R^ntiR^+x…xR^*は超準解析(norstandaxd analysis)の観点で見ると超実数の集合となっている。楔R^n+iR^+x…xR^+で定義される正則関数によって超関数が定まる。このことは超関数と一般関数(generalized functions, ^<n*>Map(R^<12>,C)の元)の関連的研究の有効性を示唆しているものと考えられる。
(3)超実数はR^+x…xR^+で定義される実数値関数の同値類である。従って超実数は曲面的構造をもつものと考えられる。すなわち実数は点と考えられるのに対して,超実数は構造をもちその構造は曲面的であるということができる。これは興味ある事実といえよう。
(4)次に超関数と一般関数の関連を示す例をあげる。
超関数1/(x_1+io)^x…x1/(xm+io)は領域R^n+iR^+x…xR^+で定義される正測関数1/Z_1x…x1/znの境界値と考えられる。これを一般関数の観点で見ると次のようになる。
[1/(x1+iy_1)x…x1/(xm+iyn)]=[(x_1ーiy_1)/(x1^2+y^12)]x…x[(xmーiym)/xn^2+yn^2)] -
擬リ-マン空間上の大域解析学
Grant number：02640126 1990
日本学術振興会 科学研究費助成事業 一般研究(C)
熊原 啓作, 赤井 逸, 下村 克巳, 和泉澤 正隆, 小島 政利, 若山 正人
Grant amount：\1500000 （ Direct Cost: \1500000 ）
n次元ユ-クリッド空間のフ-リエ変換はラドン変換の1次元フ-リエ変換である。S.Helgasonはこの方法をリ-マン対称空間に適応して種々の結果を得た。本研究の主題は擬リ-マン空間であるが,熊原と若山は擬ユ-クリッド空間が符号数が(n,1)のロ-レンツ空間の場合にラドン変換及び双対ラドン変換を定義し,その性質を研究した。ラドン変換はロ-レンツ空間上の関数を超平面上で種分して,超平面全体のなす多様体上の関数を対応させる。その主な結果はダランベ-ル作用素と双対ラドン変換を用いたラドン変換の逆変換公式をnが4の倍数のとき(例えばn=4ならばドゥ・ジッタ空間)に求めたことである。
また若山は局所リ-マン対称空間上調和解析で重要なセルバ-グのゼ-タ関数の研究を行い,ゼ-タ関数の対数微分に関する重要な分式およびラプラス作用素の平方根によって定義されるエ-タ不変量とスピン表現の間の関係式を得た。
さらに若山は、q類似微分作用素を発見し,古典不変式論で重要なキセペリ恒等式の類似式を量子群に対して証明した。このq類似微分作用素は量子群の理論に新しい局面を開くと確信されるもので,継続した研究による結果が期待される。
和泉澤は調和解析的方法によって右側連続確率過程のなす束の閉苞によって新しい右側連続確率過程が得られることを示した。
下村はブラウン・ピ-タ-ソンスペクトルを中心に種々のホモトピ-を計算した。 -
Application of representation theory to number theory
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Analysis and invariant theory based on group symmetries
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微分方程式系のスペクトルの研究
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表現論の数論，ゼータ間数論への応用
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群対称性に基づく不変式論と解析学
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Spectral problems of system of differential equations
Grant type：Competitive