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Kiyoshi Takeuchi

Section

Graduate School of Science

Job title

Professor

Degree

博士（数理科学）（東京大学）
修士（理学）（東京大学）

researchmap

https://researchmap.jp/read0046104

J-GLOBAL ID

200901038891748944

Professional Memberships 1

日本数学会

Research Areas 1

Natural sciences / Basic analysis /

Papers 48

On a Bernstein-Sato polynomial of a meromorphic function Peer-reviewed

K. Takeuchi

Nagoya Math. Journal　2023/09
On the monodromies and the limit mixed Hodge structures of families of algebraic varieties Peer-reviewed

T. Saito, K. Takeuchi

Michigan Math. J.　2023/09
Meromorphic nearby cycle functors and monodromies of meromorphic functions (with Appendix by T. Saito) Peer-reviewed

T.T. Nguyen and K. Takeuchi

Revista Matematica Complutense　2023/02
On the monodromy conjecture for non-degenerate hypersurfaces Peer-reviewed

A. Esterov, A. Lemahieu, K. Takeuchi

J. of European Mathematical Society　24　3873-3949　2022/01
The bifurcation set of a rational function via Newton polytopes Peer-reviewed

T.T. Nguyen, T. Saito, K. Takeuchi

Math. Zeitschrift　298　3873-3949　2021/01
On irregularities of Fourier transforms of regular holonomic D-modules Peer-reviewed

Takeuchi, Kiyoshi

Advances in Math.　2020/06
On some topological properties of Fourier transforms of regular holonomic D-modules Peer-reviewed

Takeuchi,Kiyoshi

Canadian Mathematical Bulletin　63　(2)　454-468　2020/06
Bifurcation values of polynomial functions and perverse sheaves Peer-reviewed

Takeuchi,Kiyoshi

Annales de l'Institut Fourier　2019
Hyperbolic localization and Lefschetz fixed point formulas for higher-dimensional fixed point sets

Ike, Yuichi, Matsui, Yutaka, Takeuchi, Kiyoshi

Int. Math. Res. Not.　(15)　4852-4898　2017
Publisher: OXFORD UNIV PRESS
DOI： 10.1093/imrn/rnx030 　

ISSN： 1073-7928

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We study Lefschetz fixed point formulas for constructible sheaves with higher-dimensional fixed point sets. Under fairly weak assumptions, we prove that the local contributions from them are expressed by some constructible functions associated to hyperbolic localizations. This gives an affirmative answer to a conjecture of Goresky-MacPherson [8] in particular for smooth fixed point components (see [9, page 9, (1.12) Open problems]). In the course of the proof, the new Lagrangian cycles introduced in our previous article [20] will be effectively used. Moreover, we show various examples for which local contributions can be explicitly determined by our method.
On the sizes of the Jordan blocks of monodromies at infinity Peer-reviewed

Yutaka Matsui, Kiyoshi Takeuchi

HOKKAIDO MATHEMATICAL JOURNAL　44　(3)　313-326　2015/10

ISSN： 0385-4035
CONFLUENT A-HYPERGEOMETRIC FUNCTIONS AND RAPID DECAY HOMOLOGY CYCLES Peer-reviewed

Alexander Esterov, Kiyoshi Takeuchi

AMERICAN JOURNAL OF MATHEMATICS　137　(2)　365-409　2015/04

ISSN： 0002-9327

eISSN： 1080-6377
Monodromies at infinity of confluent A-hypergeometric functions Peer-reviewed

Kama Ando, Alexander Esteroy, Kiyoshi Takeuchi

ADVANCES IN MATHEMATICS　272　1-19　2015/02

DOI： 10.1016/j.aim.2014.10.024 　

ISSN： 0001-8708

eISSN： 1090-2082
Motivic Milnor Fibers and Jordan Normal Forms of Milnor Monodromies Peer-reviewed

Yutaka Matsui, Kiyoshi Takeuchi

PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES　50　(2)　207-226　2014/06

DOI： 10.4171/PRIMS/130 　

ISSN： 0034-5318
INVERTIBLE POLYNOMIAL MAPPINGS VIA NEWTON NON-DEGENERACY Peer-reviewed

Ying Chen, Luis Renato G. Dias, Kiyoshi Takeuchi, Mihai Tibar

ANNALES DE L INSTITUT FOURIER　64　(5)　1807-1822　2014

ISSN： 0373-0956
MEROMORPHIC CONTINUATIONS OF LOCAL ZETA FUNCTIONS AND THEIR APPLICATIONS TO OSCILLATING INTEGRALS Peer-reviewed

Toshihisa Okada, Kiyoshi Takeuchi

TOHOKU MATHEMATICAL JOURNAL　65　(2)　159-178　2013/06

ISSN： 0040-8735
Monodromy at infinity of polynomial maps and newton polyhedra (with an Appendix by C. Sabbah) Peer-reviewed

Yutaka Matsui, Kiyoshi Takeuchi

International Mathematics Research Notices　2013　(8)　1691-1746　2013
Publisher: OXFORD UNIV PRESS
DOI： 10.1093/imrn/rns092 　

ISSN： 1073-7928 1687-0247
多項式写像と A-超幾何関数の無限遠点におけるモノドロミー Peer-reviewed

竹内潔, 松井優

數學　64　(3)　225-253　2012/07
Publisher: 日本数学会
DOI： 10.11429/sugaku.0643225 　

ISSN： 0039-470X
Motivic Milnor Fibers over Complete Intersection Varieties and their Virtual Betti Numbers Peer-reviewed

Alexander Esterov, Kiyoshi Takeuchi

INTERNATIONAL MATHEMATICS RESEARCH NOTICES　2012　(15)　3567-3613　2012

DOI： 10.1093/imrn/rnr154 　

ISSN： 1073-7928
Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves Peer-reviewed

Yutaka Matsui, Kiyoshi Takeuchi

MATHEMATISCHE ZEITSCHRIFT　268　(1-2)　409-439　2011/06

DOI： 10.1007/s00209-010-0678-5 　

ISSN： 0025-5874
MILNOR FIBERS OVER SINGULAR TORIC VARIETIES AND NEARBY CYCLE SHEAVES Peer-reviewed

Yutaka Matsui, Kiyoshi Takeuchi

TOHOKU MATHEMATICAL JOURNAL　63　(1)　113-136　2011/03

ISSN： 0040-8735
A geometric degree formula for A-discriminants and Euler obstructions of toric varieties Peer-reviewed

Yutaka Matsui, Kiyoshi Takeuchi

ADVANCES IN MATHEMATICS　226　(2)　2040-2064　2011/01

DOI： 10.1016/j.aim.2010.08.020 　

ISSN： 0001-8708
Monodromy at infinity of A-hypergeometric functions and toric compactifications Peer-reviewed

Kiyoshi Takeuchi

MATHEMATISCHE ANNALEN　348　(4)　815-831　2010/12

DOI： 10.1007/s00208-010-0501-y 　

ISSN： 0025-5831
Microlocal Study of Lefschetz Fixed-Point Formulas for Higher-Dimensional Fixed Point Sets Peer-reviewed

Yutaka Matsui, Kiyoshi Takeuchi

INTERNATIONAL MATHEMATICS RESEARCH NOTICES　2010　(5)　882-913　2010

DOI： 10.1093/imrn/rnp163 　

ISSN： 1073-7928
Monodromies at infinity of polynomial maps and A-hypergeometric functions Peer-reviewed

K., Takeuchi

Proceedings of the Centre for Mathematics and its Applications　43　141-174-174　2010/01
A-discriminants and Euler obstructions of toric varieties (Differential Equations and Exact WKB Analysis) Peer-reviewed

MATSUI, Yutaka, TAKEUCHI, Kiyoshi

RIMS Kokyuroku Bessatsu　10　(0)　149-165　2008/11
Publisher: Kyoto University
ISSN： 1881-6193
Topological radon transforms and degree formulas for dual varieties Peer-reviewed

Yutaka Matsui, Kiyoshi Takeuchi

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY　136　(7)　2365-2373　2008

ISSN： 0002-9939
Topological Radon Transforms and Their Applications (Algebraic Analysis and the Exact WKB Analysis for Systems of Differential Equations) Peer-reviewed

MATSUI, Yutaka, TAKEUCHI, Kiyoshi

RIMS Kokyuroku Bessatsu　5　(0)　225-240　2008/01
Publisher: Kyoto University
ISSN： 1881-6193
Microlocal study of topological Radon transforms and real projective duality Peer-reviewed

Yutaka Matsui, Kiyoshi Takeuchi

ADVANCES IN MATHEMATICS　212　(1)　191-224　2007/06

DOI： 10.1016/j.aim.2006.10.001 　

ISSN： 0001-8708
Perverse sheaves and Milnor fibers over singular varieties Peer-reviewed

K., Takeuchi

Advanced Studies in Pure Mathematics　46　211-222-222　2007/01
Generalized Plucker-Teissier-Kleiman formulas for varieties with arbitrary dual defect Peer-reviewed

Yutaka Matsui, Kiyoshi Takeuchi

REAL AND COMPLEX SINGULARITIES　248-+　2007
Topological Radon Transforms and Projective Duality (Recent Topics on Real and Complex Singularities)

Matsui,Yutaka, Takeuchi,Kiyoshi

RIMS Kokyuroku　1501　(0)　132-146　2006/07
Publisher: Kyoto University
ISSN： 1880-2818
Characteristic cycles of perverse sheaves and Milnor fibers Peer-reviewed

P Nang, K Takeuchi

MATHEMATISCHE ZEITSCHRIFT　249　(3)　493-511　2005/03

DOI： 10.1007/s00209-004-0712-6 　

ISSN： 0025-5874
Dimension formulas for the hyperfunction solutions to holonomic D-modules Peer-reviewed

K., Takeuchi

Advances in Math　180　134-145　2003/01
Notes on the Canchy-Kowalevski theorem for E-modules Peer-reviewed

Y., Sugiki, K., Takeuchi

J. Funct. Anal　181　1-13　2001/01
Microlocal vanishing cycles and ramified Cauchy problems in the Nilsson class Peer-reviewed

K Takeuchi

COMPOSITIO MATHEMATICA　125　(1)　111-127　2001/01

ISSN： 0010-437X
On the solvability of operators with multiple characteristics Peer-reviewed

H Koshimizu, K Takeuchi

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS　26　(9-10)　1691-1720　2001

ISSN： 0360-5302
Extension theorems for the distribution solutions to D-modules with regular singularities Peer-reviewed

H Koshimizu, K Takeuchi

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY　128　(6)　1685-1690　2000

ISSN： 0002-9939
A Hartogs-type theorem for solutions to systems with regular singularities Peer-reviewed

K., Takeuchi

Arch der Math　73　(5)　390-393　1999/01
Microlocal inverse image and bimicrolocalization Peer-reviewed

Kiyoshi Takeuchi

Publications of the Research Institute for Mathematical Sciences　34　(2)　135-153　1998
Publisher: Kyoto University
DOI： 10.2977/prims/1195144758 　

ISSN： 0034-5318
Edge-of-the-wedge type theorems for hyperfunction solutions Peer-reviewed

Kiyoshi Takeuchi

Duke Mathematical Journal　89　(1)　109-132　1997

DOI： 10.1215/S0012-7094-97-08907-9 　

ISSN： 0012-7094
On higher-codimensional boundary value problems

K., Takeuchi

New trends in microlocal analysis　1997/01
Publisher: Springer
On the solvability of partial differential equations Peer-reviewed

H., Koshimizu, K., Takeuchi

Proc. Japan Academy　72　131-133　1996/01
Microlocal boundary value problem in higher codimensions Peer-reviewed

K Takeuchi

BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE　124　(2)　243-276　1996

ISSN： 0037-9484
Binormal deformation and bimicrolocalization Peer-reviewed

Kiyoshi Takeuchi

Publications of the Research Institute for Mathematical Sciences　32　(2)　277-322　1996
Publisher: Kyoto University
DOI： 10.2977/prims/1195162965 　

ISSN： 0034-5318
Etude microlocale des problemes aux limites en codimension superieure

K., Takeuchi

Compte Rendu Acad, Sci.　320　441-443　1995/01
Theoremes de type edge of the wedge pour les solutions hyperfonctions

K., Takeuchi

Compte Rendu Acad, Sci.　321　1333-1336　1995/01
Deformation binormale et bispecialisation

P., Schapira, K., Takeuchi

Compte Rendu Acad, Sci.　319　707-712　1994/01
On the second microlocalization along isotropic submanifolds Peer-reviewed

K., Takeuchi

Proc. Japan Academy　69　(5)　136-139　1993/01

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Misc. 2

Higher codimensional BVP for $\mathcal{D}$-modules (Complex Analysis and Microlocal Analysis)

Takeuchi Kiyoshi

RIMS Kokyuroku　1090　87-99　1999/04
Publisher: Kyoto University
ISSN： 1880-2818
Cauchy Problems for Sheaves and its Applications(Study of Partial Differential Equations by means of Functional Analysis)

KOSHIMIZU HIROSHI, TAKEUCHI KIYOSHI

RIMS Kokyuroku　969　58-68　1996/10
Publisher: Kyoto University
ISSN： 1880-2818

Books and Other Publications 2

D-modules

竹内,潔

共立出版　2017/08
D-modules, perverse sheaves and representation theory

R., Hotta, K., Takeuchi, T., Tanisaki

Birkhauser　2007/01

Presentations 14

On irregularities of Fourier transforms of regular holonomic D-modules International-presentation Invited

Takeuchi,Kiyoshi

27th ICFIDCAA Krasnoyarsk　2019/08
Meromorphic nearby cycle functors and monodromies of meromorphic functions International-presentation Invited

Takeuchi,Kiyoshi

Non-isolated singularities and derived geometry　2019/07/29
Exponential factors and Fourier transforms of D-modules International-presentation Invited

Takeuchi,Kiyoshi

Higgs bundles and D-modules　2019/06/03
On irregularities of Fourier transforms of regular holonomic D-modules International-presentation Invited

Takeuchi,Kiyoshi

Riemann-Hilbert correspondences　2018/02/06
On the monodromy conjecture for non-degenerate hypersurfaces International-presentation Invited

Takeuchi,Kiyoshi

3rd PRIMA congress in Oaxaca　2017/08
On the monodromies and the limit mixed Hodge structures of families of algebraic varieties International-presentation Invited

Takeuchi,Kiyoshi

Singularity theory conference　2017/07/12
On the monodromies and the limit mixed Hodge structures of families of algebraic varieties International-presentation Invited

Takeuchi,Kiyoshi

Iberian Meeting on Algebraic Analysis　2016/06/09
多項式写像の無限遠点における特異性とモノドロミー Invited

竹内,潔

トポロジーシンポジウム　2014/07/28
Monodromies and asymptotic expansions at infinity of confluent A-hypergeometric functions International-presentation Invited

竹内,潔

First Joint International Meeting RSME-SCM etc.　2014/06/30
Bifurcation points of polynomial functions and perverse sheaves Invited

Takeuchi,Kiyoshi

超平面と超曲面特異点のトポロジー　2014/04/26
合流型A-超幾何関数のモノドロミーについて Invited

竹内,潔

超幾何方程式研究会 2014　2014/01/07
Toric compactifications for polynomial maps and their applications International-presentation Invited

Takeuchi,Kiyoshi

The 1st Franco-Japanese-Vietnamese Symposium on Singularities　2013/09/19
On the monodromies of complex polynomials International-presentation Invited

Takeuchi,Kiyoshi

Australian-Japanese workshop on real and complex singularities　2013/09/10
Confluent A-hypergeometric functions and rapid decay homology cycles Invited

Takeuchi,Kiyoshi

Weekend workshop on computational approaches to D-modules and hypergeometric systems　2013/04/20

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Research Projects 24

Study of D-modules with irregular singularities and singularity theory

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Tohoku University

2024/04/01 - 2028/03/31
不確定特異点を持つD-加群と幾何学的モノドロミーの研究

竹内潔

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 基盤研究(B)

2017/04/01 - 2022/03/31

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望月拓郎と Kedlayaの理論により、不確定特異点を持つホロノミー D-加群の理論は劇的な発展を遂げた。特に D'Agnoloと柏原は、不確定特異点を持つホロノミー D-加群に対するリーマンヒルベルト対応を確立した。また柏原と Schapiraは、これをホロノミー D-加群のフーリエ変換に応用した。フーリエ変換は D-加群の理論で基本的な対象だが、高次元の場合の詳しい性質はほとんど未解明である。我々は、柏原-Schapiraの理論を用いて正則ホロノミー D-加群のフーリエ変換の詳しい性質を解明した。特にその特異集合を具体的に記述し、それに沿う指数因子や不確定度が元の正則ホロノミー D-加群の特性多様体を用いて記述できることを示した。我々は、さらにある特別な形の不確定特異点を持つホロノミーD-加群にたいしても、同様の結果が成り立つことを示した。これらの結果は、約30年前の Brylinskiの結果を拡張するものであり、さらなる発展が期待される。幾何学的モノドロミーの研究については、これまで得られた結果を有理（型）関数のミルナーモノドロミーやその定める写像のモノドロミーに一般化した。そのために、Deligneにより定義された nearby cycle函手を有理型関数の場合に一般化し、基礎理論を整備した。また有理関数の定める複素平面への写像について、その分岐点集合のニュートン多面体を用いた上からの評価を得た。これは Nemethi-Zahariaの古典的な結果の拡張である。さらに有理型関数にたいするBerstein-佐藤多項式を発見し、その根と有理型関数のミルナーモノドロミーに関して柏原-Malgrangeの定理の類似を証明した。またこの新しいBerstein-佐藤多項式の理論を用いて、有理型関数の定める乗法的イデアル層の跳躍数の集合の上からの評価を得た。
General studies on L-class, cobordism theory, bivariant theory and related topics

Yokura Shoji

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Kagoshima University

2016/04/01 - 2019/03/31

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We extended Levine-Morel's algebraic cobordism to S-schemes. While we were aiming to complete a bivariant algebraic cobordism, Toni Annala (University of British Columbia) completed it in November 2018. Now we are working together with Toni Annala on a bivariant algebraic cobordism of vector bundles. While we were looking for a bivariant L-class, we noticed that Hirzebruch chi-y genus is multiplicative for a fiber bundle. With this unexpected discovery as a starting point, we obtained the multiplicativity mod 8 of chi-y genus and homological congruence formulae of the motivic characteristic class and so on. Based on the idea of bivariant theory, we also obtained interesting results on the homotopy set of mappings and so on.
Residue theory on singular varieties and its applications

SUWA Tatsuo, ASUKE Taro, OHMOTO Toru, OKA Mutsuo, TAKEUCHI Kiyoshi, TAJIMA Shinichi, NAKAMURA Yayoi, YOKURA Shoji

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Hokkaido University

2012/04/01 - 2017/03/31

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The localization theory of characteristic classes developed by the principal investigator turned out to be very effective in a wide range of problems related to characteristic classes mainly in complex analytic geometry. During the period, we obtained the following results. (1) As to the degeneracy loci problem of vector bundle homomorphisms, we constructed a universal localization. (2) We generalized the Lefschetz coincidence point formula. For this, the local and global homology classes we introduced played key roles. (3) We developed the theory of relative Bott-Chen cohomology and give some applications. (4) We discovered a simple way of expressing Sato hyperfunctions. For this we strengthened the theory of relative Dolbeault cohomology. We also gave simple expressions of fundamental operations on the hyperfunctions.
Study on polynomial maps and hypergeometric functions of several variables

Takeuchi Kiyoshi, MATSUI Yutaka

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: University of Tsukuba

2013/04/01 - 2016/03/31

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We studied monodromies at infinity of polynomial maps. Especially for the maps which are not tame at infinity, by proving a vanishing theorem on the cohomology groups of generic fibers, we described the Jordan normal forms of their monodromies at infinity in many cases. As a byproduct of this study, we obtained also a description of the bifurcation sets of polynomial maps. Moreover, a formula for the characteristic polynomials of the monodromies at infinity of confluent A-hypergeometric functions was obtained. As for the monodromy conjecture, we confirmed it for polynomials which are non-degenerate at the origin in many cases.
New developments of singularity theory of mappings

SAEKI OSAMU, OHMOTO Toru, YOKURA Shoji, IWASE Norio, KAMADA Seiichi, SAKUMA Kazuhiro, ISHIKAWA Masaharu, FUKUI Toshizumi, ISHIKAWA Goo, YAMAMOTO Minoru, TAKASE Masamichi, ASHIKAGA Tadashi, KATANAGA Atsuko, KOBAYASHI Mahito, YAMAMOTO Takahiro, TAKEUCHI Kiyoshi, TAKATA Toshie

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (A)

Institution: Kyushu University

2011/04/01 - 2016/03/31

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The main purpose of this project was to indicate new directions of the singularity theory of differentiable mappings in our era of modern Mathematics. First, we completely solved affirmatively the long-standing classical problem of Milnor about the existence of nontrivial polynomial map germs of dimension 6 to 3, by utilizing the topology of configuration spaces. We also completely determined the conditions for a special generic map to be desingularized from a viewpoint of immersions and embeddings. We classified singular fibers of stable maps on 3-manifolds with boundary, established the notion of cobordism group of Morse functions on surfaces with boundary, and showed that it is a cyclic group of order two. Finally, we applied such a theory of singular fibers to the data visualization and developed a new user interface. Summarizing, we could broadly show the new directions for the development of the singularity theory of differentiable mappings.
Study of geometric and analytic monodromies and local zeta functions

TAKEUCHI Kiyoshi, MATSUI Yutaka

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: University of Tsukuba

2010 - 2012

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By using the theory of motivic Milnor fibers, we obtained some formulas which express the Jordan normal forms of monodromies at infinity of polynomial maps. Moreover we generalized this result to the case of polynomial maps from complete intersection varieties. By using the theory of rapid decay homologies we also obtained the integral representations of confluent A-hypergeometric functions. By this result we calculated their asymptotic expansions and Stokes multipliers at infinity.
Localization theory of Atiyah classes and its applications

SUWA Tatsuo, OHMOTO Toru, OKA Mutsuo, KAWAZUMI Nariya, TAKEUCHI Kiyoshi, TAJIMA Shinichi, NAKAMURA Yayoi, YOKURA Shoji, YOSHIKAWA Kenichi

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Hokkaido University

2009 - 2011

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(1) Concerning the localization theory of Atiyah classes, with collaboration of M. Abate, F. Bracci and F. Tovena, we established the following fundamental theories : (1) a simple definition of Atiyah classes suitable for the localization theory,(2) Cech-Dolbeault cohomology theory,(3) introduction of the complex analytic Thom class,(4) proof of a Bott type vanishing theorem in terms of Atiyah forms. (2) Concerning the degeneracy loci problem of a homomorphism of vector bundles, with collaboration of T. Ohmoto, we started to try to prove the Thom-Porteous formula localized at the degeneracy loci. This is done by constructing a universal localization of a Schur polynomial of Chern. It is a vast generalization of the Thom class of a vector bundle.
Radon transforms on homogeneous spaces and their application to harmonic analysis

KAKEHI Tomoyuki, ISOZAKI Hiroshi, TAKEUCHI Kiyoshi, KINOXHITA Tamotsu

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: University of Tsukuba

2007 - 2009

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We studied Radon transforms on homogeneous spaces, and their applications to harmonic analysis. Our results are as follows. (1) We proved that the ranges of generalized matrix Radon transforms are characterized by Pfaffian type invariant differential operators. In addition, we also obtained the inversion formulas. (2) We proved that under some condition the support of the fundamental solution to the Schroedinger equation on a compact symmetric space becomes a lower dimensional subset at a rational time and that it coincides with the whole symmetric space at an irrational time.
Applications of algebraic analysis to geometry

TAKEUCHI Kiyoshi, SUWA Tatsuo, TAJIMA Shinichi, TANISAKI Toshiyuki

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: University of Tsukuba

2007 - 2009

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We studied Lefschetz fixed point formulas for maps with higher-dimensional fixed point sets and obtained a formula which expresses their fixed point indices explicitly. We also obtained formulas describing the dimensions and the degrees of A-discriminant varieties introduced by Gelfand etc. in terms of the geometric data of the configuration A.Moreover, as byproducts of this research, various results on the monodromies at infinity of polynomial maps, the analytic continuations of A-hypergeometric functions and the poles of local zeta functions etc. are also obtained.
Applications of algebraic analysis to singularity theory Competitive

Institution: University of Tsukuba.

2004/04 - 2007/03
Complex analysis of residues currents and computational algebraic analysis

TAJIMA Shinichi, YOSHIHARA Hisao, KOJIMA Hideo, TAKEUCHI Kiyoshi, NAKAMURA Yayoi

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Niigata University

2005 - 2007

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Hypersurface isolated singularities and associated residues currents are considered in the context of algebraic analysis. ・An efficient algorithm that computes bases of a dual vector space of a Milnor algebra associated to a singular point has been constructed. ・A new method for computing standard bases of a zero-dimensional ideal in a power series ring has been proposed. The key ingredient in this approach is the concept of algebraic local cohomology and the Grothendieck local duality. ・An algorithm for construction holonomic D-modules attached to hypersurface isolated singularities has been derived and the structure of these holonomic D-modules have been investigated. ・An algorithmic method for computing homological indices of holomorphic vector fields has been proposed.
Lie algebras and quantum groups via algebraic analysis

TANISAKI Toshiyuki, KASHIWARA Msaki, SHOJI Toshiaki, SAITO Yoshihisa, KANEDA Msaharu, NAKAJIMA Hiraku

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Osaka City University

2005 - 2006

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1. Tanisaki investigated the action of the braid groups on the zero-weight spaces of the integral modules over quantized enveloping algebras. Under a certain condition on the representation the braid group action descends to the Hecke algebra, and a module over the Hecke algebra is obtained. In the case of type A all irreducible modules over the Hecke algebra is derived. Using the Kazhdan-Lusztig bases for the Hecke algebra modules and the global bases for the modules over the quantized enveloping algebras we can show that the construction above makes sense over rings. Hence it is possible to investigate modular representations by this method. Tanisaki gave a new proof of the conjecture by Lascoux-Leclerc-Thibon. 2. Tanisaki investigated applications of the geometric Langlands correspondence to representation theory. Especially, he tried to obtain the twining character formula due to Naito etc. using geometric Langalands correspondence. 3. Kashiwra investigated level zero fundamental modules over affine quantum algebras and Demazure modules. 4. Shoji investigated representations of modified Ariki-Koike algebras introduced by himself. He also obtained interesting results on the corresponding q-Shur algebras. 5. Naito investigated crystal bases of the extremal weight modules. Especially, he considered about the action of the diagram automorphisms and obtained results about the elements of the crystal base fixed by this group. 6. Satio investigated Macdonald polynomials using Cherednik algebras. 7. Kashiwara investigated representations of affine Hecke algebras of type B. 8. Shoji investigated generalized Green functions and obtained results about some constant.
Harmonic analysis on Grassmann manifolds and its applications to Radon transforms and inverse problems

KAKEHI Tomoyuki, TAIRA Kazuaki, TAKEUCHI Kiyoshi, KINOSHITA Tamotsu, MORIYA Katsuhiro, TERUI Akira

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: University of Tsukuba

2004 - 2006

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In this research project, we studied the following (1), (2) and (3). (1) Dual Radon transforms on affine Grassmann manifolds. (2) Moment conditions and support theorems for Radon transforms on affine Grassmann manifolds. (3) Range characterization of the matrix Radon transform. (1) The main result is as follows. Let G(d,n) be the affine Grassmann manifold of d-dimensional planes in the n-dimensional Euclidian space. We assume that q<p and dim(G(p,n))<dim(G(q,n)). Let R be the Radon transform from the space of smooth functions on G(p,n) to that on G(q,n). Then the range of the Radon transform R is characterized by the system of Pfaffian equations. (2) The main result is as follows. We assume that p<q and dim(G(p,n))=dim(G(q,n)). The Radon transform R associated with the inclusion incidence relation maps the Schwartz space on G(p,n) to that on G(q,n). Let f be a Schwartz class function on G(p,n). If the image Rf is compactly supported, then the function f is also compactly supported. In addition, we proved that the range of R is characterized by generalized moment conditions. (3) The main result is as follows. Let M be the space of n×k matrices, and let Ξ be the space of matrix planes in M. The matrix Radon transform from functions on M to functions on Ξ is defined as the integral of a function on each matrix plane. Then the range of the matrix radon transform is characterized as the kernel of a generalized Pfaffian type operator arising from the corresponding Cartan motion group.
Representation theory of algebraic groups via algebraic analysis

TANISAKI Toshiyuki, KASHIWARA Masaki, SHOJI Toshiaki, SAITO Yoshihisa, KANEDA Masaharu, TAKEUCHI Kiyoshi

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Osaka City University

2003 - 2004

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1. Tanisaki investigated on the quantized flag manifolds, especially at roots of 1. He has formulated a conjecture which can be regarded as an analogue of the result of Bezrukavnikov-Mirkovic-Ryuminin about the correspondence of representations and D-modules on the flag manifold in positive characteristics. This is different from recent works of Backelin-Kremnitzer and Mirkovic. It should be solved in the near future although there are some problems to be overcome. He also extended a result of Soergel about the ring of differential operators on the partial flag manifold of reductive algebraic groups and obtained similar results for differential operators acting on vector bundles. Furthermore, he considered about parabolic analogue of Soergel's result on the center of category O. 2. Kashiwara showed that the crystal base of some finite dimensional representation of affine quantum group with fundamental weight as its external weight is isomorphic to that of the Demazure module of irreducible module with level 1 highest weight. 3. Ariki has shown that the representation types of the classical Hecke algebras are governed by the Poincare polynomials. 4. Nakajima proved that the first tern of the Necrasov partition function coincides with the pro-potential of Seidberg-Written. 5. Shoji has solved Lusztig's conjecture on the characters of the special linear groups over finte fields. Moreover, he determined the scalar appearing in the conjecture. 6. Kaneda investigated on the correspondence between D-modules on flag varieties in positive characteristics and representations of the corresponding algebraic groups. He formulated a certain derived equivalence in terms of arithmetic differential operators due to Berthelot., and obtained some results in the case of the projective space. 7. Ichino investigated on the diagonal restriction of Saito-Kurokawa lift, and proved the algebraicity of a special value of a certain L-function.
Toward to a generalization of modular invariance of vertex operator algebras into Hilbert type and Siegel type.

MIYAMOTO Masahiko, MORITA Jun, KIMURA Tatsuo, NAITOU Satoshi, TAKEUCHI Kiyoshi, KITAZUME Masaaki

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: University of Tsukuba

2001 - 2004

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A concept of vertex operator algebras (VOA shortly) has originated from the moonshine vertex operator algebra, which was constructed in order to explain a mysterious relation (the moonshine conjecture) between Monster simple finite group (the largest sporadic finite simple group) and the classical elliptic modular function. Our purpose of this project is to clarify the modular invariance property of VOAs and extend it in multivarables. (1)We found a new construction of the moonshine vertex operator algebra by using Ising models, which offers a new modular invariance in multivariables. Compared with the original construction, our construction is easy and we can apply our construction for many other VOAs. (2)We have shown that C2-condition is enough to get, a modular invariance. Classically, the rationality (completely reducibility of modules) was considered to be more important than C2-condition, but our research has shown that we don't need rationality. (3)We construct an infinitely many VOAs with Euclidian Jordan Algebras as Griess algebras for any complex central charge c. So we construct a candidate of Siegel modular invariance. (4)We found an order formula to determine the automorphism group of VOAs. In our construction, Miyamoto involution plays an essential role and so we can easily get information about the centralizer of Involution in the full automorphism group. The question is if we can determine the automorphism group from it.
D-加群の解の構造の幾何学的研究

竹内潔

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 若手研究(B)

Institution: 筑波大学

2002 - 2003

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孤立特異点を持つ複素超曲面は、ミルナーらによる重要な研究以後多くの数学者により研究された。しかしそのミルナーファイバーのコホモロジーやそのモノドロミーは、重み付き斉次多項式など特別な場合をのぞいて良くわかっていないのが現状である。私はナング氏と共同で孤立特異点型の特異性を持ったD-加群を研究し、既約なD-加群すべての特性サイクルを求めることに成功した。興味深いことに、特性サイクルの係数に特異点のモノドロミーの情報が自然に現われる。この事を逆に利用して、モノドロミーの各固有空間の次元を上から押さえる不等式を得た。さらにD-加群の指数定理や偏屈層の理論を援用することにより、特異点集合が孤立していない場合にたいする一般化も行った。また東京大学院生の松井優氏とグラスマン多様体上の構成可能関数のラドン変換の基礎的研究を行い、エルンストレムの射影空間の場合の定理の一般化がグラスマンでは成り立たない事などを示した。これまでの代数解析の研究と並行して今後はこうした複素特異点論や積分幾何への応用の研究も行っていきたい。以上の研究のために今年度は計算機を買い揃え、専門書等を購入して周辺分野の知見を広めるととに努め、さらに最新の研究成果を知るために国内の研究集会に出張した。特に特異点の代数幾何や表現論などの本来他分野の研究者と研究連絡を取り合った。今後これらの研究者と活発に共同研究をしていく予定である。また堀田良之・谷崎俊之氏と代数解析とその表現論、交叉コホモロジー理論への応用に関する専門書の執筆を行った。
Global properties of differential operators of subdeterminantal type and integral geometry on symmetric spaces

KAKEHI Tomoyuki, TAIRA Kazuaki, SASAKI Tateaki, KAJITANI Kunihiko, NAITO Satoshi, MIYAMOTO Masahiko

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: University of Tsukuba

2001 - 2002

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1. Pfaffian type operators and Radon transforms on affine Grassmann manifolds : Let G(d, n) be the affine Grassmann manifolds of all d-dimensional planes in R^n". Then the Radon transform R^p_q is defined as the transform from smooth functions on G(p,n) to smooth functions on G(q,n] arising from the inclusion incidence relation. Then our results are stated as follows. (1) In the case p < q. Let s and r be the rank of G(p, n) (resp. G(q, n) ). We assume that s < r. Then the range of R^p_q is characterized as the kernel of a single Pfaffian type invariant differenial operator of order 2s + 2. (2) In the case p < q. We assume that s 【less than or equal】 r. Then the inversion formula for R^p_q is given as DR^p_qR^p_q = I, where D is the reproducing operator consisting of Pfaffian type operators. (3) In the case p > q. We assume that s < r. Then the range of R^p_q is characterized as the kernel of an invariant system of differential equations of order s + 1, which consists of two different kinds of Pfaffians. This research was done in collaboration with F. Gonzalez. 2. Sobolev estimates for Radon transforms : Basically a Radon transform is an integration of a function over a submanifold. So it is expected that a Radon transform regularizes a function to some extent, and in fact, it was shown by Strichartz that the q-plane transform R^0_4 maps a function on L^2 to a funtion on H^<(9)/(2)> the Sobolev space of order 9/2. In this case, the gain of regularity is proportional to the demension of the fiber of the corresponding double fibration. However, in the case of R^p_q for general p and q, we discovered that R^p_q does not regularize a function so much in the sense that the gain of regularity is no longer proportional to the dimension of the fiber.
Representation theory of algebraic groups through algebraic analysis

TANISAKI Toshiyuki, SHOJI Toshiaki, KASHIWARA Masaki, SAITO Yashihisa, KAWANAKA Norkki, KANEDA Masahara

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B).

Institution: Hiroshima University

1999 - 2000

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1. Highest weight modules over Kac-Moody Lie algebras Kashiwara and Tanisaki tried to determine characters of irreducible modules over affine Lie algebras with critical highest weights. In particular, they considered relations of representations of affine Lie algebras with D-modules on the semi-infinite flag manifold. Through the investigation they noticed that the behavior of the equivariant line bundles is quite different from those on the ordinary flag manifold. 2. Flag manifold for quantum groups Tanisaki constructed generalized flag manifold corresponding to general parabolic subgroups. The treatment of the unipotent radical is more delicate than the ordinary case. Besides Morita and Tanisaki tried to find a good formulation for the sheaf cohomologies and D-modules on the quantized flag manifold from the view point of non-commutative schemes. 3. Representations of toroidal Lie algebras Saito constructed certain representations of toroidal Lie algebras usig Boson. He also investigated the automorphisms of the toroidal Lie algebras and found a connection with the moudular groups. 4. Representations of quantum groups over Laurent polynimial rings Kaneda investigated representations of quantum groups over the Laurent polynimial rings, and have proved a version of a theorem of Kempf. 5. Solvable games Kawanaka found a generalization of Sato's game, and investigated on it from the view point of representation theory. 6..Representations of comples reflection groups and their Hecke algebras Shoji tried to extend the Frobenius formua for the Hecke algebra of type A to the Hecke algebras for complex reflection groups, and succeed in it in the case of Ariki-Koike algebra.
有限体およびP進体上のKZ方程式とその表現論への応用

谷崎俊之, 柏原正樹, 竹内潔, 斉藤義久, 三町勝久, 兼田正治, 都築暢夫

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 萌芽的研究

Institution: 広島大学

1998 - 1999

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1.アフィン・リー代数の最高ウェイト表現の研究研究代表者と柏原正樹(分担者)は,アフィン・リー代数の最高ウェイト表現の研究を行なった.特に臨界レベルのウェイトに関して研究を行ない,カジュダン・ルスティック型指標公式の予想を定式化した. 2.量子群の旗多様体の研究研究代表者は,引き続き量子群の旗多様体の研究を行ない,特に非可換スキーム論の観点から,層係数コホモロジー・微分作用素・D加群等の非可換版について,それらのよい定式化を求める試みを行なった. 3.ラドン・ペンローズ変換の研究研究代表者は,旗多様体上での拡張された意味でのラドン変換の研究を行ない,いわゆるBGG分解を用いて,ラドン変換の満たすスペクトル系列を得た. 4.正標数におけるD加群の研究分担者の兼田正治は,正標数における簡約代数群の旗多様体上のD加群について考察した.特にそのコホモロジー群の消滅・代数群の表現との関係に関して考察を行なった. 5.バーンズ型積分の研究分担者の三町勝久は,超幾何方程式・KZ方程式に関する研究を行なった.特にバーンズ型積分と種々の特殊多項式の関連に関するに詳しい考察を行ない,新たな結果を得た. 6.トロイダル代数の研究分担者の斉藤義久は,引き続きトロイダル代数に関する研究を行ない,アフィンリー代数との関係をさらに明らかにした.またこの観点から,ソリトン型方程式との関連について考察した. 7.高次元境界値問題の研究分担者の竹内潔は,代数解析的手法による微分方程式の研究を行なった.特に確定特異点型方程式の場合に高次元境界値問題に関する詳しい考察を行なった.
高余次元境界値問題と大域解析

竹内潔

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 奨励研究(A)

Institution: 広島大学

1997 - 1998

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高余次元境界値問題の研究においては、偏微分方程式の解の延長問題を研究してきた。すなはち、楕円型方程式の解の定義域が自動的に延長するという、柏原ー河合の定理をより一般の方程式系へ拡張する計画である。今年度は、当研究者が予想した最も広いクラスの系にまで結果を拡張することができ、しかもdistribution解の接続についての新しい結果や、確定特異点的な方程式系についての対応する結果を証明することができた(二つの結果とも掲載予定)。また、これらの研究に使用された層のマイクロ台をカットする手法やD-加群の消滅輪体の理論を利用して、Ramified Cauchy問題やE-加群に対するCauchy-Kowalevski型定理の研究を開始した。前者の研究では、D'Agnolo-Schapiraの結果の証明の簡易化に成功し、その様々なバリエーションが得られただけでなく、D-加群の正則関数解の複体の消滅輪体の超局所的視点からの研究が今後可能になると期待され、現在鋭意研究中である。またE-加群に対するCauchy-Kowalevski型定理の研究では、石村氏の定理の証明で不明解であった箇所に正しい証明を与えることに成功した。この結果はE-加群の逆像に関するものだが、順像すなはちE-加群の積分についての結果を最終目標としている。研究実施計画の「D-加群の積分変換」に取り組むための最初の一歩になることが期待される。また佐藤超関数解についてよく知られた偏微分方程式の解の消滅定理を、Andronikof氏の理論を用いることで関数空間がdistributionの場合にも証明した。これについては、フランスのColin氏が構成した柏原-Schapiraの函手の積分変換の理論を応用して、無限回微分可能関数での解の研究を開始した。以上の研究の他、D-加群の積分変換や特性サイクルの理論の指数定浬への応用、表現論とD-加群等の分野について見識をひろめるために、他の研究者と研究連絡をおこなった。特に最近のSchmid-Vilonenらによる表現論への応用についての学習に努めた。
代数群の表現論の代数解析学的研究

谷崎俊之, 竹内潔, 吉岡康太, 都築暢夫, 菅野孝史, 隅廣秀康

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 基盤研究(C)

Institution: 広島大学

1996 - 1996

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主として,筆者の定義したゲルファント超幾何方程式の拡張と,それに関連して生ずる表現論の問題について考察し,結果を得た. 以前の研究で,この方程式がホロノミー系になるためのひとつの十分条件は得られていたが,これについて更にリー群論的立場からの考察を行い,巾零軌道との関係を明らかにした. また,解の積分表示の存在について更に研究を進めた.どのような場合に解の積分表示がある種のラドン変換で与えられるかを定め,その場合のラドン変換の一般的性質について考察した.柏原正樹との共同研究で扱ったラドン変換は,射影直線に関するものであったが,今回は半単純リー群のある種の一般型放物部分群に関するものを対象とした.特に,ラドン変換の像がどのような方程式を満たすかという問題をD加群論の立場から研究した.ラドン変換の像が常にある種の方程式系(ヴァーマ加群と関連して具体的に書ける)をみたすことは証明できた.予想は,ラドン変換がもとの関係空間をその方程式系を満たす関数の全体に1対1にうつすということだが,全射性は有る意味で(D加群論的定式化のもとで)一般に言えた.単射性は,現在のところ一般線形群の場合のみが言えている.なお,この結果は実リー群の場合の大島利雄・関口英子の結果の複素ヴァージョンである.ある種のコホモロジー群の消滅が言えば,我々の結果から大島・関口の結果は従うのであるが,それは今後の問題である. その他,我々の超幾何方程式を定義する際の前提になっている,ヴァーマ加群の最大真部分加群についての有る事実の量子群版についても研究を行った.これは,今後我々の方程式を差分化する際に必要になるであろう基本的結果である.
Higher codimensional Boundary Valne Problems and Second Microlocalization Competitive

Institution: University of Tsukuba.
Algebraic analysis and its applications to representation theory Competitive

Institution: University of Tsukuba.

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