Study of the Analysis on Manifolds

• Project/Area Number: 17K05284
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Osaka City University (2020-2021); Tokyo University of Science (2017-2019)
• Principal Investigator: Furutani Kenro 大阪市立大学, 数学研究所, 特別研究員 (70112901)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2017: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: Clifford module / pseudo H-type Lie環(群) / exotic sphere / sub-Riemann構造 / 劣楕円型作用素 / Fourier積分作用素 / spectral zeta関数 / Maslov 量子化条件 / Cayley射影平面 / Calabi-Yau構造 / 幾何学的量子化 / Maslov index / Lagrangian distribution / pseudo H-type Lie代数 / spectral zeta 関数 / Clifford algebra / admissible module / pseudo H-type algebra / spectral inverse problem / Riemann zeta function / Calabi-Yau structure / 国際共同研究 / 大域解析学 / sub-Riemann 構造 / pseudo H type Lie 環 / isospectral problem / sub-Riemann structure / Quantization condition / Fourier 積分作用素 / Weyl asymptotics / heat kernel / non-holonomic structure / sub-Laplacian / Grushin type operator / bi-characteristic flow / Popp’s measure / 関数解析学 / 熱核

Outline of Final Research Achievements

(1) We proved an existence of a sub-Riemanian structure of co-dimension 3 on the Gromoll-Meyer exotic sphere. (2) We made clear that the Maslov quantization condition works also for sub-Riemanian cases. (3) We classified Lie algebras attached to Clifford algebras and determined all their automorphism groups. (4) We determine the spectral zeta functions for these class of nil-manifords and also find a difference from the Riemannian case on the behavior of residues. (5) We constructed a Bargmann type transformation on the Cayley projective plane.

Academic Significance and Societal Importance of the Research Achievements

本研究の対象である幾何構造(= sub-Riemann構造)と付随する劣楕円型作用素(= sub-Laplacian)の大域的な研究の歴史はそう古くはないので、以前の研究からも引き続き研究対象がどの辺にあるのかの研究も継続し、豊富な例が多方面にあることが徐々に分かって来た。特にこの期間の研究で具体例として重要な、Clifford代数に付随するベキ零Lie環(群)についての基礎的構造の研究はほぼ完成し、更に今後続けるべく多くの問題、例えば本研究はコロナで延長期であるので既に次期の科研費研究で同時に、この群の一様離散部分群の研究を始め、それはspectral zeta関数の研究にも繋がっている。

Research Products

• [Int'l Joint Research] University of Bergen(ノルウェー)
• [Int'l Joint Research] Leibniz University Hannover(ドイツ)
• [Int'l Joint Research] University of Bergen(ノルウェー)
• [Int'l Joint Research] Leibniz university of Hannover(ドイツ)
• [Int'l Joint Research] Roskilde university(デンマーク)
• [Int'l Joint Research] University of Bergen(ノルウェー)
• [Int'l Joint Research] Leibniz University of Hannover(ドイツ)
• [Int'l Joint Research] Roskilde University(デンマーク)
• [Int'l Joint Research] University of Bergen(ノルウェー)
• [Int'l Joint Research] Universidad de La Frontera(チリ)
• [Int'l Joint Research] Leibniz University of Hannover(ドイツ)
• [Int'l Joint Research] University of Bergen(Norway)
• [Int'l Joint Research] Leibniz University of Hannover(Germany)
• [Int'l Joint Research] Universidad de la Frontera(チリ)
• [Int'l Joint Research] Georgetown University(米国)
• [Int'l Joint Research] Roskilde University(Denmark)
• [Journal Article] AUTOMORPHISM GROUPS OF PSEUDO H-TYPE ALGEBRAS (2021)
  • Author(s): Furutani Kenro、Markina Irina
  • Journal Title: Journal of Algebra Volume: 20 Issue: 14 Pages: 1-37
  • DOI: 10.1016/j.jalgebra.2020.09.038
  • URL: https://ocu-omu.repo.nii.ac.jp/records/2016902
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Spectral theory of a class of nilmanifolds attached to Clifford modules (2020)
  • Author(s): Wolfram Bauer, Kenro Furutani, Chisato Iwasaki and Abdellah Laaroussi
  • Journal Title: Mathematische Zeitshirift Volume: - Issue: 1-2 Pages: 557-583
  • DOI: 10.1007/s00209-020-02525-5
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Residue of the spectral zeta function of a sub-Laplacian on a certain class of nilmanifolds (2020)
  • Author(s): Kenro Furutani
  • Journal Title: Applied Analysis and Optimization Volume: -
  • Peer Reviewed
• [Journal Article] Riemann submersion and Maslov quantization condition (2018)
  • Author(s): Kenro Furutani, Mitsuji Tamura
  • Journal Title: Applied Analysis and Optimization Volume: 2 Pages: 373-401
  • Peer Reviewed / Open Access
• [Journal Article] Complete classification of pseudo H-type algebras:II (2018)
  • Author(s): Kenro Furutani, Irina Markina
  • Journal Title: Geometriae Dedicata Volume: - Pages: 1-31
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Lie algebras attached to Clifford modules and simple graded Lie algebras (2018)
  • Author(s): Kenro Furutani, Irina Markina, Mauricio Godoy Molina, Tohru Morimoto, Alexander Vasiliev
  • Journal Title: Journal of Lie Theory Volume: 28 Pages: 843-864
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Double submersions and Hamilton flows (2017)
  • Author(s): Kenro Furutani & Mitsuji Tamura
  • Journal Title: Applied Analysis and Optimization Volume: 1 Pages: 381-400
  • Peer Reviewed
• [Journal Article] First Integrals of bi-characteristic curves of a sub-Laplacian and related Grushin type operators (2017)
  • Author(s): Wolfram Bauer, Kenro Furutani & Mitsuji Tamura
  • Journal Title: Journal of Non-linear and Convex Analysis Volume: 18 Pages: 893-917
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Sub-Riemannian structures in a principal bundle and their Popp measures (2017)
  • Author(s): Wolfram Bauer, Kenro Furutani & Chisato Iwasaki
  • Journal Title: Applicable Analysis Volume: 96 Issue: 14 Pages: 2390-2407
  • DOI: 10.1080/00036811.2017.1333602
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A codimension 3 sub-Riemannian structure on Gromoll-Meyer exotic sphere (2017)
  • Author(s): Wolfram Bauer, Kenro Furutani & Chisato Iwasaki
  • Journal Title: Differential Geometry and its Applications Volume: 53 Pages: 114-136
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Calabi-Yau structure and Bargmann type transformation on the Cayley projective plane (2021)
  • Author(s): 古谷賢朗
  • Organizer: 立命館大学幾何学セミナー
  • Invited
• [Presentation] Calabi-Yau structure and Bargmann type transformation on the Cayley projective plane (2021)
  • Author(s): Kenro Furutani
  • Organizer: Microlocal and Global Analysis, Interactions with Geometry at Potsdam University , Germany
  • Int'l Joint Research / Invited
• [Presentation] Calabi-Yau structure and Bargmann type transformation on the Cayley projective plane (2021)
  • Author(s): Kenro Furutani
  • Organizer: Himeji conference on Partial Differential Equations
  • Int'l Joint Research / Invited
• [Presentation] Complete classification of Lie algebras attached to Clifford modules and an application (2019)
  • Author(s): 古谷賢朗
  • Organizer: 信州大学topology seminar
  • Invited
• [Presentation] Eigenvalue theorem for sub-Laplacians and Lagrangian submanifolds satisfying Maslov quantization condition (2019)
  • Author(s): Kenro Furutani
  • Organizer: 2019 NCTS International Workshop on Geometric and Harmonic Analysis at National Taiwan University
  • Int'l Joint Research / Invited
• [Presentation] Weyl-type law for a sub-Laplacian (2019)
  • Author(s): Kenro Furutani
  • Organizer: International Workshop on Applied Analysis and Optimization 2019 at China Medical University, Taichung, Taiwan
  • Int'l Joint Research / Invited
• [Presentation] Complete classification of nilpotent Lie algebras attached to a certain kind of Clifford modules and an application (2019)
  • Author(s): 古谷賢朗
  • Organizer: 広島大学 Topology-Geometry seminar
  • Invited
• [Presentation] Eigenvalue theorem for sub-Laplacian and Lagrangian submanifolds satisfying Maslov quantization condition (2019)
  • Author(s): Kenro Furutani
  • Organizer: Geometry and Mathematical Physics seminar at Copenhagen university, Denmark
  • Int'l Joint Research / Invited
• [Presentation] Eigenvalue theorem for sub-Laplacian and behavior of Lagrangian submanifolds under submersion (2019)
  • Author(s): Kenro Furutani
  • Organizer: Workshop Analysis and PDE, Leibniz University at Hannover, Germany
  • Int'l Joint Research / Invited
• [Presentation] From elliptic operators to sub-elliptic operators (2019)
  • Author(s): 古谷賢朗
  • Organizer: Memorial workshop at Kyoto University for late professor H. Yoshizawa
  • Invited
• [Presentation] Lie algebras attached to a class of Clifford modules; existence of lattices, their classification and automorphism groups (2019)
  • Author(s): 古谷賢朗
  • Organizer: 立命館大学幾何学セミナー
  • Invited
• [Presentation] Complete classification of pseudo H-type algebras and an application to Kac's problem with respect to Laplacian and sub-Laplacian (2019)
  • Author(s): 古谷賢朗
  • Organizer: 研究集会：多様体上の微分方程式 at 金沢大学
  • Invited
• [Presentation] Riemann submersion and Maslov quantization condition (2019)
  • Author(s): 古谷賢朗
  • Organizer: 金沢大学数理学談話会
  • Invited
• [Presentation] Riemann submersion and Maslov quantization condition (2019)
  • Author(s): Kenro Furutani
  • Organizer: 2018 Naha Workshop on Mathematical Analysis
  • Int'l Joint Research / Invited
• [Presentation] Submersion and Maslov quantization condition (2018)
  • Author(s): Kenro Furutani
  • Organizer: 2018 IWAAO at China Medical University
  • Int'l Joint Research / Invited
• [Presentation] Automorphism groups of pseudo H-type groups (2018)
  • Author(s): Kenro Furutani
  • Organizer: 2018 NCTS Geometric and Harmonic Analysis
  • Int'l Joint Research / Invited
• [Presentation] Geometry and Analysis of non-holonomic structure (2018)
  • Author(s): 古谷賢朗
  • Organizer: 靜岡大学複素解析セミナー
  • Invited
• [Presentation] Complete classification of pseudo H-type algebras and an application (2017)
  • Author(s): 古谷賢朗
  • Organizer: 微分方程式と幾何学、立命館大学研究集会
  • Invited
• [Presentation] Sub-Riemannian structures in a principal bundle and their Popp mesures (2017)
  • Author(s): Kenro Furutani
  • Organizer: 2017 International Workshop on Applied Analysis and Optimization, China medical university, Taichung, Taiwan
  • Int'l Joint Research / Invited
• [Presentation] Complete classification of pseudo H-type algebras and an application (2017)
  • Author(s): Kenro Furutani
  • Organizer: 2017 NCTS International Workshop on Harmonic Analysis and Geometric Analysis, National Taiwan university, Taipei, Taiwan
  • Int'l Joint Research / Invited
• [Funded Workshop] 幾何構造と微分方程式ー対称性・特異性及び量子化の視点からー (2022)
• [Funded Workshop] 幾何構造と微分方程式ー対称性と特異点の視点からー (2018)