Applied Singularity Theory in Information Science and Stractural Design

• Project/Area Number: 18K18714
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: Hokkaido University
• Principal Investigator: Ohmoto Toru 北海道大学, 理学研究院, 教授 (20264400)
• Project Period (FY): 2018-06-29 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥6,500,000 (Direct Cost: ¥5,000,000、Indirect Cost: ¥1,500,000); Fiscal Year 2020: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000); Fiscal Year 2019: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000); Fiscal Year 2018: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
• Keywords: 特異点論 / ルジャンドル変換 / 接触幾何学 / 情報幾何学 / 応用幾何学 / Line Geometry / 双対平坦構造 / 統計多様体 / ラグランジュ・ルジャンドル特異点論 / カタストロフ理論 / パーシステントホモロジー / ルジャンドル特異点論 / 特異モデル / 機械学習 / リー環 / 情報幾何 / 応用特異点論 / ヘッセ多様体 / Line geometry / 特異統計モデル / アーキテクチュアル幾何 / トポロジカル絶縁体 / 応用特異点 / トポロジカル物性

Outline of Final Research Achievements

Based on Legendre singularity theory, we generalize the theory of dual flat structures (Amari-Nagaoka) which takes an important role in information geometry. This leads to a new theoretical approach to singular models in Deep Learning. We also provide a singularity-theoretic approach to applied geometry in structural design. This is a new development of local theory in classical projective differential geometry which has long been forgotten for hundred years after Darboux and Wilczynski.

Academic Significance and Societal Importance of the Research Achievements

本研究課題は，機械学習分野あるいはアーキテクチュラル幾何など，数学の外の世界に向けた応用特異点論の新しい萌芽的研究である．応用分野の諸課題に《特異点》は本質的に現れるにも関わらず，本格的な研究は未開拓であった．その端緒をひらく本研究の学術的意義は大きい．また，大学院生や若手研究者を積極的に共同研究に巻き込むことで，次世代に向けた新しい学際的研究の雛形を提示するものと考える．

Research Products

• [Int'l Joint Research] Warsawa Tech. University(ポーランド)
• [Int'l Joint Research] Jamia Millia Islamia(インド)
• [Int'l Joint Research] サンパウロ大学サンカルロス校/サンタカタリーナ連邦大学(ブラジル)
• [Int'l Joint Research] ワルシャワ工科大学(ポーランド)
• [Int'l Joint Research] サンパウロ大学サンカルロス校/サンタカタリーナ連邦大学(ブラジル)
• [Journal Article] Local normal forms of em-wavefronts in flat affine coordinates (2022)
  • Author(s): N. Nakajima
  • Journal Title: 京都大学数理解析研究所考究録 Volume: -
• [Journal Article] The dually flat structure for singular models (2021)
  • Author(s): N. Nakajiama and T. Ohmoto
  • Journal Title: Information Geometry Volume: 4 Issue: 1 Pages: 31-64
  • DOI: 10.1007/s41884-021-00044-8
  • Peer Reviewed
• [Journal Article] Symplectic singularities of differentiable mappings (2021)
  • Author(s): G. Ishikawa and S. Janeczko
  • Journal Title: Surveys in Differential Geometry Volume: 24 Issue: 1 Pages: 117-172
  • DOI: 10.4310/sdg.2019.v24.n1.a4
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Normal and tangent maps to frontals (2021)
  • Author(s): G. Ishikawa
  • Journal Title: Journal of Mathematical Sciences Volume: 255 Issue: 5 Pages: 664-677
  • DOI: 10.1007/s10958-021-05403-5
  • Peer Reviewed
• [Journal Article] A characterization of the alpha-connections on the statistical manifold of normal distributions (2021)
  • Author(s): H. Furuhata, J. Inoguchi, S. Kobayashi
  • Journal Title: Information Geometry Volume: 4 Issue: 1 Pages: 177-188
  • DOI: 10.1007/s41884-020-00037-z
  • Peer Reviewed
• [Journal Article] Statistical submanifolds from a viewpoint of the Euler inequality (2021)
  • Author(s): N. Satoh, H. Furuhata, I. Hasegawa, T. Nakane, Y. Okuyama, K. Sato, M. Hasan Shahid, A. Naaz Siddiqui
  • Journal Title: Information Geometry Volume: 4 Issue: 1 Pages: 189-213
  • DOI: 10.1007/s41884-020-00032-4
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Singular mappings and their zero-forms (2020)
  • Author(s): Goo Ishikawa, Stanislaw Janeczko,
  • Journal Title: Pure and Applied Mathematics Quarterly Volume: 16 Pages: 1643-4657
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Recognition Problem of Frontal Singularities (2020)
  • Author(s): Ishikawa Goo
  • Journal Title: Journal of Singularities Volume: 21 Pages: 149-166
  • DOI: 10.5427/jsing.2020.21i
  • Peer Reviewed / Open Access
• [Journal Article] Geometric algebra and singularities of ruled and developable surfaces (2020)
  • Author(s): J. Tanaka, To. Ohmoto
  • Journal Title: Journal of Singularities Volume: 21 Pages: 249-267
  • DOI: 10.5427/jsing.2020.21o
  • Peer Reviewed / Open Access
• [Journal Article] On families of Lagrangian submanifolds (2020)
  • Author(s): S. Izumiya, M. Takahashi
  • Journal Title: Journal of Singularities Volume: 21 Pages: 188-207
  • DOI: 10.5427/jsing.2020.21j
  • Peer Reviewed / Open Access
• [Journal Article] Application of Singularity Theory to Bifurcation of Band Structures in Crystals (2020)
  • Author(s): Teramoto H.、Tsuchida A.、Kondo K.、Izumiya S.、Toda M.、Komatsuzaki T.
  • Journal Title: Journal of Singularities Volume: 21 Pages: 268-281
  • DOI: 10.5427/jsing.2020.21p
  • Peer Reviewed / Open Access
• [Journal Article] Leibniz complexity of Nash functions on differentiations (2019)
  • Author(s): G Ishikawa, T. Yamashita
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 71
  • Peer Reviewed
• [Journal Article] Cofrontals (2019)
  • Author(s): G. Ishikawa
  • Journal Title: Romanian Journal of Pure and Applied Mathematics Volume: 60
  • Peer Reviewed / Open Access
• [Journal Article] Geometric invariants of 5/2-cuspidal edges (2019)
  • Author(s): A. Honda, K. Saji
  • Journal Title: Kodai Math. J. Volume: 42
  • Peer Reviewed
• [Journal Article] Singularities of a surface given by Kenmotsu-type formula in Euclidean three-space (2019)
  • Author(s): L. Martins, K. Saji, K. Teramoto
  • Journal Title: Sao Paulo J. Math. Sci. Volume: 13
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Evoltoids and pedaloids of plane curves (2019)
  • Author(s): S. Izumiya, N. Takeuchi
  • Journal Title: Note di Matematica Volume: 39
  • Peer Reviewed / Open Access
• [Journal Article] Lightcone dualities for submanifolds in the sphere (2019)
  • Author(s): Y. Jiang, S. Izumiya
  • Journal Title: Houston Journal of Mathematics Volume: 45
  • Peer Reviewed / Open Access
• [Journal Article] Caustics and Maxwell sets of world sheets in anti-de Sitter space (2018)
  • Author(s): S. Izumiya
  • Journal Title: Advances in Theoretical and Mathematical Physics Volume: 22 Issue: 3 Pages: 759-802
  • DOI: 10.4310/atmp.2018.v22.n3.a5
  • Peer Reviewed
• [Journal Article] Evolutes of curves in the Lorentz-Minkowski plane (2018)
  • Author(s): S. Izumiya, M. C. Romero Fuster, M. Takahashi
  • Journal Title: Adv. Stud. Pure Math. Volume: 78 Pages: 313-330
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The theory of graph-like Legendrian unfoldings : Equivalence relations (2018)
  • Author(s): Izumiya Shyuichi
  • Journal Title: Adv. Stud. Pure Math. Volume: 78 Pages: 107-162
  • Peer Reviewed
• [Journal Article] Duals of timelike Sabban curves in de Sitter n-space (2018)
  • Author(s): S. Izumiya、Y. Wang
  • Journal Title: Journal of Geometry Volume: 109 Issue: 2
  • DOI: 10.1007/s00022-018-0438-y
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Singularities of tangent surfaces to directed curves (2018)
  • Author(s): Ishikawa G.、Yamashita T.
  • Journal Title: Topology and its Applications Volume: 234 Pages: 198-208
  • DOI: 10.1016/j.topol.2017.11.018
  • Peer Reviewed / Open Access
• [Journal Article] Singularities of frontals (2018)
  • Author(s): G. Ishikawa
  • Journal Title: Adv. Stud. Pure Math. Volume: 78 Pages: 55-106
  • Peer Reviewed
• [Journal Article] Fold singularities on spacelike CMC surfaces in Lorentz-Minkowski space (2018)
  • Author(s): A. Honda, M. Koiso and K. Saji
  • Journal Title: Hokkaido Math. J. Volume: 47 Pages: 245-267
  • Peer Reviewed
• [Journal Article] Normal form of the swallowtail and its applications (2018)
  • Author(s): K. Saji
  • Journal Title: Internat. J. Math. Volume: 29 Issue: 07 Pages: 1850046-1850046
  • DOI: 10.1142/s0129167x18500465
  • Peer Reviewed
• [Journal Article] On the geometry of folded cuspidal edges (2018)
  • Author(s): Oset Sinha Raul、Saji Kentaro
  • Journal Title: Revista Matematica Complutense Volume: - Issue: 3 Pages: 627-650
  • DOI: 10.1007/s13163-018-0257-6
  • Peer Reviewed / Int'l Joint Research
• [Presentation] 双極小統計部分多様体といくつかの問題 (2022)
  • Author(s): 古畑 仁
  • Organizer: Tsudoi KK
• [Presentation] 統計部分多様体に対する不等式 (2022)
  • Author(s): 古畑 仁
  • Organizer: 福岡大学微分幾何セミナー
  • Invited
• [Presentation] Persistent characteristic classes and Riemann-Roch (2021)
  • Author(s): T. Ohmoto
  • Organizer: Asia Pacific Seminar on Applied Topology and Geometry (Online)
  • Int'l Joint Research / Invited
• [Presentation] Singularities of parallels to tangent developable surfaces (2021)
  • Author(s): G. Ishikawa
  • Organizer: 24-th Petrovskii Conference, Differential Equations and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] 統計多様体の部分多様体論 (2021)
  • Author(s): 古畑 仁
  • Organizer: 日本数学会２０２１年度秋季総合分科会幾何学分科会
  • Invited
• [Presentation] アファイン座標系における em-波面の局所的標準形 (2021)
  • Author(s): 中島直道
  • Organizer: 研究集会「特異点論の未来」
• [Presentation] アファイン座標系における e/m-波面の局所的標準形 (2021)
  • Author(s): 中島直道
  • Organizer: 第68回幾何学シンポジウム
  • Invited
• [Presentation] Local normal forms of em-wavefronts in flat affine coordinates (2021)
  • Author(s): 中島直道
  • Organizer: RIMS短期共同研究集会「可微分写像の特異点論及びその応用」
  • Invited
• [Presentation] Applied Singularity Theory -- an updated new catastrophe theory (2021)
  • Author(s): T. Ohmoto
  • Organizer: International Symposium on Big-Data, Cybersecurity and IoT (Online)
  • Int'l Joint Research / Invited
• [Presentation] Information Geometry from Singularity Theory Viewpoint (2020)
  • Author(s): 大本 亨・中島 直道
  • Organizer: 研究集会「接触構造、特異点、微分方程式及びその周辺」(オンライン)
• [Presentation] 情報幾何への特異点論的アプローチ (2020)
  • Author(s): 大本 亨・中島 直道
  • Organizer: ワークショップ「統計多様体の幾何学とその周辺」(オンライン)
  • Invited
• [Presentation] Information Geometry from Singularity Theory Viewpoint (2020)
  • Author(s): N. Nakajima and T. Ohmoto
  • Organizer: 16th International Workshop on Real and Complex Singularities
  • Int'l Joint Research
• [Presentation] Singularities of parallels to tangent maps of frontals (2020)
  • Author(s): Goo Ishikawa
  • Organizer: 16th International Workshop on Real and Complex Singularities
  • Int'l Joint Research
• [Presentation] Tangential and Normal Maps to Frontals (2020)
  • Author(s): Goo Ishikawa
  • Organizer: The International Conference on Differential Equations and Dynamical Systems 2020
  • Int'l Joint Research
• [Presentation] On K[SO(p)]-equivalence among map germs (2019)
  • Author(s): S. Izumiya
  • Organizer: 6th International Workshop on Singularities in Generic Geometry and its Applications (Valencia VI)
  • Int'l Joint Research / Invited
• [Presentation] Singularities of Cauchy horizons in globally hyperbolic space-times (2019)
  • Author(s): S. Izumiya
  • Organizer: Singularity Theory and its Applications to Differential Equations and Differential Geometry (Moscow)
  • Int'l Joint Research / Invited
• [Presentation] Several problems on singularities of implicit differential equations and related topics (2019)
  • Author(s): G. Ishikawa
  • Organizer: Singularity Theory and its Applications to Differential Equations and Differential Geometry (Moscow)
  • Int'l Joint Research / Invited
• [Presentation] Geometric invariants of lightlike points of surfaces in 3-dimensional Lorentzian manifolds (2019)
  • Author(s): K. Saji
  • Organizer: 6th International Workshop on Singularities in Generic Geometry and its Applications
  • Int'l Joint Research / Invited
• [Presentation] Contact-like relation of vector fields and functions and its applications (2019)
  • Author(s): K. Saji
  • Organizer: Hyperplane arrangements and Japanese Australian workshop on Real and Complex Singularities
  • Int'l Joint Research / Invited
• [Presentation] 滑らかな写像芽の間の幾何学的同値関係 ーThom-Mather理論へのオマージュー (2019)
  • Author(s): 泉屋周一
  • Organizer: 「日本数学会年会」総合講演（東京工業大学）
  • Invited
• [Presentation] Topology of complements to real affine space line arrangements (2019)
  • Author(s): Goo Ishikawa, Motoki Oyama,
  • Organizer: Western Joint Sectional Meeting, American Mathematical Society. Hawaii, USA.
  • Invited
• [Presentation] Statistical submanifolds and warped product spaces (2019)
  • Author(s): H. Furuhata
  • Organizer: Information Geometry and Affine Differential Geometry III
  • Invited
• [Presentation] Topics on frontal singularities (2018)
  • Author(s): G. Ishikawa
  • Organizer: 15th International Workshop on Real and Complex Singularities, Sao Carlos, Brazil
  • Invited
• [Presentation] SO(3) normal forms and differential geometric invariants of singular surfaces (2018)
  • Author(s): K. Saji
  • Organizer: 15th International Workshop on Real and Complex Singularities, Sao Carlos, Brazil
  • Invited
• [Funded Workshop] The 14th Algebra-Analysis-Geometry Seminar, Kagoshima (2019)