Geometry for Materials based on discrete surface theory

• Project/Area Number: 15K13432
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tohoku University
• Principal Investigator: Kotani Motoko 東北大学, 理学研究科, 教授 (50230024)
• Co-Investigator(Renkei-kenkyūsha): Naito Hisashi 名古屋大学, 多元数理科学研究科, 准教授 (40211411)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: 離散幾何学 / スペクトル幾何 / 物性物理

Outline of Final Research Achievements

We call a trivalent graph in the 3-dimensional Euclidean space a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. We develop a discrete surface theory on trivalent graphs, and define several geometric notions such as area, area variation formula, curvature, mean curvature. To abstract a continuum object hidden in the discrete surface, we introduce a subdivision method by applying the Goldberg-Coxeter subdivision, and discuss the convergence of a sequence of discrete surfaces defined inductively by the subdivision. We also study the limit set as the continuum geometric objects associated with the given discrete surface.

Research Products

• [Journal Article] A discrete surface theory (2017)
  • Author(s): Motoko Kotani, Hisashi Naito, Toshiaki Omori
  • Journal Title: Computer Aided Geometric Design Volume: 58 Pages: 24-54
  • DOI: 10.1016/j.cagd.2017.09.002
  • Peer Reviewed / Open Access
• [Journal Article] Asymptotic expansion of the transition probability for non-symmetric random walks on crystal lattices（スペクトル・散乱理論とその周辺 (2017)
  • Author(s): Ishiwata Satoshi, Kawabi Hiroshi, Kotani Motoko
  • Journal Title: 数理解析研究所講究録 Volume: 2023 Pages: 15-28
  • Peer Reviewed
• [Journal Article] Long time asymptotics of non-symmetric random walks on crystal lattices (2017)
  • Author(s): Satoshi Ishiwata, Hiroshi Kawabi, Motoko Kotani
  • Journal Title: Journal of Functional Analysis Volume: 272 Issue: 4 Pages: 1553-1624
  • DOI: 10.1016/j.jfa.2016.11.011
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Analysis of Periodic Atomic Structures in Grain Boundaries by Number Theory (2017)
  • Author(s): 井上和俊、斎藤光浩、小谷元子、幾原雄一
  • Journal Title: Materia Japan Volume: 56 Issue: 10 Pages: 589-596
  • DOI: 10.2320/materia.56.589
  • NAID: 130006109914
  • ISSN: 1340-2625, 1884-5843
  • Peer Reviewed / Open Access
• [Journal Article] Structure of <110>-tilt boundaries in cubic zirconia (2017)
  • Author(s): Kazutoshi Inoue, Bin Feng, Naoya Shibata, Motoko Kotani, Yuichi Ikuhara
  • Journal Title: Journal of Materials Science Volume: 52 Issue: 8 Pages: 4278-4287
  • DOI: 10.1007/s10853-016-0682-1
  • Peer Reviewed
• [Journal Article] Stereoisomerism in Nanohoops with Heterogeneous Biaryl Linkages of E/Z- and R/S-Geometries (2016)
  • Author(s): P. Sarkar, Z. Sun, T. Tokuhira, M. Kotani, S. Sato, and H.Isobe
  • Journal Title: ACS Cent. Sci. Volume: 2 Issue: 10 Pages: 740-747
  • DOI: 10.1021/acscentsci.6b00240
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Stereoisomerism, Crystal Structures, and Dynamics of Belt-shaped Cyclonaphthylenes (2016)
  • Author(s): Z. Sun, T. Suenaga, P. Sarkar, S. Sato, M. Kotani, and H.Isobe
  • Journal Title: Proc. Natl. Acad. Sci. U.S.A. Volume: 113 Issue: 29 Pages: 8109-8114
  • DOI: 10.1073/pnas.1606530113
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Materials inspired by mathematics (2016)
  • Author(s): Motoko Kotani & Susumu Ikeda
  • Journal Title: Science and Technology of Advanced Materials(STAM) Volume: 17(1) Issue: 1 Pages: 253-259
  • DOI: 10.1080/14686996.2016.1180233
  • Peer Reviewed
• [Journal Article] Phase behavior of a binary fluid mixture of quadrupolar molecules (2016)
  • Author(s): M. Toda, S. Kajimoto, S. Toyouchi, T. Kawakatsu, Y. Akama, M. Kotani, H. Fukumura
  • Journal Title: Phys. Rev. E, Volume: 94 Issue: 5 Pages: 052601-052601
  • DOI: 10.1103/physreve.94.052601
  • Peer Reviewed
• [Journal Article] Construction of Negatively Curved Cubic Carbon Crystals via Standard Realizations (2016)
  • Author(s): Hisashi Naito
  • Journal Title: Springer Proc. Math. Stat. Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Numerical studies of the optimization of the first eigenvalue of the heat diffusion in inhomogenoues media (2015)
  • Author(s): Kagame Matsue and Hisashi Naito
  • Journal Title: Japan Journal of Industrial and Applied Mathematics Volume: 32 Issue: 2 Pages: 489-512
  • DOI: 10.1007/s13160-015-0177-5
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] 化学と幾何学 : 離散幾何学と炭素構造 (2015)
  • Author(s): 内藤 久資
  • Journal Title: 数理科学 Volume: 624 Pages: 42-47
• [Journal Article] The Decomposition Formula of &lang;001&rang; Symmetrical Tilt Grain Boundaries (2015)
  • Author(s): K. Inoue, M. Saito, Z. C. Wang, M. Kotani, Y. Ikuhara
  • Journal Title: MATERIALS TRANSACTIONS Volume: 56 Issue: 12 Pages: 1945-1952
  • DOI: 10.2320/matertrans.M2015277
  • NAID: 130005110139
  • ISSN: 1345-9678, 1347-5320
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] On the Periodicity of &lang;001&rang; Symmetrical Tilt Grain Boundaries (2015)
  • Author(s): K. Inoue, M. Saito, Z. C. Wang, M. Kotani, Y. Ikuhara
  • Journal Title: MATERIALS TRANSACTIONS Volume: 56 Issue: 3 Pages: 281-287
  • DOI: 10.2320/matertrans.M2014394
  • NAID: 130004825278
  • ISSN: 1345-9678, 1347-5320
  • Peer Reviewed / Open Access
• [Journal Article] Visualization of topological landscape in shear-flow dynamics of amorphous solids (2015)
  • Author(s): Takeshi Fujita, Koji Ohara, Keiji Miura, Akihiko Hirata, Motoko Kotani, Yasumasa Nishiura, Mingwei Chen
  • Journal Title: EPL (Europhysics Letters) Volume: 110 Issue: 3 Pages: 38002-38002
  • DOI: 10.1209/0295-5075/110/38002
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] Mathematical challenges for materials design (2018)
  • Author(s): Motoko Kotani
  • Organizer: Total Energy and Force Methods 2018 Workshop
  • Int'l Joint Research / Invited
• [Presentation] Analysis of phase transitions using minimal surfaces (2018)
  • Author(s): Hisashi Naito
  • Organizer: Discrete Geometric Analysis for Materials Design
  • Int'l Joint Research / Invited
• [Presentation] Structure understanding of materials by discrete geometric analysis (2017)
  • Author(s): Motoko Kotani
  • Organizer: Novel Optical Materials
  • Place of Presentation: Keller Hall, The IMA (Minneapolis, USA)
  • Year and Date: 2017-03-16
  • Int'l Joint Research / Invited
• [Presentation] 三分岐離散調和曲面の収束 (2017)
  • Author(s): 内藤久資
  • Organizer: 多様体上の微分方程式
  • Invited
• [Presentation] 炭素構造と離散幾何解析 (2017)
  • Author(s): 内藤久資
  • Organizer: 幾何学を指標とする化学構造と機能物性
  • Invited
• [Presentation] 数学による構造・機能相関の解明 (2017)
  • Author(s): 小谷 元子
  • Organizer: 17-1ポリマーフロンティア21
  • Invited
• [Presentation] Structure understanding of materials by discrete geometric analysis (2017)
  • Author(s): Kotani, Motoko
  • Organizer: International Workshop on Industrial Mathematics
  • Int'l Joint Research / Invited
• [Presentation] Mathematical Challenge to Understanding of Material (2017)
  • Author(s): Motoko Kotani
  • Organizer: International Conference on Research and Education in Mathematics 2017 (ICREM8)
  • Int'l Joint Research / Invited
• [Presentation] Discrete surface theory for trivalent graphs and its application to materials science (2017)
  • Author(s): Motoko Kotani
  • Organizer: Geometry Workshop in Obergurgl 2017
  • Int'l Joint Research / Invited
• [Presentation] 次世代物質う探索のための離散幾何学 領域紹介 (2017)
  • Author(s): 小谷 元子
  • Organizer: 第11回 物性科学領域横断研究会（領域合同研究会）
  • Invited
• [Presentation] 数学から材料科学へのアプローチ (2017)
  • Author(s): 小谷 元子
  • Organizer: シンポジウム「高分子未来サミット」
  • Invited
• [Presentation] Mathematical challenge for materials science (2017)
  • Author(s): Motoko Kotani
  • Organizer: 2017 Dasan Conference Mathematical Challenges and Industrial Problem Solving
  • Int'l Joint Research / Invited
• [Presentation] Carbon structures and trivalent discrete surfaces (2016)
  • Author(s): Hisashi Naito
  • Organizer: A3 Foresight Program Modeling and Simulation of Hierarchical and Heterogeneous Flow System with Applications to Materials Science
  • Place of Presentation: Tohoku University(仙台)
  • Year and Date: 2016-11-16
  • Int'l Joint Research / Invited
• [Presentation] Carbon structures and trivalent discrete surfaces (2016)
  • Author(s): Hisashi Naito
  • Organizer: GEMS 2016 Geometry and Materials Sciences
  • Place of Presentation: OIST(沖縄県)
  • Year and Date: 2016-10-16
  • Int'l Joint Research / Invited
• [Presentation] Topology in Materials (2016)
  • Author(s): Motoko Kotani
  • Organizer: IGA/AMSI Workshop Topological Matter, Strings, K-theory and related areas
  • Place of Presentation: The University of Adelaide(Adelaid, Australia)
  • Year and Date: 2016-09-26
  • Int'l Joint Research / Invited
• [Presentation] Figure Materials (2016)
  • Author(s): Motoko Kotani
  • Organizer: 16th International Conference on Computational and Mathematical Methods in Science and Engineering
  • Place of Presentation: BARCELO COSTA BALLENA HOTEL ( Cadiz, Spain)
  • Year and Date: 2016-07-05
  • Int'l Joint Research / Invited
• [Presentation] Mathematical Challenge to a New Phase of Materials Science (2016)
  • Author(s): 小谷 元子
  • Organizer: Workshop on nonlinear partial differential equations and related topics - In honor of Professor Atsushi Tachikawa on his 60th birthday
  • Place of Presentation: 石川県文教会館
  • Year and Date: 2016-05-09
  • Invited
• [Presentation] Discrete geometric analysis and fullerene structures (2016)
  • Author(s): Hisashi Naito
  • Organizer: The A3 mini-workshop on soft matter
  • Place of Presentation: Beihang University, China
  • Year and Date: 2016-03-24
  • Int'l Joint Research / Invited
• [Presentation] Mathematical challenge to structural understanding of materials (2015)
  • Author(s): Motoko Kotani
  • Organizer: 2nd International Symposium on Frontiers in Materials Science (FMS)
  • Place of Presentation: 東京 早稲田大学西早稲田キャンパス
  • Year and Date: 2015-11-21
  • Int'l Joint Research / Invited
• [Presentation] Mathematical challenge to structural understanding of Materials (2015)
  • Author(s): Motoko Kotani
  • Organizer: Geometric analysis in geometry and topology 2015
  • Place of Presentation: 東京理科大学（神楽坂），森戸記念館
  • Year and Date: 2015-11-10
  • Int'l Joint Research / Invited
• [Presentation] Mathematical challenge to structural understanding and structure of materials (2015)
  • Author(s): Motoko Kotani
  • Organizer: 15th International Conference Computational and Mathematical Methods in Science and Engineerin
  • Place of Presentation: Costa Ballena, Cadiz, Spain
  • Year and Date: 2015-07-07
  • Int'l Joint Research / Invited
• [Presentation] Mathematical Challenge to Structural Understandingof Materials (2015)
  • Author(s): Motoko Kotani
  • Organizer: Mathematics of NovelMaterials
  • Place of Presentation: Mittag-Leffler Institute, Djursholm, Sweden
  • Year and Date: 2015-06-01
  • Int'l Joint Research / Invited