| Project/Area Number |
15H02055
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Meiji University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
楯 辰哉 東北大学, 理学研究科, 教授 (00317299)
樋口 雄介 昭和大学, 教養部, 講師 (20286842)
赤間 陽二 東北大学, 理学研究科, 准教授 (30272454)
内藤 久資 名古屋大学, 多元数理科学研究科, 准教授 (40211411)
阿原 一志 明治大学, 総合数理学部, 専任教授 (80247147)
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥18,200,000 (Direct Cost: ¥14,000,000、Indirect Cost: ¥4,200,000)
Fiscal Year 2018: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2017: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2016: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2015: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
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| Keywords | 位相的結晶 / 一般化されたリーマン和 / 算術的準結晶 / 結晶的tight frame / 準結晶 / 算術的離散集合 / 和公式 / 結晶構造の標準的実現 / 離散幾何解析 / 包含排除の原理 / 原始的格子点 / 原始的ピタゴラス数 / 一様配置 / tight frame / crystallography / グラスマン多様体 / 有理点 |
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| Outline of Final Research Achievements |
In this research, I treated several topics in mathematical crystallography. Especially motivated by the recent development in systematic design of crystal structures, I discussed interesting relationships among seemingly irrelevant subjects; say, standard crystal models, tight frames in the Euclidean space, rational points on Grassmannian, and quadratic Diophantine equations. The central object in this study is what I call crystallographic tight frames, which are considered a generalization of root systems. I also made a remark on the connections with tropical geometry, a relatively new area in mathematics, specifically with combinatorial analogues of the Abel-Jacobi map and Abel's theorem.
What is more, I explained how the idea of Riemann sum is linked to other branches of mathematics; for instance, some counting problems in elementary number theory and the theory of quasicrystals, the former having a long history and the latter being an active field still in a state of flux.
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| Academic Significance and Societal Importance of the Research Achievements |
本研究は、物質科学に密接に関連しており、特に結晶と準結晶の数学的理論を通して、結晶デザインの系統的方法を与えることにより、現実の社会に貢献している。
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