The development of discrete geometric analysis

2018 Fiscal Year Final Research Report

• Project/Area Number: 15H02055
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Meiji University
• Principal Investigator: Sunada Toshikazu 明治大学, 総合数理学部, 専任教授 (20022741)
• 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)
• Project Period (FY): 2015-04-01 – 2019-03-31
• 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

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.

Free Research Field

離散幾何解析学

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。