2011 Fiscal Year Final Research Report
Development and applications of discrete geometric analysis
| Project/Area Number |
21340039
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Meiji University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
TATE Tatsuya 名古屋大学, 多元数理科学研究科, 准教授 (00317299)
HIGUCHI Yusuke 昭和大学, 教養部, 講師 (20286842)
AHARA Kazushi 明治大学, 理工学部, 准教授 (80247147)
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| Project Period (FY) |
2009 – 2011
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| Keywords | 離散幾何解析学 / 標準的実現 / 位相的結晶 |
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| Research Abstract |
The primarily purpose of this project was to provide a mathematical insight into the modern crystallography, a typical practical science that originated in the classification of the observed shapes of crystals. The tools we employed are adopted from algebraic topology, a field in pure mathematics cultivated during the first half of the last century. More specifically the elementary theory of covering spaces and homology is effectively used in the study of 3D networks associated with crystals. Further we formulate a minimum principle for crystals in the framework of discrete geometric analysis, which provides us with the concept of standard realizations, a canonical way to place a given crystal structure in space so as to produce the most symmetric microscopic shape. In spite of its pure-mathematical nature, this concept combined with homology theory turns out to fit with a systematic design and enumeration of crystal structures, an area of considerable scientific interest for many years. Meanwhile, standard realizations show up in asymptotic behaviors of random walks on topological crystals, the abstraction of crystal structures, and are closely related to a discrete analogue of Abel-Jacobi maps in algebraic geometry.
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