2007 Fiscal Year Final Research Report Summary
Nonlinear partial differntial equations related to geometric variational problems
| Project/Area Number |
16540188
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Nagoya University |
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Principal Investigator |
NAITO Hisashi Nagoya University, Graduate School of Mathematics, Associate Professor (40211411)
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| Co-Investigator(Kenkyū-buntansha) |
NAYATANI Shin Nagoya University, Graduate School of Mathematics, Professor (70222180)
MAEDA Yoshiaki Keio University, Department of Mathematics, Professor (40101076)
TACHIKAWA Atsushi Tokyo Science University, Department of Mathematics, Professor (50188257)
KUBO Masashi Nagoya University, Graduate School of Mathematics, Associate Professor (20319148)
ISHIGE Kazuhiro Tohoku University, Department of Mathematics, Associate Professor (90272020)
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| Project Period (FY) |
2004 – 2007
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| Keywords | Differential Geometry / Harmonic Map / Variational Problems / Crystal Lattices |
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| Research Abstract |
The head investigator researches the geometric visualization of standard realized crystal lattices, which related with harmonic maps as an example of geometric variational problems. The standard realization of crystal lattices is defined by Kotani-Sunada, and it is considered as a realization of real crystals in nature. The definition of crystal lattice is an abelian covering of finite graph. The covering transformation group and/or the first fundamental group of a finite graph is its the first homology group and it is abelian. So, the crystal lattice is an abelian convering graph of a finite graph. The standard realization of a crystal lattices is defined using harmonic maps into Albanese Torus. Hence the definition is very abstract.
The head investigator construct an application to visualize the standard realization of crystal lattices. Sunada constructs K4 Crystal Lattice as the standard realization of K4 Graph in 3-dimensional Euclidean space. Our application plays important role in calculations of the K4 real crystal with Carbon atoms
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