2004 Fiscal Year Final Research Report Summary
Commutative and Non-commutative Bloch Theory
| Project/Area Number |
13304012
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (A)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Global analysis
|
|---|
| Research Institution | Meiji University (2003-2004)
Tohoku University (2001-2002) |
|---|
Principal Investigator |
SUNADA Toshikazu Meiji U., Math., Professor, 理工学部, 教授 (20022741)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
MORIMOTO Hiroko Meiji U., Math., Professor, 理工学部, 教授 (50061974)
MASUDA Kyuya Meiji U., Math., Professor, 理工学部, 教授 (10090523)
TSUSHIMA Ryuji Meiji U., Math., Professor, 理工学部, 教授 (20118764)
SATO Atsuchi Meiji U., Math., Associate Professor, 理工学部, 助教授 (70178705)
AHARA Kazushi Meiji U., Math., Assistant Professor, 理工学部, 講師 (80247147)
|
|---|
| Project Period (FY) |
2001 – 2004
|
| Keywords | describe geometric analysis / crystal lattice / random walk / large deviation / 多面体 |
|---|
| Research Abstract |
The present research is the Bloch theory for crystal lattices, which are infinite graphs to be introduced to describe, in an abstract way, how the atoms in a crystal are bound to each other by internal force. Crystalline structure is embodied by the presence of certain abelian group actions on graphs. Our interest is in geometry and analysis of the discrete Laplacians (and their magnetic or elastic versions) on crystal lattices as a special class of difference operators. Especially we have studied asymptotic behaviors of random walks, and established a large deviation property in view of geometry. The method we took up is a real version of Bloch theory. We could relate the large deviation property to a rational convex polyhedron in the first homology group of a finite graphs, which has remarkable combinatorial features, and show up also in the Gromov-Hausdorff limit of a crystal lattice.
|
