2007 Fiscal Year Final Research Report Summary
Noncomutative Discrete Geometric Analysis
| Project/Area Number |
18540221
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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 | Meiji University |
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Principal Investigator |
SUNADA Toshikazu Meiji University, Faculty of Engneering and Science, Professor (20022741)
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| Co-Investigator(Kenkyū-buntansha) |
KOTANI Motoko Tohoku University, Mathematics, Professor (50230024)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Discrete Laplacian / Graph / Crystal lattice / Diamond twin / Random walk |
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| Research Abstract |
Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. As indicated by the title, this field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". Edges in a graph as 1-dimensional objects, however, do not play a substantial role except when we discuss geometry of graphs. Therefore our view is different from, for instance, the case of quantum graphs where differential operators on edges are vital. Actually the role of edges in analysis is just to give a neighboring relation among vertices, and difference operators linked with this relation replace differential operators. We thus do not need to worry about irregularity of functions which, for differential operators, may cause some trouble, if not serious. Instead, the combinatorial aspect of graphs creates a different kind of technical complication. Furthermore, analysis on both infinite graphs and non-compact manifold
… More
s involves much the same degree of difficulty.
The "protagonist" in our research is a discrete analogue of Laplacians on manifolds (and related operators), which appears in many parts of mathematical sciences. In the nature of things, ideas cultivated in global analysis provide us a good guiding principle on the one hand, and the practical motivation is the moving force of theoretical progress on the other..
Discrete Laplacians appear in both geometric crystallography and probability.
In our research, we applied a remarkable result in the theory of random walks on crystal lattices to pin down a diamond crystal. It is interesting to rephrase the result such as "A random walker on a crystal lattice may detect the most natural way for his crystal lattice to sit in space". A diamond twin means a hypothetical crystal which shares the property of symmetry satisfied by the diamond crystal. Our result claims that there is only diamond twin, which is obtained as the standard realization of the maximal abelian covering graph of the complete graph K_4.
As for the activity, I was a member of organizers of the special project held at Newton Institute, Cambridge University which stated from January and ended up in June, 2007. Less
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