Applications of the concentration of measure phenomenon to analysis and geometry of Laplacian
| Project/Area Number |
25800042
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Tohoku University (2016)
Kyoto University (2013-2015) |
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Principal Investigator |
Funano Kei 東北大学, 情報科学研究科, 准教授 (40614144)
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| Project Period (FY) |
2013-04-01 – 2017-03-31
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| Project Status |
Completed (Fiscal Year 2016)
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| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | Laplacianの固有値 / Ricci曲率 / 凸体 / 普遍不等式 / ham sandwichの定理 / 最適輸送 / Neumann条件 / ラプラシアンの固有値 / ラプラシアン / 測度集中 / 曲率次元条件 / 測度の集中現象 / リッチ曲率 |
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| Outline of Final Research Achievements |
I obtained some upper bound estimates of eigenvalues of the Laplacian on closed Riemannian manifolds of nonnegative Ricci curvature. These estimates state that one can estimate eigenvalues in terms of infomation of finite number of subsets of the manifold. The method I used in the proof is the theory of optimal transportation.
I also studied domain monotonicity/reverse domain monotonicity for Neumann eigenvalues of the Laplacian on convex domains in a Euclidean space. Furthermore I got nontrivial universal inequalities among eigenvalues of the Laplacian. In the proof I used the ham sandwich theorem coming from algebraic topology. These studies are valuable.
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Report
(5 results)
Research Products
(23 results)
