2018 Fiscal Year Final Research Report
Geometric study of a sequence of spaces with unbounded dimension
| Project/Area Number |
26400060
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Tohoku University |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2019-03-31
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| Keywords | 測度の集中現象 / 測度距離空間 / オブザーバブル直径 / オブザーバブル距離 / セパレーション距離 / 等周不等式 |
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| Outline of Final Research Achievements |
Based on the study of concentration of measure phenomenon, Gromov proposed a new geometric theory for metric measure spaces. One of main motivation to study this theory is to investigate a sequence of spaces with unbounded dimension. In our study, we develop and deepen the theory. The concentration of measure phenomenon can be considered as a geometric variant of the law of large numbers. We study a geometric variant of the central limit theorem, which appears as an analog of phese transition phenomenon. One of examples is the sequence of spheres with unbounded dimension. If the radius of the sphere has small order, then we observe the concentration of measure phenomenon. If the radius has large order, then we observe the dissipation phenomenon. If the radius has the order of the square root of the dimension, then we see that the sphere converges to the infinite-dimensional space with Gaussian measure. We have proved this kind of phenomenon for many other spaces.
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| Free Research Field |
幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
空間列の収束や漸近的挙動を調べた幾何学的な研究である．従来の空間のグロモフ・ハウスドルフ収束の研究では，幾何群論と微分幾何の問題であった．しかし本研究は，幾何学のみならず，確率論，統計学，解析学，統計力学などと関係した非常に興味深い研究であり，国内外で学術的に高く評価されている．従来の研究とは異なった独創的な方法で高次元および無限次元空間へアプローチするものであり，今後の進展が期待されている．
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