Noncommutative Analysis and Functional Analytic Group Theory
| Project/Area Number |
17K05277
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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 |
Basic analysis
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| Research Institution | Kyoto University |
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Principal Investigator |
Ozawa Narutaka 京都大学, 数理解析研究所, 教授 (60323466)
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| Project Period (FY) |
2017-04-01 – 2022-03-31
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| Project Status |
Completed (Fiscal Year 2021)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| Keywords | 函数解析 / 解析的群論 / 作用素環論 / 関数解析 / 群論 / ランダムウォーク / Banach空間論 / 関数解析的群論 / 半正定値問題 |
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| Outline of Final Research Achievements |
A group is the mathematical term to study the symmetry that given objects have. The notion of a group naturally arises in various scientific studies and are considered one of the most basic and important objects of study. In literature, groups are mostly studied with algebraic or more recently geometric methods. The present research has focused on developing analytic methods for the study of groups. Analytic methods are used to connect the rough but robust geometric information of groups to the precise but fragile algebraic structural investigation. Our methods has connected previously unrelated areas of study and open up a new study horizon. In particular, we have solved the well-known longstanding problem about the rigidity aspect of the automorphism group of a free group, which is the group of the symmetry that the universal symmetry carries.
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| Academic Significance and Societal Importance of the Research Achievements |
自由群自己同型群は最も普遍的な対称性そのものに内在する対称性を記述する数学的言語である。この群が剛性を持つか否かは著名な未解決問題であったが、本研究計画ではそれを肯定的に解決しており、研究成果の学術的価値は極めて高い。この研究成果には、実験数学及び工業数学において使われている積置換アルゴリズムに関する応用も存在する。積置換アルゴリズムは与えられた群においてランダムサンプリングを行うアルゴリズムとして高性能であることが経験上知られてきたが、自由群自己同型群の剛性定理により、積置換アルゴリズムが実際に高性能であることの数学的な保証が得られた。
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Report
(6 results)
Research Products
(32 results)
