2023 Fiscal Year Final Research Report
Development of convergence theory for sequences of spaces whose unbounded dimension from the aspect of geometric analysis
| Project/Area Number |
22K20338
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Multi-year Fund |
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| Review Section |
0201:Algebra, geometry, analysis, applied mathematics,and related fields
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| Research Institution | Kyushu University |
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Principal Investigator |
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| Project Period (FY) |
2022-08-31 – 2024-03-31
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| Keywords | 測度距離空間 / 集中位相 / 測度の集中現象 / 次元が無限大に発散する空間列 / ピラミッド / 無限次元極限 / 主束構造 / Cauchy分布 |
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| Outline of Final Research Achievements |
In this study, we consider the convergence of sequences of metric measure spaces with respect to the concentration topology, which is based on the concept of concentration of measure phenomenon, and investigate the behavior of geometric analytic quantities. The convergence notion induced by the concentration topology is effective in capturing the high-dimensional aspects of spaces, and it is also interesting as a generalization of the celebrated measured Gromov-Hausdorff convergence. In our research, we develop a general theory of invariants for metric measure spaces and prove that the variance, the Poincare constant, etc., are “good” invariant. We applied them to a classification problem for the infinite-dimensional limit spaces. We also determine the limit of the generalized Cauchy distribution as the dimension diverges to infinity under suitable scaling. This discovery extends the scope of this study beyond the Riemannian geometry.
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| Free Research Field |
測度距離空間の収束理論
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| Academic Significance and Societal Importance of the Research Achievements |
本研究では，空間列の高次元挙動および無限次元極限を調査し，不変量を用いた無限次元極限の区別や新しいモデルの無限次元極限の決定などを行った．これらの成果は新たな研究の流れを構築し，研究領域を広げたとして学術的に高く評価されている．特にリーマン幾何学的な対象のみならず，高次元の確率分布やブラウン運動などを収束理論の対象として取り入れられることを指摘した意義は非常に大きい．従来以上に確率論や統計学とも密接に関係した理論に発展することが今後期待される．また本研究では，従来にない斬新な結果のみならずそれらを含む基礎理論の見直しにも取り組み，理論の体系化にも力を入れている．
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