2022 Fiscal Year Final Research Report
Ultimate analysis of hierarchies in computability theory, descriptive set theory, and general topology
| Project/Area Number |
19K03602
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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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| Review Section |
Basic Section 12030:Basic mathematics-related
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| Research Institution | Nagoya University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Keywords | 記述集合論 / 計算可能性理論 / 決定性公理 / 構成的逆数学 / 次数スペクトル / 実効トポス / ベター擬順序 / 高階計算可能性 |
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| Outline of Final Research Achievements |
I have conducted research aimed at analyzing hierarchical structures in computability theory, descriptive set theory, and general topology. Major results include, in the theory of Wadge degrees in descriptive set theory, structural analysis of Borel measurable functions by measurable reduction, solution of Fournier's problem around Wadge rank ω_2, and extension of Louveau's theorem to BQO-valued functions. On the topological side, I analyzed the degree spectra of various topological spaces, in particular, gave a numbering of ω-continuous domains, and studied a new innovation of de Groot duals using higher-order computability. Other topics include the solution of Lee-van Oosten's problem on the LT-topologies on the effective topos, its connection with synthetic descriptive set theory and the construction of a convenient construction method for models in constructive reverse mathematics, and so on.
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| Free Research Field |
計算可能性理論
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| Academic Significance and Societal Importance of the Research Achievements |
計算可能性理論および記述集合論に関する様々な問題の解決や新機軸の研究を行うことで，計算可能性，記述複雑性，定義複雑性に関する学術的知見を深めた．また，本研究は世界中の様々な研究者に波及し，Lutz-Siskindによる保測写像および順序保存写像に対する第一マーティン予想の解決，Day-Marksによる分解可能性予想の解決など，長きに亘り未解決であった大未解決問題の進展に繋がったなど，学術的影響は大きい．
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