2018 Fiscal Year Final Research Report
Aspects of computability-theoretic structures as topological invariants
| Project/Area Number |
17H06738
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Single-year Grants |
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| Research Field |
Foundations of mathematics/Applied mathematics
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| Research Institution | Nagoya University |
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Principal Investigator |
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| Project Period (FY) |
2017-08-25 – 2019-03-31
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| Keywords | 計算可能性理論 / 記述集合論 / 逆数学 / 計算可能解析学 / 再帰理論 / コルモゴロフ複雑性 / 位相次元論 / フラクタル次元 |
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| Outline of Final Research Achievements |
By analyzing computability-theoretic structures from the topological viewpoint, we obtained a large number of results on the relationship among computability theory, descriptive set theory, and general topology. Most of results concern on extracting the computability theoretic nature from topological structures and set-theoretic structures, and then clarify these structures.
For example, by understanding the principle of computable enumerations as some sorts of nonmetrizable topology, we established a new theory unifying the previous works on enumeration degrees. For another example, applying computability theory on uncountable cardinals, we succeeded to obtain a complete description of the bqo-valued Borel functions on nonseparable complete ultrametric spaces. Furthermore, we also solved several open problems, e.g., in reverse mathematics.
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| Free Research Field |
数学基礎論
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| Academic Significance and Societal Importance of the Research Achievements |
計算可能性の理論は，コンピュータ科学の理論的基礎を支えるものである．本研究は，『空間』『次元』『ランダム性』といった素朴で重要な数学的構造が内部に孕む計算論的構造を明示化するという点で意義深い．また，このような発想は，逆に，非計算論的数学の概念を用いた計算論的概念の再理解にも繋がる．したがって，報告者の手法の応用範囲の拡大は，計算論と応用対象の分野双方に新たな手法と視点を持ち込むこととなり，広範な分野にインパクトを与えるものとなる．
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