| Project/Area Number |
16K21115
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Algebra
Theory of informatics
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| Research Institution | Kyoto University (2019)
Tohoku University (2016-2018) |
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Principal Investigator |
Uramoto Takeo 京都大学, 高等研究院, 研究員 (40759726)
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | semigalois category / Witt vector / duality / class field theory / automaton / regular language / profinite monoid / integral Witt vector / state complexity / 谷山志村予想 / Bost-Connes system / Bost-Connes系 / Christolの定理 / 有限オートマトン / 副有限モノイド / semigalois圏 / 類体論 / オートマトン / ガロア理論 / 擬ガロア圏 / 正規言語 |
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| Outline of Final Research Achievements |
In this research we studied an axiomatization of Eilenberg theory, which classically concerns classifications of regular languages, finite monoids, and finite automata. We proved that this theory can be axiomatized in terms of the duality theory of semigalois categories, which clarified that this theory is essentially an extension of classical galois theory. After this axiomatization, we further studied an application of this theory to number theory. In particular, we proved an arithmetic analogue of Christol's theorem and related this theorem to the theory of semigalois categories. Moreover, we further studied Bost-Connes' C* dynamical systems, and proved that the algebra of integral Witt vectors gives an arithmetic subalgebra of the system. This suggests that Witt vectors can be realized by special values of certain deformation families of modular functions, on which we also obtained certain observations.
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| Academic Significance and Societal Importance of the Research Achievements |
正規言語の分類理論であるEilenberg理論は元々形式言語理論における一分野でしかなかったが、その圏論的公理化によって、応用範囲が広がった。特に整数論に対する応用があったことは、さらなる研究の方向性を示唆しており、このEilenberg理論と整数論の繋がりはそれ自体で一つの研究領域になりうる。中でも古典的な類体論に対して新しい見方を与えることができたことが最も意味があるように思われる。この観察によって古典類体論の新しい非可換拡張の方向性も示唆されているように思う。
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