| 研究課題/領域番号 |
18K11166
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| 研究機関 | 京都大学 |
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研究代表者 |
ディブレクト マシュー 京都大学, 人間・環境学研究科, 准教授 (20623599)
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| 研究期間 (年度) |
2018-04-01 – 2024-03-31
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| キーワード | quasi-Polish space / duality / algebraic geometry / topology / descriptive set theory / domain theory / computability theory |
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| 研究実績の概要 |
Our main results this year were applications of the general theory that we developed during this project.
At CCA 2022, we presented some preliminary results about coPolish rings (topological rings whose topology is coPolish), the main result being the construction of a contravariant functor from the category of coPolish rings and continuous ring homomorphisms to the category of quasi-Polish spaces that agrees with the standard construction of the spectrum of a ring for (countable) discrete rings, but results in spaces that are more manageable for computability and foundational applications when the ring is non-discrete.
At CTS 2022, we presented an effective version of the classical descriptive set theory result that adding countably many closed sets to the topology of a quasi-Polish space results in a quasi-Polish space. Our construction takes a c.e. transitive relation encoding a quasi-Polish space and an enumeration of co-c.e. closed subsets of the space and outputs a c.e. transitive relation encoding the quasi-Polish space with the refined topology. Although the classical result is well-known for (quasi-) Polish spaces, to our knowledge this is the first effective proof.
At CiE 2022, we presented joint work with T.Kihara and V.Selivanov that gives a detailed analysis of the enumerability of various classes of effective quasi-Polish spaces, which supersedes known results on enumerating certain classes of domains. We also showed that the subclasses of effective quasi-Polish spaces corresponding to the T1,T2, and T3 separation axioms are each (lightface) coanalytic complete.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
3: やや遅れている
理由
Although we have produced several interesting applications over the past year, travel restrictions as well as increases in work-related (but non-research related) responsibilities has made it difficult to carefully discuss the details of some of our deeper research topics (e.g. quasi-Polish frames and applications to algebraic geometry) with other researchers that are sufficiently knowledgeable of those areas. Since the restrictions have now been mostly lifted, we are more optimistic about the possibility of discussing research with other specialists in person this year.
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| 今後の研究の推進方策 |
Our main goal this year is (still) to collect our results on quasi-Polish frames and quasi-Polish regular frames into a paper (or two). While writing up parts of the papers last year, we made some critical progress towards a complete characterization of quasi-Polish frames, and redirected our efforts in that direction. There are still several related issues in that area that we will investigate, and after wrapping that up we will return to writing up the papers. At the same time, if time permits, we will also continue looking at applications to the algebraic geometry of coPolish rings.
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| 次年度使用額が生じた理由 |
Our main use of research funds is to cover travel expenses and conference fees, and it was still difficult to travel last year. Since restrictions have been mainly lifted, we expect this year we will be able to meet with other researchers by traveling overseas or inviting them to Japan.
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