| 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 – 2023-03-31
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| キーワード | quasi-Polish space / topology / duality / valuations / measure theory / domain theory / category theory |
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| 研究実績の概要 |
We continued doing work on the characterization of quasi-Polish spaces as spaces of ideals of a countable transitive relation. We focused on two main topics (1) the construction of the powerspace of valuations of a quasi-Polish space, and (2) on the functorial aspects of these powerspace constructions, and interpreting them as computable maps on the category of quasi-Polish spaces viewed as a represented space.
Polish spaces are often used for advanced probability and measure theory, and it has been known for a while that many measure theoretic results for Polish spaces extend to quasi-Polish spaces. "Valuations" are a notion originating in domain theory for providing semantics of probabilistic programming languages. It is known that there is a close correspondence between valuations and Borel measures on quasi-Polish spaces, and that this correspondence is a bijection when restricted to probabilistic valuations and Borel probability measures. Our results provide an extremely elegant construction of the space of valuations on a quasi-Polish space in terms of their spaces of ideals. This result extends old work from domain theory, and our approach is compatible with the usual way a Polish topology is placed on the space of probabilistic measures on a Polish space.
The powerspace constructions are well-known to be functorial, and we showed that our constructions in terms of spaces of ideals extend naturally to the continuous maps between these spaces. We showed these functorial aspects are computable if the category of quasi-Polish spaces is viewed as a represented space.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
Actually, the status is something between (2) "Progressing smoothly" and (3) "Slightly delayed", but being optimistic I choose (2).
We have accumulated a lot of nice results, and new research directions come up with each new discovery, so in that sense I think "Progressing smoothly" is appropriate. On the other hand, after two years of travel restriction due to the novel coronavirus, and now new complications due to the war in Ukraine, it has been difficult to have discussions and collaborations with other experts in the field. This makes it difficult to share details of proof techniques, to discuss the philosophical significance and applications of the acquired results, and to develop new techniques to attack unsolved problems. It is clear that much more progress could have been made if there had not been all these obstacles, so in that sense I think progress is "Slightly delayed".
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| 今後の研究の推進方策 |
The main goal for this year is to collect our unpublished results on quasi-Polish frames and quasi-Polish regular frames into a paper or two. In particular, we have a result on completing a sober countably based regular frame into a quasi-Polish regular frame, and we found a simplification in the classification of quasi-Polish algebras of the double powerspace monad. We have also begun looking at possible applications to algebraic geometry, since important examples of topological rings (such as the ring of polynomials with real coefficients) have natural coPolish topologies, and have a dual interpretation in terms of quasi-Polish spaces.
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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 again this year it has been impossible to travel overseas or to invite collaborators to Japan because of the coronavirus situation. We plan to participate in multiple international conferences and workshops in the upcoming year, and we will use the remaining funds to cover these expenses. We are also considering inviting collaborators to Japan if the situation continues to improve.
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