Theory and applications of Stone-duality for quasi-Polish spaces
| Project/Area Number |
18K11166
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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 60010:Theory of informatics-related
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| Research Institution | Kyoto University |
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Principal Investigator |
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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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Project Status |
Granted (Fiscal Year 2022)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2022: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | quasi-Polish space / duality / algebraic geometry / topology / descriptive set theory / domain theory / computability theory / valuations / measure theory / category theory / locale theory / Stone-duality / frames / computable analysis / geometric logic / quasi-Polish spaces / logic / computation |
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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
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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| Strategy for Future Research Activity |
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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Report
(5 results)
Research Products
(25 results)
