2023 Fiscal Year Research-status Report
Theory and applications of Stone-duality for quasi-Polish spaces
| Project/Area Number |
18K11166
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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 – 2025-03-31
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| Keywords | quasi-Polish space / duality / valuations / computability theory / descriptive set theory |
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| Outline of Annual Research Achievements |
This year we continued finding applications and presenting results of the general theory developed during this project.
At CCR 2023, we presented our work on the valuations powerspace functor on the category of quasi-Polish spaces. There is a close correspondence between valuations and Borel measures on quasi-Polish spaces, and this correspondence is a bijection when restricted to probabilistic valuations and Borel probability measures. Our result shows how to computably convert codes for a continuous map between quasi-Polish spaces into codes for the corresponding spaces of valuations and a code for the continuous map that sends a valuation to its pushforward valuation along the original map. The construction is simple and formalizable within second order arithmetic, but general enough for applications involving Polish spaces (e.g., random dynamical systems) and continuous domains (e.g. probabilistic programming languages).
We also published a journal article containing joint work with T. Kihara and V. Selivanov, which contained and extended results we presented earlier at CiE 2022. The new results included work on effectively extending quasi-Polish topologies, and new results on effective continuous domains, such as enumerating continuous domains, on ideal presentations of effective domains, and on the degree spectra of continuous domains.
We also gave lectures about computable topology as part of a summer school for young researchers and students studying mathematical logic (数学基礎論サマースクール2023). The participants were very active and talented.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We have been able to attend international conferences in person again this year, after several years of only being able to attend online. This has allowed us to get valuable feedback and ideas for future research. Our joint work with T. Kihara and V. Selivanov was accepted to a journal, and it contains many important results and applications to computability theory and computable topology. Participating in the summer school allowed us to share some of the latest ideas on computable topology with the next generation of Japanese mathematicians and logicians.
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| Strategy for Future Research Activity |
This year we will continue developing applications of the results achieved during this project, as well as presenting our findings at international conferences and papers. We have already started looking at generalizations of the result on effectively extending topologies that was published in joint work with T. Kihara and V. Selivanov. We are also interested in applications of this work to better understanding some basic results in computability, such as the low basis theorem. In addition, we hope to look at more examples of coPolish rings and their spectra.
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| Causes of Carryover |
Funds for travel expenses remained because we were unable to travel in previous years due to the pandemic. We will use the remaining funds this year to attend international conferences.
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