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Theory and applications of Stone-duality for quasi-Polish spaces

Research Project

Project/Area Number 18K11166
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 60010:Theory of informatics-related
Research InstitutionKyoto University

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)

Project Period (FY) 2018-04-01 – 2025-03-31
Project Status Granted (Fiscal Year 2023)
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)
Keywordsquasi-Polish space / duality / valuations / computability theory / descriptive set theory / algebraic geometry / topology / domain theory / measure theory / category theory / locale theory / Stone-duality / frames / computable analysis / geometric logic / quasi-Polish spaces / logic / computation
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.

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.

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.

Report

(6 results)
  • 2023 Research-status Report
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (28 results)

All 2024 2023 2022 2021 2020 2019 2018

All Journal Article (6 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 6 results,  Open Access: 5 results) Presentation (22 results) (of which Int'l Joint Research: 16 results,  Invited: 9 results)

  • [Journal Article] Ideal presentations and numberings of some classes of effective quasi-Polish spaces2024

    • Author(s)
      de Brecht Matthew、Kihara Takayuki、Selivanov Victor
    • Journal Title

      Computability

      Volume: Pre-press Pages: 1-24

    • DOI

      10.3233/com-230442

    • Related Report
      2023 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Enumerating Classes of Effective Quasi-Polish Spaces2022

    • Author(s)
      M. de Brecht, T. Kihara, & V. Selivanov
    • Journal Title

      Revolutions and Revelations in Computability (CiE 2022), Lecture Notes in Computer Science

      Volume: 13359 Pages: 88-102

    • DOI

      10.1007/978-3-031-08740-0_8

    • ISBN
      9783031087394, 9783031087400
    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Some Notes on Spaces of Ideals and Computable Topology2020

    • Author(s)
      de Brecht Matthew
    • Journal Title

      Beyond the Horizon of Computability, CiE 2020 Proceedings (LNCS)

      Volume: 12098 Pages: 26-37

    • DOI

      10.1007/978-3-030-51466-2_3

    • ISBN
      9783030514655, 9783030514662
    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On the commutativity of the powerspace constructions2019

    • Author(s)
      de Brecht Matthew、Kawai Tatsuji
    • Journal Title

      Logical Methods in Computer Science

      Volume: 15 Pages: 1-13

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Domain-complete and LCS-complete Spaces2019

    • Author(s)
      de Brecht Matthew、Goubault-Larrecq Jean、Jia Xiaodong、Lyu Zhenchao
    • Journal Title

      Electronic Notes in Theoretical Computer Science

      Volume: 345 Pages: 3-35

    • DOI

      10.1016/j.entcs.2019.07.014

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Overt choice2019

    • Author(s)
      de Brecht Matthew、Pauly Arno、Schroder Matthias
    • Journal Title

      Computability

      Volume: Pre-press Issue: 3-4 Pages: 1-23

    • DOI

      10.3233/com-190253

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Constructing the valuations powerspace functor2023

    • Author(s)
      Matthew de Brecht
    • Organizer
      16th International Conference on Computability, Complexity and Randomness
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research
  • [Presentation] 計算可能位相空間論2023

    • Author(s)
      Matthew de Brecht
    • Organizer
      数学基礎論サマースクール2023
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] A note on the closed prime spectrums of coPolish commutative rings2022

    • Author(s)
      M. de Brecht
    • Organizer
      Nineteenth International Conference on Computability and Complexity in Analysis (CCA 2022)
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research
  • [Presentation] Effectively Extending Topologies2022

    • Author(s)
      M. de Brecht
    • Organizer
      Computing in topological structures: Foundations and implementations (CTS 2022)
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The category of quasi-Polish spaces as a represented space2022

    • Author(s)
      M. de Brecht
    • Organizer
      Online Seminar on Computability Theory and Applications
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On constructing the space of valuations of a quasi-Polish space2021

    • Author(s)
      M. de Brecht
    • Organizer
      RIMS symposium Research Trends on General Topology and its Related Fields
    • Related Report
      2021 Research-status Report
  • [Presentation] Computable functors on the category of quasi-Polish spaces.2021

    • Author(s)
      M. de Brecht
    • Organizer
      2021 North American Annual Meeting of the Association for Symbolic Logic
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The category of quasi-Polish spaces as a represented space2021

    • Author(s)
      M. de Brecht
    • Organizer
      日本数学会トポロジー分科会 第68回トポロジーシンポジウム
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Quasi-Polish spaces as spaces of ideals2021

    • Author(s)
      M. de Brecht
    • Organizer
      Continuity, Computability, Constructivity - From Logic to Algorithms (CCC 2021)
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] Tutorial on Quasi-Polish spaces2021

    • Author(s)
      M. de Brecht
    • Organizer
      Dagstuhl Seminar 21461
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Representing quasi-Polish spaces as spaces of ideals, with applications to computable topology2021

    • Author(s)
      M. de Brecht
    • Organizer
      Japan-Russia workshop on effective descriptive set theory, computable analysis, and automata
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the transferability of results between subcategories of spaces and locales2021

    • Author(s)
      M. de Brecht
    • Organizer
      Fourth Workshop on Mathematical Logic and its Applications
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] Some notes on spaces of ideals and computable topology2020

    • Author(s)
      M. de Brecht
    • Organizer
      Computability in Europe 2020 (online)
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] Some results on countably based consonant spaces2019

    • Author(s)
      de Brecht Matthew
    • Organizer
      Recent Developments in General Topology and its Related Fields RIMS Symposium
    • Related Report
      2019 Research-status Report
  • [Presentation] Overt Choice2019

    • Author(s)
      de Brecht Matthew、Pauly Arno、Schroder Matthias
    • Organizer
      Sixteenth International Conference on Computability and Complexity in Analysis
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] A note on the spatiality of localic products of countably based sober spaces2019

    • Author(s)
      de Brecht Matthew
    • Organizer
      Computability, Continuity, Constructivity (CCC 2019)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Representing spaces using rounded ideals2019

    • Author(s)
      de Brecht Matthew
    • Organizer
      2019年度ジェネラルトポロジーシンポジウム
    • Related Report
      2019 Research-status Report
  • [Presentation] Extending descriptive set theory to non-metrizable spaces and represented spaces2018

    • Author(s)
      M. de Brecht
    • Organizer
      Computability in Europe (CiE 2018)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A note on the descriptive complexity of the upper and double powerspaces2018

    • Author(s)
      M. de Brecht
    • Organizer
      Fifteenth International Conference on Computability and Complexity in Analysis (CCA 2018)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Some properties of the countable space S02018

    • Author(s)
      M. de Brecht
    • Organizer
      Dagstuhl-Seminar 18361 (Measuring the Complexity of Computational Content)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Continuous valuations on quasi-Polish spaces2018

    • Author(s)
      M. de Brecht
    • Organizer
      Workshop on Computability Theory and Foundations of Mathematics 2018
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Some results on quasi-Polish semi-lattices and lattices2018

    • Author(s)
      M. de Brecht
    • Organizer
      General Topology Symposium 2018
    • Related Report
      2018 Research-status Report

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Published: 2018-04-23   Modified: 2024-12-25  

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