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Research to elucidate Shelah's conjecture and its related branches in model theory

Research Project

Project/Area Number 17K05342
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionUniversity of Tsukuba

Principal Investigator

Tsuboi Akito  筑波大学, 数理物質系(名誉教授), 名誉教授 (30180045)

Co-Investigator(Kenkyū-buntansha) 塩谷 真弘  筑波大学, 数理物質系, 准教授 (30251028)
Project Period (FY) 2017-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywordsモデル理論 / 有限数学 / 数理論理学 / 無限組み合わせ論 / 無限組み合わせ / 有限モデル理論 / コンパクト性 / 組み合わせ論 / Model Theory
Outline of Final Research Achievements

Model theory is a research area that considers the mathematical structure from meta standpoints, and is studied using saturated extensions of the structure. Arguments in the extended structure are finally applied to the original structure, giving a novel approach to the study of mathematical structures.
The important thing in studying mathematical structures is to discover hidden structures that cannot be seen from the surface. We focused on the invariant by Shelah. Intuitively, this invariant represents the number of mutually independent strict orders in the n-th product of M, where M is a saturated model of the theory T. We have succeeded to show that this invariant has a certain additivity.

Academic Significance and Societal Importance of the Research Achievements

モデル理論における安定性理論と呼ばれる学術分野は安定性と非安定性について研究している.構造の理論に安定性があれば,抽象的独立概念が定義される.非安定性を持つ場合は,どの程度非安定性があるかを判断することが重要となる.本研究ではその非安定性の度合をShelahの定義したκ_srdを用いて研究した.その系として,κ_srdを決定するためには,基本的に1変数の場合を考えれば十分であることが分かった.その結果、不変量計算の複雑さが大幅に軽減され、応用面でも大きな成果となった.

Report

(7 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (17 results)

All 2023 2022 2021 2020 2019 2018 2017

All Journal Article (3 results) (of which Peer Reviewed: 1 results,  Open Access: 1 results) Presentation (13 results) (of which Int'l Joint Research: 2 results,  Invited: 2 results) Book (1 results)

  • [Journal Article] Some results related to Keisler-Shelah isomorphism theorem2022

    • Author(s)
      Akito Tsuboi
    • Journal Title

      RIMS Kokyuroku

      Volume: 2218

    • Related Report
      2022 Annual Research Report
  • [Journal Article] On the number of independent orders2021

    • Author(s)
      Kota Takeuchi and Akito Tsuboi
    • Journal Title

      Annals of Pure and Applied Logic

      Volume: vol. 172, issue 2 Issue: 2 Pages: 102886-102886

    • DOI

      10.1016/j.apal.2020.102886

    • Related Report
      2021 Research-status Report 2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Non-expressibility of a class of finite graphs2019

    • Author(s)
      坪井 明人 (Tsuboi,Akito )
    • Journal Title

      RIMS Kokyuroku

      Volume: No.2119

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Presentation] Ramsey’s theorem and coheir sequences2023

    • Author(s)
      坪井明人
    • Organizer
      日本数学会年会
    • Related Report
      2022 Annual Research Report
  • [Presentation] Some comments on the difference between forking and dividing2022

    • Author(s)
      坪井明人
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2022 Annual Research Report
  • [Presentation] Colored Random Graphs and the Order Property2022

    • Author(s)
      Akito Tsuboi
    • Organizer
      Model theoretic aspects of the notion of independence and dimension, RIMS Model Theory Workshop 2022
    • Related Report
      2022 Annual Research Report
  • [Presentation] First course of model theory2022

    • Author(s)
      坪井明人
    • Organizer
      数学基礎論サマースクール
    • Related Report
      2022 Annual Research Report
  • [Presentation] Keisler’s Isomorphism Theorem, CH and Instability2021

    • Author(s)
      Akito Tsuboi
    • Organizer
      RIMS model theory workshop
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] On coloring of Fraisse limits2020

    • Author(s)
      Akito Tsuboi
    • Organizer
      RIMS model theory workshop
    • Related Report
      2020 Research-status Report
  • [Presentation] A remark on Ehrenfeucht theories2019

    • Author(s)
      坪井明人
    • Organizer
      日本数学会秋季総合分科会 (金沢大学 9月19日)
    • Related Report
      2019 Research-status Report
  • [Presentation] Compactness argument for finite structures2019

    • Author(s)
      坪井明人
    • Organizer
      研究集会「Model Theory Summer School 2019」(東海大学 山中湖)
    • Related Report
      2019 Research-status Report
  • [Presentation] On the number of independent strict orders2019

    • Author(s)
      Akito Tsuboi
    • Organizer
      Model Theory Workshop at RIMS (京都大学数理解析研究所 12月3日)
    • Related Report
      2019 Research-status Report
  • [Presentation] Finite combinatorics and compactness2018

    • Author(s)
      坪井明人
    • Organizer
      北陸先端科学技術大学院大学情報科学系セミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] A proof of Hales-Jewett theorem using a nonstandard method2018

    • Author(s)
      Akito Tsuboi
    • Organizer
      組合せ論・モデル理論セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] On the notion of stability2017

    • Author(s)
      坪井明人
    • Organizer
      モデル理論「夏の学校」研究集会
    • Related Report
      2017 Research-status Report
  • [Presentation] "Hales-Jewett theorem and model theory"2017

    • Author(s)
      Akito Tsuboi
    • Organizer
      2017 RIMS Model Theory Workshop
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Book] 集合入門2019

    • Author(s)
      坪井明人 塩谷真弘 佐垣大輔
    • Total Pages
      155
    • Publisher
      牧野書店
    • ISBN
      9784434257964
    • Related Report
      2018 Research-status Report

URL: 

Published: 2017-04-28   Modified: 2024-01-30  

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