2022 Fiscal Year Final Research Report
Research to elucidate Shelah's conjecture and its related branches in model theory
| Project/Area Number |
17K05342
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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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| Research Field |
Foundations of mathematics/Applied mathematics
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| Research Institution | University of Tsukuba |
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Principal Investigator |
Tsuboi Akito 筑波大学, 数理物質系(名誉教授), 名誉教授 (30180045)
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| Co-Investigator(Kenkyū-buntansha) |
塩谷 真弘 筑波大学, 数理物質系, 准教授 (30251028)
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| Project Period (FY) |
2017-04-01 – 2023-03-31
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| Keywords | モデル理論 |
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| 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.
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| Free Research Field |
数理論理学
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| 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．
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