| 研究課題/領域番号 |
18K13448
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| 研究機関 | 静岡大学 |
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研究代表者 |
メヒア ディエゴ 静岡大学, 理学部, 准教授 (70777961)
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| 研究期間 (年度) |
2018-04-01 – 2023-03-31
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| キーワード | Forcing Theory / Forcing Iterations / Creature Forcing / Bounded Arithmetic / 連続体上の組合せ論 / Ultrafilters / 多次元反復強制法 / Strong Measure Zero Sets |
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| 研究実績の概要 |
The main goal of the project consists in developing the following forcing iteration techniques and apply them to solve open problems about combinatorics of the real line: (i) Multidimensional iterations with ultrafilter limits; (ii) Multidimensional template iterations; and (iii) Weak creature forcing.
The purpose for fiscal year 2021 was to continue to develop the methods number (ii) and (iii).
Although the main objectives of (ii) were already achieved in the fiscal year 2020, extensions and generalizations of these results are in process. For instance, we developed methods to combine preservation techniques for cardinal characteristics of the generalized Baire space with proper forcing methods to solve problems concerning the combinatorics of strong measure zero sets (雑誌論文1). These results and new still unpublished results were presented in invited lectures (学会発表1,3-6).
We achieved representative advances concerning (iii). We developed simple creature forcing techniques to construct models where many parametrized cardinal characteristics of the continuum can assume different values (雑誌論文2). This work was considerably expanded, where we simplified much more sophisticated proper forcing techniques (学会発表2). Concerning arithmetic, we published research about solutions of higher power residues modulo prime (雑誌論文3).
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
We managed to solve the main problems of (ii) and succeeded with great advances of (iii). However, due to the Covid-19 pandemics restrictions, research visits were restricted locally to Japan, so the conclusion of (iii) is still pending.
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| 今後の研究の推進方策 |
Fiscal year 2022 is dedicated to produce expansions of topic (ii) and to conclude the part concerning topic (iii) about connections of creature forcing with arithmetic. Now that face-to-face research meetings have been reactivated around the world, I expect to restart active collaboration with researchers in Europe (Austria, Slovakia, Spain) and give a very satisfactory conclusion to this project.
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| 次年度使用額が生じた理由 |
Due to international restrictions from Covid-19 pandemics, only local travel expenses were used. In Fiscal Year 2022, I plan to use the rest of the grant mainly for research collaboration with researchers abroad (Austria, Slovakia, Spain, USA) and, in case Japan restrictions are lifted, to invite researcher to collaborate with me in Shizuoka.
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