| 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 – 2022-03-31
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| キーワード | 強制法理論 / 反復強制法 / Creature forcing / 限定算術 / 連続体上の組合せ / 超フィルター / 多次元反復強制法 / 強測度ゼロ |
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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: (1) Multidimensional iterations with ultrafilter limits; (2) Multidimensional template iterations; and (3) Weak creature forcing.
The purpose for fiscal year 2018 was to develop method number (1), and the results were quite satisfactory: (1.1) Generalization of preservation methods along two dimensional iterations and new characterizations and consistency results related to strong measure zero sets (雑誌論文2); (1.2) Successful development of ultrafilter limits in two dimensional iterations. The main application of this method is a new consistent constellation of Cichon's diagram into 7 values, which can be used to split the whole diagram modulo three compact cardinals. This result was presented in two international conferences (学会発表1,2).
One initial result about (2) was obtained: (2.1) Creation of two dimensional iterations with support restriction, which allows to manipulate the cofinality of cardinal characteristics of the continuum (雑誌論文1). Furthermore, a generalization of Mostowski's and Shoenfield's absoluteness theorems for arbitrary Polish spaces was obtained (雑誌論文3).
During this fiscal year there were two invited lectures at international conferences (学会発表1,2), one contributed lecture (学会発表3), and one research visit to UNAM Morelia (Mexico).
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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
The results mentioned in 研究実績の概要(1.1)(1.2) fulfills the objective in 研究実績の概要(1) and it was already presented at international conferences in Colombia (学会発表2) and in Austria (学会発表1), however, the corresponding paper is still under review. On the other hand, the initial development 研究実績の概要(2.1) is a very good start towards the objective in 研究実績の概要(2), which is already published (雑誌論文1).
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| 今後の研究の推進方策 |
The current fiscal year will be focused to develop new techniques of (2) multidimensional template iterations, with applications to solve open problems about combinatorics of the real line. It is expected to expand the support restriction techniques presented in 雑誌論文1 to three or more dimensions, even with a more powerful structure. On the other hand, considering the power of the techniques obtained in 研究実績の概要(1.2), it is planned to bring more applications and solve more open consistency problems, particularly about the real line and topology. Initial advances on 研究実績の概要(3) are considered, at least in the context of strong creature forcing.
To guarantee success of the aforementioned plan, the following research visits are planned: (i) two times to Kobe University (Joerg Brendle); (ii) one visit to TU Wien and University of Vienna in Vienna, Austria (Jakob Kellner, Martin Goldstern, Vera Fischer, Miguel Cardona); (iii) one invited lecture at Casa Matematica in Oaxaca, Mexico; (iv) National University of Colombia (Pedro Zambrano) and Pascual Bravo (Ismael Rivera) in Colombia (v) one contributed lecture at "Set Theory and Infinite" conference at Kyoto University RIMS.
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| 次年度使用額が生じた理由 |
Due to financial support from the host at University of Vienna in the trip to Vienna on September 2018, some travel expenses remained. Since the same support cannot be guaranteed for the research visits in fiscal year 2019, the incurred amount will be used to help cover travel expenses to attend a conference (invited lecture) in Oaxaca, Mexico, and for the research visits to Mexico and Colombia.
Concerning article costs, the small amount remaining on 2018 will be used to complete expenses destined to bibliographical references.
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