2019 Fiscal Year Research-status Report
New developments in iterated forcing
| Project/Area Number |
18K13448
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| Research Institution | Shizuoka University |
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Principal Investigator |
メヒア ディエゴ 静岡大学, 理学部, 准教授 (70777961)
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| Project Period (FY) |
2018-04-01 – 2022-03-31
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| Keywords | 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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| Outline of Annual Research Achievements |
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 2019 was to develop methods number (1) and (2).
(1.1) We successfully produced two dimensional iterations with ultrafilter limits（雑誌論文3）and one of its applications is the consistency of Cichon's diagram separated in the maximum number of possible different values（雑誌論文2）, which solves the first of the three main problems of the project. A new method of intersection with σ-closed models was also created for this purpose. This results were presented in all the invited and contributed lectures of this fiscal year（学会発表1-4）
(1.2) Advances on the second main problem of the project where made by including the combinatorial concepts of Martin's axiom, its σ-centered version, and the distributivity number（雑誌論文4）. In addition, applications of forcing axioms to versions of Whitehead's problem for path algebra modules where obtained（雑誌論文1）.
During this fiscal year there were two research visits to Colombia, one to UNAM Morelia (Mexico), and one to TU Wien and University of Vienna in Austria.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
研究実績の概要(1.1) solves the first main problem of the project, and the new methods developed where not as complex as researchers were imagining for many years. This new method promises to have several applications in this area of research.
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| Strategy for Future Research Activity |
The current fiscal year will be focused to develop new techniques of (2) multidimensional template iterations and (3) Weak creature forcing. Concerning (2), we plan to expand the methods from 雑誌論文2-4 to several dimensions and bring stronger applications to combinatorics of the real, in particular, to advance in results towards the second main question of the project.
The work on (3) is expected to take two years of the project. Research collaboration (online in case visiting is not yet allowed by the covid-19 situation) with researchers in Barcelona (Spain) will be crucial to the development of this part of the project.
To advanced in this plan, research collaboration (online or visits) is scheduled with members of the following universities: TU Wien and University of Vienna (Austria), National University of Colombia and Pascual Bravo (Colombia), UNAM Morelia (Mexico), and Polytechnic University of Catalonia (Spain).
The researcher is scheduled to host the Kyoto University RIMS Set Theory Conference on Novermber 2020.
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