2020 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 / 連続体上の組合せ論 / 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: (i) Multidimensional iterations with ultrafilter limits; (ii) Multidimensional template iterations; and (iii) Weak creature forcing.
The purpose for fiscal year 2020 was to develop methods number (ii) and (iii).
(1) We developed a forcing technique that is very template-like but in another direction (雑誌論文1): support restriction is allowed, but not in every set. Under certain closure properties, this allows more control on the combinatorics of the forcing. With this technique we solved critical instances of the second main problem of the project, namely, we showed that the combinatorial notions of splitting and reaping are independent from other classical combinatorial notions of the reals, like Cichon's diagram and Martin's axiom. This result was presented in the invited lecture 学会発表1.
(2) In connection with the second main problem of the project, we developed additional forcing techniques related with collapsing forcing, which allows to obtain singular values in the constellations of classical cardinal characteristics of the continuum (雑誌論文2). However, these techniques still rely on the use of large cardinals. Related results were presented in 学会発表2.
(3) In connection with (iii), we have submitted one paper about advances in creature forcing (still unpublished).
The researched hosted the RIMS Set Theory Workshop at Kyoto University.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
Although the advances in the second main problem of the project are very satisfactory, the progress about weak creature forcing has been slower than expected. Research collaboration with researchers from the Polytechnic University of Catalonia (Spain) and TU Wien (Austria) are essential for this part, but due to the Covid-19 pandemics research visits has been impossible to plan. We have been progressing through online meetings, but advances has not been as smoothly as desired.
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| Strategy for Future Research Activity |
The current fiscal year will be devoted to develop new techniques of (ii) multidimensional template iterations and (iii) Weak creature forcing.
For (ii) we plan to build multidimensional versions of template iterations, also in the strong sense as in 雑誌論文2, to solve more instances of the second main problem of the project. We also consider creating new forcing techniques (even different from templates) towards solving this problem. Collaboration with researchers from TU Wien (Austria) and the Hebrew University of Jerusalem (Israel) is expected for this part.
For the work on (iii), collaboration with researchers from the Polytechnic University of Catalonia (Spain) and TU Wien (Austria) is crucial to advance properly. Although we keep working through online meetings, we expect that the Covid-19 situation improves so that research visits become possible.
The lack of research visits has represented delays on the project, specially concerning (iii), and the research funds destined to this part has not been used. For this reason, the researcher is considering to ask an extension of this project to one more year, so that the funds can be used for research visits accordingly.
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| Causes of Carryover |
Due to the Covid-19 pandemics, no travel expenses were used at all. I plan to use them in Fiscal Year 2021 (hoping that traveling will become allowed at some point), and request an extension of the project for one more year to be able to use all the funds and conclude the project properly.
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