2021 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 – 2023-03-31
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| Keywords | 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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| 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 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 These results and new still unpublished results were presented in invited lectures (学会発表1,3-6).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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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
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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| Causes of Carryover |
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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