| Project/Area Number |
23KF0257
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| Research Category |
Grant-in-Aid for JSPS Fellows
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| Allocation Type | Multi-year Fund |
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| Section | 外国 |
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| Review Section |
Basic Section 12030:Basic mathematics-related
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| Research Institution | Kobe University |
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Principal Investigator |
Brendle Jorg 神戸大学, システム情報学研究科, 教授 (70301851)
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| Co-Investigator(Kenkyū-buntansha) |
PARENTE FRANCESCO 神戸大学, システム情報学研究科, 外国人特別研究員
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| Project Period (FY) |
2023-11-15 – 2026-03-31
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| Project Status |
Granted (Fiscal Year 2023)
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| Budget Amount *help |
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2025: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2024: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2023: ¥200,000 (Direct Cost: ¥200,000)
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| Keywords | Set theory / Boolean algebras / Topology / Ultrafilters / Forcing |
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| Outline of Research at the Start |
We will investigate the structure of ultrafilters on Boolean algebras, with particular emphasis on certain orderings on such ultrafilters (like the Rudin-Frolik ordering) as well as on the relationship between combinatorial properties of ultrafilters and partitions of Boolean algebras. While the methods will mainly come from combinatorial set theory, this research has important repercussions on areas like topology and model theory.
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| Outline of Annual Research Achievements |
We investigated the structure of ultrafilters on Boolean algebras and obtained several interesting results, which can roughly be divided into two topics (and will be written up as two research papers).
(1) In joint work of Brendle and Parente with Michael Hrusak, we established a close connection between cardinal invariants of the meager ideal, the order structure of partitions of the Cohen algebra, and ultrafilters on the latter algebra. In particular, we expressed the reaping number of a reduced power of the Cohen algebra as the maximum of the classical reaping number and the cofinality of the meager ideal cof(M), and obtained as a consequence that the ultrafilter number of the Cohen algebra is larger or equal than cof(M) in ZFC. This answers a question we addressed several years ago.
(2) We analyzed two definitions of the Rudin-Frolik ordering for ultrafilters on Boolean algebras, one by Murakami and one by Balcar and Dow, and proved that Murakami's definition is stronger than the one by Balcar and Dow. They are known to be equivalent for ultrafilters over the natural numbers, and we proved this is still true for the Cohen algebra. On the other hand we showed that (consistently) there are ultrafilters on the power set algebra of the first uncountable cardinal that are Balcar-Dow-comparable but not Murakami-comparable.
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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
While we had a rocky start with few results in the first few months, things improved drastically in February and March, and right now we are progressing more smoothly than originally planned.
In particular, after some preparatory work in the first months, we recently obtained most of the above mentioned results, namely the theorem saying that the ultrafilter number on the Cohen algebra is larger or equal than the cofinality of the meager ideal, as well as the results about the Rudin-Frolik ordering for the power set algebra of uncountable cardinals and for the Cohen algebra.
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| Strategy for Future Research Activity |
We plan to continue our research on the structure of ultrafilters on Boolean algebras, with focus on (1) the Rudin-Frolik ordering and other orderings on such ultrafilters and (2) the connection between combinatorial properties of such ultrafilters and the order structure of partitions of the underlying Boolean algebra. For topic (1), the ultimate goal is an extension of the Ultrapower Axiom to ultrafilters on Boolean algebras, and for (2) we first plan to concentrate on establishing a connection between cardinal invariants of the null ideal and the order of partitions of the random algebra.
As explained in the original project all costs are for travel expenses. The research fellow will attend the Young Set Theory Workshop in Budapest in June and the European Set Theory Conference in Muenster in September, to present our research results.
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