| 研究課題/領域番号 |
23KF0257
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| 研究種目 |
特別研究員奨励費
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| 配分区分 | 基金 |
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| 応募区分 | 外国 |
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| 審査区分 |
小区分12030:数学基礎関連
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| 研究機関 | 神戸大学 |
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研究代表者 |
Brendle Jorg 神戸大学, システム情報学研究科, 教授 (70301851)
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| 研究分担者 |
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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| 研究期間 (年度) |
2023-11-15 – 2026-03-31
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| 研究課題ステータス |
交付 (2023年度)
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| 配分額 *注記 |
2,000千円 (直接経費: 2,000千円)
2025年度: 600千円 (直接経費: 600千円)
2024年度: 1,200千円 (直接経費: 1,200千円)
2023年度: 200千円 (直接経費: 200千円)
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| キーワード | Set theory / Boolean algebras / Topology / Ultrafilters / Forcing |
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| 研究開始時の研究の概要 |
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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| 研究実績の概要 |
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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| 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
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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| 今後の研究の推進方策 |
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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