2006 Fiscal Year Final Research Report Summary
Combinatorics related to the continuum
| Project/Area Number |
17540116
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kobe University |
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Principal Investigator |
BRENDLE Jorg Kobe U, Graduate School of Science and Technology, Associate Professor, 大学院自然科学研究科, 助教授 (70301851)
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| Co-Investigator(Kenkyū-buntansha) |
FUCHINO Sakae Chubu University, Faculty of Engineering, Professor, 工学部, 教授 (30292098)
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| Project Period (FY) |
2005 – 2006
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| Keywords | Foundations of Mathematics / Topology / Set Theory / Forcing Theory / Infinitary Combinatorics / Descriptive Set Theory / Cardinal Invariants of the Continuum |
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| Research Abstract |
We investigate the combinatorial structure of the reals and its interplay with forcing theory, as well as other areas of set theory. Particular focus is put on structures like P(ω)/fin.
(1) Distributivity numbers. Using finite support iteration of Laver forcing L_F with respect to a filter F, we prove the consistency of η(P(ω)/fin x P(ω)/fin) < η(C^ω/fin) where η(A) is the distributivity number of a Boolean algebra A, and C is the Cohen algebra. This answers a question of Dow.
(2) Groupwise density numbers. Let g be the groupwise density number, and let g_f be the groupwise density number for ideals. We show the consistency of g < g_f, thus answering a question of Mildenberger.
(3) Topological groups. Let G = ([ω]^<<ω>, Δ) be the group of finite subsets of the natural numbers ω equipped with symmetric difference Δ as group multiplication. In joint work with Michael Hrusak, we obtain the consistency of the statement for all ω_1-generated filters F on ω, the group topology on G corresponding to F is not Frechet.
(4) Forcing theory. Using a novel iteration technique for ccc forcing, we obtain a new proof of Shelah's result saying that u < a is consistent where u is the ultrafilter number and a, the almost disjointness number.
(5) Mad families with strong combinatorial properties. In joint work with Greg Piper, we construct, under the continuum hypothesis CH, a maximal almost disjoint (mad) family which is simultaneously a ο-set, as well as a mad family which is concentrated on a countable subset. This confirms two conjectures of A. Miller.
(6) Homogeneity properties of product-like models. In joint work with Sakae Fuchino, we investigate several combinatorial principles which hold in generic extensions by partial-orders which admit many automorphisms. In particular, we show the homogeneity principle HP(κ) implies the combinatorial principle C^s(κ) of Juhasz, Soukup and Szentmiklossy, and we prove HP(N_2) holds in product models like the Cohen model.
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Research Products
(26 results)
