2003 Fiscal Year Final Research Report Summary
A research on generic constructions
| Project/Area Number |
14540146
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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 | Hosei University (2003)
Toyota National College of Technology (2002) |
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Principal Investigator |
IKEDA Koichiro Hosei University, Faculty of Business Administration, Associate Professor, 経営学部, 助教授 (60332029)
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| Project Period (FY) |
2002 – 2003
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| Keywords | generic structure / stable theory / Lachlan's conjecture / pseudoplane / simple theory / omega-categorical |
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| Research Abstract |
The aim of this project was to get a clue to a solution of the following three problems :
Lachlan's conjecture : For any stable theory T, the number of countable models of T is one or infinite. Hodges' problem : Is there an omega-categorical projective plane?
Baldwin's problem : Is any superstable generic structure omega-stable?
In 2002, we obtained the following proposition ;
Proposition 1 : If a generic pseudoplane M is strictly stable, then cl(A) is finite for any finite subset A of M. The result was published in Kokyuroku of the Research Institute of Mathematical Sciences in Kyoto. In August 2002, we held a reserch meeting in Yamanakako, and I discussed with model theorists. After this meeting, I got the following two theorems :
Theorem 2 : There is no omega-categorical generic projective plane.
Theorem 3 : Any generic superstable pseudoplane is omega-stable.
Theorem 2 and 3 are partial results of Hodges' problem and Baldwin's problem respectively. The former result was published in Notre Dame Journal of Formal Logic. The latter result was published in Kokyuroku of the Research Institute of Mathematical Sciences in Kyoto. In July 2003, we held a research meeting in Hachioji, and I participated a project organized by Byunghan Kim. After that, I got some proposition result characterizing a generic structure whose theory is simple. This result was published in Kokyuroku of the Research Institute of Mathematical Sciences in Kyoto.
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