2006 Fiscal Year Final Research Report Summary
Structures of sets related to differentiability
| Project/Area Number |
16540087
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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 |
Geometry
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| Research Institution | Kinki University |
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Principal Investigator |
IZUMI Shuzo Kinki University, Science and engineering, professor, 理工学部, 教授 (80025410)
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| Co-Investigator(Kenkyū-buntansha) |
KOIKE Satoshi Hyogo University of Teacher Education, Science, Technology, and Human Life, professor, 学校教育研究科, 教授 (60161832)
FUKUI Toshizumi Saitama University, Science, professor, 理学部, 教授 (90218892)
SAKUMA Kazuhiro Kinki University, Science and engineering, assistant-professor, 理工学部, 助教授 (80270362)
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| Project Period (FY) |
2004 – 2006
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| Keywords | paratangent bundle / smooth function / Whitney's extension problem / interpolation |
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| Research Abstract |
Our main problem is an answer to Whitney's problem 1934, the problem of finding a condition for a function defined on a closed subset of a Euclidean space to be extendible to smooth function on the ambient space. We have given a partial answer to this problem in the paper Ann Inst. Fourier, Grenoble 2004. Namely we have shown that a certain kind of self similar sets have full higher order tangent space. Then, by a conjecture posed by Bierstone-Milman-Pawlucki, we get an positive answer to Whitney's problem for these sets.
Later, this problem has been completely solved by Fefferman (Annals of Mathematics 2007). Hence we have shifted the subject a little. We studied the higher order tangent space of a set itself. We have found that the interpolation theory plays an important role in the study of smooth functions. This yields a survey paper on interpolation theory in the book.
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