2006 Fiscal Year Final Research Report Summary
Algebra, Geometry, Analysis in non-linear equations
| Project/Area Number |
15340004
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Nagoya University |
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Principal Investigator |
UMEMURA Hiroshi Nagoya University, Graduate school of mathematics, Professor, 大学院多元数理科学研究科, 教授 (40022678)
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| Co-Investigator(Kenkyū-buntansha) |
FUJIWARA Kazuhiro Nagoya University, Graduate school of mathematics, Professor, 大学院多元数理科学研究科, 教授 (00229064)
OKADA Soichi Nagoya University, Graduate school of mathematics, Professor, 大学院多元数理科学研究科, 教授 (20224016)
OKAMOTO Kazuo Tokyo University, Graduate school of Mathematical Science, Professor, 大学院数理科学研究科, 教授 (40011720)
MUKAI Shigeru Kyoto University, Research Institute of Mathematical Science, Professor, 数理解析研究所, 教授 (80115641)
NOUMI Masatoshi Kobe University, Faculty of Science department of Mathematics, Professor, 理学部, 教授 (80164672)
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| Project Period (FY) |
2003 – 2006
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| Keywords | Differential Galois theory / Groupoid / Painleve equations / Elliptic Painleve equations / Lie pseudo-group |
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| Research Abstract |
One of the central theme of this research is the differential Galois theory of infinite dimension. The head investigator presented in 1996 one such theory. Malgrange interested in this theory himself proposed a general differential Galois theory. Umemura's differential Galois theory is a Galois theory of differential field extension. Namely when a differential field extension L/K is given, we construct a kind of Galois closure of the extension, The Galois group p is th infinitesimal automorphism group of this Galois closure. On the other hand, Malgrange's Galois roupois is attached to a foliation on a variety. Namely when a foliation F on a variety is given, the Galois groupoid is the smallest algebraic Lie groupoid whose Lie algebra contains the tangent vectors to the foliation F.
These two definitions are seemingly different but specialists observed that they coincide in Examples. In recent three years the development in this direction was remarkable. On can show in the absolute case L/K, by which we mean the base field K is a subfield in the constant field of L, these two definitions are equivalent. The proof is done through the universal
On the other hand one of the investigator of this project, M. Noumi at Kobe University studied the most general Painleve equation or the Master equation called the elliptic Painleve equation. He showed among other things that as Riccati solutions to the elliptic Painleve equation, there appear hyperelliptic geometric functions. This result is one of the most remarkable results in this field of research.
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Research Products
(12 results)
