2012 Fiscal Year Final Research Report
General Galois theory for differential and difference equations and its applications to dynamical systems
| Project/Area Number |
20540041
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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 |
Algebra
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| Research Institution | Nagoya University |
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Principal Investigator |
UMEMURA Hiroshi 名古屋大学, 大学院・多元数理科学研究科, 名誉教授 (40022678)
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| Co-Investigator(Kenkyū-buntansha) |
KAWAHIRA Tomoki 名古屋大学, 大学院・多元数理科学研究科, 准教授 (50377975)
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| Project Period (FY) |
2008 – 2012
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| Keywords | 代数学 / 解析学 / 関数方程式 / 幾何学 |
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| Research Abstract |
(1) Equivalence of Galois theories. We had two general differential Galois theories: (a) Malgrange’s theory proposed in 2001 and (b) ours in 1996. The first is geometric and the second is algebraic. We proved that these two theories are equivalent by algebraic construction of the jet space and thus Lie groupoid of automorphisms.
(2) Construction of general Galois theory for difference equations and its applications to dynamical systems. We succeeded in constructing general difference Galois theory for difference equations. An elegant application of the theory is discrete dynamical systems on compact Riemann surfaces. In fact, we can determine all the discrete dynamical systems on compact Riemann surfaces that have finite dimensional Galois groups. Since the Galois group of these systems are solvable, we may say we have determined all the integrable discrete dynamical systems on compact Riemann surfaces,.
(3) Quantization of differential Galois theory Our student F.Heiderich discovered that we can generalize our Galois theory beyond difference and differenctial framework. He suggests that we can establish a similar theory for functuinal equations with respect to operators, The most fascinating feature of this view point is we may quantizeGalois theory. We started a irst research in this directions exploiting examples.
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Research Products
(17 results)
