General Galois theory for differential and difference equations and its applications to dynamical systems

(1)伽罗瓦理论的等价性。我们有两个一般的微分伽罗瓦理论：(A)2001年提出的Malgrange理论和(B)1996年提出的我们的理论。第一个是几何的，第二个是代数的。通过射流空间的代数构造，证明了这两个理论是等价的。(2)差分方程广义Galois理论的构造及其在动力系统中的应用。我们成功地构造了差分方程广义差分伽罗瓦理论。该理论的一个很好的应用是紧黎曼曲面上的离散动力系统。事实上，我们可以确定具有有限维伽罗瓦群的紧黎曼曲面上的所有离散动力系统。由于这些系统的Galois群是可解的，我们可以说我们已经确定了紧黎曼曲面上的所有可积离散动力系统。(3)微分Galois理论的量子化我们的学生F.Heiderich发现，我们可以将Galois理论推广到差分和微分框架之外。他建议我们可以建立关于算子的泛函方程的类似理论，这一观点最吸引人的特点是我们可以量化伽罗瓦理论。我们首先在这个方向上进行了研究，并利用了一些例子。

(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.