Moduli spaces of meromorphic connections and the Fourier transform

斯托克斯现象出现在具有不规则奇异点的复杂微分方程理论中，众所周知，微分系统是由它的斯托克斯数据决定的。傅里叶-拉普拉斯变换是所有数学和物理中一个重要的积分变换，Stokes数据在该变换下的行为是一个活跃的研究领域。在这个项目中，我们的目标是通过结合文献中非常感兴趣的两个问题来更好地理解傅里叶-拉普拉斯变换的几何：一方面，Stokes数据的傅里叶-拉普拉斯变换的显式计算；另一方面讨论了亚纯微分方程的模空间的构造。具体地说，我们首先想对Stokes数据进行傅里叶-拉普拉斯作用的显式计算，利用最近的发展为我们提供了强大的工具来考虑这些问题：D 'Agnolo-Kashiwara的增强前轴理论。其次，我们将利用这些结果描述Stokes数据模空间上傅里叶-拉普拉斯变换引起的同构，并研究它们在这些空间上对几何结构的行为。

Stokes phenomena occur in the theory of complex differential equations with irregular singularities, and it is well-known that a differential system is determined by its Stokes data. The Fourier-Laplace transform is a prominent integral transform in all of mathematics and physics, and the behaviour of Stokes data under this transformation is an active field of research. In this project, we aim at understanding better the geometry of the Fourier-Laplace transformation by combining two questions that have been of great interest in the literature: On the one hand, the explicit computation of Fourier-Laplace transforms of Stokes data; on the other hand the construction of moduli spaces of meromorphic differential equations. Concretely, we first want to perform explicit computations of the Fourier-Laplace action on Stokes data, making use of recent developments which give us powerful tools for such considerations: the theory of enhanced ind-sheaves of D’Agnolo-Kashiwara. Secondly, we will use these results to describe the isomorphisms induced by Fourier-Laplace transform on moduli spaces of Stokes data and study their behaviour with respect to geometric structures on these spaces.