Quantization, Singularities and Holomorphic Dynamics

在这个项目中，团队的目标是将我们的专业知识与量子化，全纯动力学和叶状理论中的一些基本问题和结构做出重要贡献的目的捆绑在一起。我们将展示和利用这些领域之间的深刻联系，并使他们承担各种开放的问题。我们的目标是提供新的视角和新颖的解决问题的策略，以涵盖这些领域，并在长期内促进更广泛的研究社区更强大的统一这些部分的数学。 作为一个基石的理论发展的电流在复杂的设置和伯格曼/赛格内核（包括L^2方法）和它们在一些主题的研究中的系统开发，我们解决了以下相互关联的问题：Kähler空间和Cauchy-Riemann流形上的量子化和约化的交换; Kähler势空间和适应复结构的量子化; Bergman核渐近性、解析挠率、复空间上的Newlander-Nirenberg定理;全纯叶理的叶的奇点和聚点，特别是具有非双曲奇点的叶;奇异全纯叶理的唯一遍历性;全纯动力系统在相空间和参数空间中动力学现象的定量计数;随机全纯截面零点的等分布。

The goal of the team brought together in this project is to bundle our expertise with the aim of making important contributions to a number of fundamental problems and conjectures in Quantization, Holomorphic Dynamics and Foliation Theory. We will exhibit and exploit the deep ties between these areas and bring them to bear on diverse open questions. Our goal is to provide fresh perspectives and novel problem-solving strategies to encompass these fields and in the long-term to foster in the wider research community a stronger unification of these parts of mathematics.Using as a cornerstone the development of the theory of currents in the complex setting and of the Bergman/Szegö kernel (including L^2 methods) and their systematic exploitation in the study of a number of topics, we address the following interrelated questions: commutation of quantization and reduction on Kähler spaces and Cauchy-Riemann manifolds; hamiltonian actions; quantization of the space of Kähler potentials and of adapted complex structures; Bergman kernel asymptotics, analytic torsion, Newlander-Nirenberg theorem on complex spaces; singularities and accumulation points of a leaf of a holomorphic foliation, especially with non-hyperbolic singularities; unique ergodicity for singular holomorphic foliations; quantitative counting of dynamical phenomena for holomorphic dynamical systems, both in the phase and parameter spaces; equidistribution of zeros of random holomorphic sections.