Quantization, Singularities and Holomorphic Dynamics

在这个项目中，团队聚集在一起的目标是将我们的专业知识与对量子化，全纯动力学和叶理理论中的一些基本问题和猜想做出重要贡献的目标结合起来。我们将展示和利用这些领域之间的深厚联系，并将它们应用于各种悬而未决的问题。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数学的这些部分的更强的统一。以复杂环境下电流理论的发展和Bergman/Szegö核（包括L^2方法）及其在许多主题研究中的系统利用为基础，我们解决以下相关问题：Kähler空间和柯西-黎曼流形上的量化和约简的交换；哈密顿的行为;Kähler电位和适应复杂结构空间的量化；复空间上的Bergman核渐近性，解析扭转，Newlander-Nirenberg定理全纯叶的奇点和积累点，特别是具有非双曲奇点的；奇异全纯叶的唯一遍历性全纯动力系统的相位和参数空间动力学现象的定量计数随机全纯截面的零的等分布。

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.