Complex Analysis and Geometry

复分析和几何提出的研究涉及复分析和代数或辛几何的接口上的几个领域。概述的主要项目旨在发展一个Delzant理论，以分类完全可积系统的紧凑流形与真实的代数型奇异的作用变量。这意味着至少包括由球形变种（Gelfand-Cetlin例子）和球形变种产生的系统。关键的工具将是玻尔-索末菲水平集和它们与哈密顿动力学中矩映射几何的相互作用。其他要处理的主题包括外部Monge-Ampere解决方案的规律性和他们的关系，多能理论;渐近几何无穷的复杂的流形，并重整化陈类; Hodge猜想的一些情况与从K3曲面导出的高维复流形上的特殊Hodge类有关，以及某些Grauert管的代数化。从哲学上讲，第一个项目基于一个可以追溯到量子力学开始的想法：力学系统的玻尔-索末菲能级，经典力学系统的一组配置，这将大大有助于对系统的量子力学理解，例如，原子的光谱线后来有一种用波函数对力学的双重描述，将两者联系起来是物理学和数学中伟大洞察力的源泉。我们怀疑我们已经发现，并试图证明的是，这种玻尔-索末菲对应是经典力学系统中某些分类问题的基础：有多少个可积系统？玻尔-索末菲能级与力学系统底层空间的经典轨迹直接相关，而相应的波函数允许人们读出系统的代数几何。提出的主要问题是，使这种“对应关系”严格的，在那里的系统可以被分类的组合关系之间的玻尔-索末菲轨迹，和一些全球性的代数几何数据。

Complex Analysis and GeometryThe research proposed deals with several areas on the interface of complex analysis and algebraic or symplectic geometry. The main project outlined seeks to develop a Delzant theory to classify completely integrable systems on compact manifolds with real algebraic type singularities of their action variables. These are meant to include the systems arising from spherical varieties (Gelfand-Cetlin examples) and those from quiver varieties at a minimum. The key tool would be Bohr-Sommerfeld level sets and their interplay with the geometry of the moment map in Hamiltonian dynamics. Other topics to be treated include regularity of exterior Monge-Ampere solutions and their relation to pluripotential theory; asymptotic geometry at infinity of complex manifolds, and renormalized Chern classes on them; some cases of the Hodge conjecture related to special Hodge-classes on higher dimensional complex manifolds derived from K3 surfaces, and the algebraicization of certain Grauert tubes.Philosophically, the first project is based on an idea going back to the beginning of quantum mechanics: the Bohr-Sommerfeld levels of a mechanical system, the set of configurations of a classical mechanical system which would contribute significantly to a quantum mechanical understanding of the system, as in, e.g., spectral lines of atoms. There was a later dual description of mechanics in terms of wave functions, and relating the two has been a source of great insight in physics and mathematics. What we suspect we have found, and are trying to prove, is that this Bohr-Sommerfeld correspondence underlies certain classification questions in classical mechanicalsystems: how many integrable systems are there? The Bohr-Sommerfeld levels are related directly to classical loci on the underlying space of the mechanical system, while the corresponding wave functions allow one to read off the algebraic geometry of the system. The main problem proposed is to make this "correspondence" rigorous, to the point where the systems can be classified by the combinatorial relations among the Bohr-Sommerfeld loci, and some global algebraic geometric data.