Research in Several Complex Variables and Applications

提案号DMS-0203072PI: Laszlo lempertabstract本项目由两个部分组成。一个是关于无限维流形的复杂分析，所提出的问题在本质上大多是上同调的。它们围绕着Dolbeault和sheaf上同群的计算。本文将研究两种类型的流形：有限维紧致复流形的环空间和Stein流形。一个（更长期的）目标将是证明循环空间上的Hirzebruch—Riemann—Rochtype指标定理，另一个目标是发现一种合适的轴类，其上同调可以在无限维Stein流形上消失（相干轴概念的推广）。我还将致力于在希尔伯特空间的伪凸域中求解柯西-黎曼非齐次方程。有限维分量是将全纯运动的概念从复平面推广到高维复流形及其子集的运动。这将有望为高维复杂动力学提供一个有用的工具。一般来说，数学阐明了构成我们探索世界的基础的形式结构。这种结构的一个例子是一个函数，它形式化了变量之间的“依赖”概念。例如，地球上某个地点的温度取决于该地点的纬度和经度，但也取决于海拔和测量时间：我们有一个由四个变量组成的函数。在另一种情况下，每摩尔气体的温度取决于每个气体分子的速度，因此取决于数万亿个变量。要理解包含如此多变量的函数，最好将其理想化为无限多变量。这些是我们提出的研究的主要对象：无穷多变量的函数。变量是复数而不是实数。引入复杂变量通常会简化问题，并且在解决了复杂问题之后，通常可以通过限制返回并获得原始实际问题的解。要考虑的问题部分是由各种数学学科和理论物理所驱动的。

Proposal Number DMS-0203072PI: Laszlo LempertABSTRACTThis project has two components. One concerns complex analysison infinite dimensional manifolds, and the proposed problems are mostly cohomological in nature. They center around thecomputation of Dolbeault and sheaf cohomology groups. Two typesof manifolds will be investigated: loop spaces of finitedimensional compact complex manifolds and Stein manifolds. One(longer term) goal will be to prove a Hirzebruch--Riemann--Rochtype index theorem on loop spaces, another to discover anappropriate class of sheaves for which cohomology can be provedto vanish on infinite dimensional Stein manifolds (generalizationof the notion of coherent sheaves). I will also work on solvingthe inhomogeneous Cauchy--Riemann equations in pseudoconvexdomains in Hilbert spaces. The finite dimensional component isabout extending the concept of holomorphic motion from the complex plane to motions of higher dimensional complex manifoldsand their subsets. This will hopefully provide a useful tool forhigher dimensional complex dynamics.Mathematics in general elucidates the formal structuresthat underlie the exploration of our world. An example ofsuch a structure is a function, which formalizes the notionof "dependence" among variable quantities. For instancetemperature at some location on Earth depends on thelatitude and the longitude of the location, but also onthe altitude, and the time of the measurement: we have a functionof four variables. In another context, the temperature of saya mole of a gas depends on the velocity of each molecule of thegas, thus on trillions and trillions of variables. Functionsof such a huge number of variables are best understood byidealizing to infinitely many variables. These are the mainobjects of the proposed research: functions of infinitelymany variables. The variables are complex rather than realnumbers. Introducing complex variables typically streamlinesproblems, and having solved the complexified problem one canoften go back and obtain, by restriction, a solution to theoriginal real problem. The problems to be considered are partlymotivated by various mathematical disciplines, and also bytheoretical physics.