Operator Theory and Complex Geometry

PI打算使用算符理论的技术来研究复杂的几何。这包括研究多变量有界域的解析子簇，其中向量值Hardy空间理论引入了几个复变量的额外技巧。PI还打算研究Dirichlet级数的Hilbert空间，目的是利用Hilbert空间测量学来阐明关于Dirichlet级数的经典问题。反馈机制在工程中已经使用了很长时间，主要是为了稳定系统。这种系统的设计是困难的，尤其是因为人们想要稳定的系统在实践中并不是确切知道的，而是大致知道的。人们使用了许多特殊的方法，但大约25年前人们意识到，许多问题可以用数学领域中称为算子论的技术系统地解决。这是成功的，工程师现在试图回答的问题的范围已经急剧增加，并且非常需要为现在正在形成的问题找到数学答案。一个重要的领域是在一个维度上被很好地理解的问题的多维概括。PI提议解决几个这样的问题。

The PI intends to study complex geometry using the techniques ofoperator theory. This includes studying analytic subvarieties ofbounded domains in several variables, where the theory ofvector-valued Hardy spaces introduces extra techniques to severalcomplex variables. The PI also intends to study Hilbert spacesof Dirichlet series, with the intention of using Hilbert spacegeometry to shed light on classical questions about Dirichletseries.Feedback mechanisms have been used in engineering for a long time,largely to stabilize systems. The design of such systems isdifficult, not least because the system that one wants tostabilize is, in practice, not known exactly, but onlyapproximately. Many ad hoc approaches wereused, but about 25 years ago it was realized that many problemscan be approached systematically using techniques froman area of mathematics called Operator Theory. This has been sosuccessful that the range of questions engineers now attemptto answer has increased dramatically, and there is a great need tofind mathematical answers to the questions now being formulated.One area of importance is the multi-dimensional generalizationsof questions that are well understood in one dimension. The PIproposes to work on several such problems.