Operator Theory and Complex Geometry

PI打算用算子理论的技术来研究复杂几何。这包括研究几个变量的有界域的解析子变种，其中向量值Hardy空间理论为几个复变量引入了额外的技术。PI还打算研究狄利克雷级数的希尔伯特空间，意图用希尔伯特空间几何来阐明狄利克雷级数的经典问题。反馈机制已经在工程中使用了很长时间，主要是为了稳定系统。设计这样的系统是困难的，尤其是因为人们想要稳定的系统在实践中是不确切的，而只是近似的。许多特别的方法被使用，但大约25年前，人们意识到许多问题可以系统地使用称为算子理论的数学领域的技术来解决。这是如此成功，以至于工程师们现在试图回答的问题的范围急剧增加，并且非常需要为现在正在制定的问题找到数学答案。一个重要的领域是在一个维度上很好地理解问题的多维概括。该计划建议着手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.