Mathematical Sciences: Projects in Modern Analysis

该奖项将支持一组以算子理论和算子代数为中心的数学家的研究项目。一些研究领域将是(i)泛函希尔伯特空间上的乘法算子，（ii）使用多变量技术的多项式次非正常，（iii）指数大于4的不可约子因子的构造，（iv）不变理论，Schur-Weyl对偶，以及单位根处量子群的张量积结构，(v)算子和协变系统代数的扩张理论，（vi）广义对易关系的单参数变形，（vii）随机符号的Toeplitz算子和奇异积分算子生成的算子代数，（viii）分析非自伴随算子代数的协调技术。这个研究项目的一个中心方面是对希尔伯特空间算子的研究。这些对象可以被认为是复数的无限矩阵，但它们通常以更适合各种应用的方式出现。在这些研究者的研究项目中，单个算子通常不是研究的对象，而是研究称为代数的大算子类。这种代数的结构是数学中最深入研究的领域之一。

This award will support the research projects of a group of mathematicians whose work centers around operator theory and operator algebras. Some of the areas of investigation will be (i) multiplication operators on functional Hilbert spaces, (ii) polynomial hyponormality using multivariable techniques, (iii) construction of irreducible subfactors with index greater than four, (iv) invariant theory, Schur-Weyl duality, and tensor product structure of quantum groups at roots of unity, (v) dilation theory for algebras of operators and covariant systems, (vi) one-parameter deformations of generalized commutation relations, (vii) operator algebras generated by Toeplitz and singular integral operators with random symbol, and (viii) coordinatization techniques to analyze non-self-adjoint operator algebras. A central aspect of this research project is an investigation of Hilbert space operators. These objects can be thought of as infinite matrices of complex numbers but they often arise in a way more suited to various applications. In the research projects of these investigators a single operator is usually not the subject of study but rather large classes of operators called algebras. The structure of such algebras is one of the most intensively studied areas of mathematics.