Mathematical Sciences: Functional Analysis

里费尔教授的项目包括综合各种想法 微分几何和算子代数理论的基础。 他将从几个角度研究变形的过程 几何结构通过传递到代数的功能或 一个李代数然后平滑地改变乘法。 研究的具体课题包括量子群， 变形量子化和杨-米尔斯理论。 Rieffel工作的基本概念是， 算子代数可以有利地看作是非交换的 类似于熟悉的几何物体。 例如， 开始与双环面，表面的内胎。 所有 关于该对象的形状的信息包含在 复值函数的代数。两个这样的 函数是那些测量相位的函数， 环面长的，短的，各一圈。 这些 函数生成连续函数的代数， 以某种普遍的方式形成环面。 改为两个（单一） 算子，一种广义函数，而不是 当相乘时，仅对一个相位进行换向。 为 相位变化，由这些生成的算子代数 射影可换么正形成一族非可换么正 tori，创造了一个几何思想可以被 在没有参考几何对象的情况下制定和研究， 通常的感觉。

Professor Rieffel's project involves a synthesis of ideas from differential geometry and the theory of operator algebras. He will study from several viewpoints the process of deforming geometric structures by passing to an algebra of functions or to a Lie algebra and then smoothly changing the multiplication. Specific subjects of investigation include quantum groups, deformation quantization, and Yang-Mills theory. Fundamental to Rieffel's work is the notion that certain algebras of operators can profitably be viewed as noncommutative analogs of familiar geometric objects. One might for instance begin with the two-torus, the surface of an inner tube. All of the information about the shape of this object is contained in the algebra of complex valued functions on it. Two such functions are the ones that measure phase as one goes around the torus the long, respectively the short, way around. These functions generate the algebra of continuous functions on the torus in a certain universal way. Replace them by two (unitary) operators, a species of generalized functions, that instead of commuting when multiplied, commute only up to a phase. As the phase varies, the operator algebras generated by these projectively commuting unitaries form a family of noncommutative tori, creating a setting in which geometric ideas can be formulated and studied without reference to a geometric object in the usual sense.