Mathematical Sciences: Functional Analysis

Rieffel教授的项目综合了微分几何和算子代数理论的思想。他将从几个角度研究通过传递到函数代数或李代数，然后平滑地改变乘法来变形几何结构的过程。具体的研究主题包括量子群、形变量子化和杨-米尔斯理论。Rieffel工作的基础是这样一个概念，即某些算子的代数可以有益地视为熟悉的几何对象的非对易类似物。例如，可以从两个圆环开始，即内管的表面。关于该物体形状的所有信息都包含在其上的复值函数的代数中。两个这样的函数是测量相位的函数，当一个函数绕环绕长圆环旋转时，分别是短圆环绕圆环旋转。这些函数以某种普遍的方式生成环面上的连续函数的代数。取而代之的是两个(么正)运算符，这是一种广义函数，乘法时不是交换，而是只交换到一个阶段。随着阶段的变化，由这些射影交换单位生成的算子代数形成了一族非交换环面，创造了一种环境，在这种环境中，几何思想可以在不参考通常意义上的几何对象的情况下被公式化和研究。

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.