Mathematical Sciences: Advanced Training in Modern Analysis

这个奖项授予Arveson教授和Wolf教授，是在数学科学研究小组的倡议下颁发的。它将支持加州大学伯克利分校现代分析专业的博士后和研究生。该奖项支持的研究将涉及现代分析的各个方面，包括李群的表示理论、算子代数理论和遍历理论。作为利用系统固有对称性的数学工具，李群的表示理论对数学本身，特别是在分析和数论方面，以及对理论物理，特别是量子力学和基本粒子物理，都产生了深远的影响。算符代数的理论始于算符的概念，算符可以被认为是复数的有限或无限矩阵。特殊类型的运算符通常放在一个代数中，自然称为运算符代数。这些抽象对象具有令人惊讶的各种应用。例如，它们在纽结理论中扮演着关键的角色，而纽结理论目前正被用于研究DNA的结构，并且它们在非对易几何中具有基本的重要性，这在物理学中变得越来越重要。一般说来，遍历理论涉及理解系统的平均行为，这些系统的动力学太复杂或太混乱，以至于无法在微观细节上进行跟踪。在标题“动力学可以被放在现代理论如何抽象变换如何作用于光滑空间。这样，遍历理论与几何联系起来，寻求在均匀空间上对流动进行分类。

This award to Professors Arveson and Wolf is made under the Mathematical Sciences Research Group initiative. It will support postdocs and graduate students in Modern Analysis at the University of California at Berkeley. The research supported by this award will deal with various aspects of Modern Analysis including the representation theory of Lie groups, the theory of operator algebras, and ergodic theory. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics. The theory of operator algebras begins with the notion of operator, which can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.