Mathematical Sciences: Toeplitz Operators, Quantum Mechanicsand Mean Oscillation

Coburn教授将继续研究与Berezin - Toeplitz量化相关的符号演算。一种特殊的设置是具有高斯测度的复n空间，并且在符号的奇异点上具有适度的生长和振荡条件。Coburn将根据符号的平均振荡来寻求在这种情况下有界Hankel算子的特征。讨论了球和多盘Hardy空间的有界性问题。Coburn期望在这些问题上发展起来的技术在研究各种自然代数的推导时是有用的。这一数学研究领域的推动力来自量子力学。在经典的，前量子世界的概观中，物理系统的可观察对象是数字，或者，如果想要跟踪这些数字对其他一些量的依赖关系，则是函数。这是常识，但在微观层面上却行不通。我们需要用算子来代替函数，通常是无限维空间的线性变换。这种替代就是数学物理学家所说的量子化。一般的想法是，除了包含重要信息的修正项外，运算符的行为几乎就像它们产生的函数一样。在这条努力的道路上，物理的洞察力常常会影响数学，反之亦然。

Professor Coburn will continue his study of the symbol calculus associated with the Berezin - Toeplitz quantization. One specific setting is complex n-space with Gaussian measure, and with moderate growth and oscillation conditions on the singularities of the symbols. Coburn will seek to characterize bounded Hankel operators in this setting in terms of mean oscillation of symbols. The corresponding boundedness problem for the Hardy spaces of the ball and polydisc will also be attacked. Coburn expects the techniques developed in working on these problems to be useful in the study of derivations on various naturally occurring algebras. The impetus for this general area of mathematical research comes from quantum mechanics. In the classical, pre-quantum overview of the world, the observables of a physical system are numbers, or, if one wants to keep track of the dependency of these numbers on some other quantities, functions. This is common sense, but at the microscopic level it doesn't quite work. One needs to replace functions by operators, linear transformations of a space that is usually infinite-dimensional. This replacement is what mathematical physicists call quantization. The general idea is that the operators behave almost like the functions from which they arise, except for correction terms which turn out to contain important information. In this line of endeavor, physical insight informs mathematics as often as the reverse.