Mathematical Sciences: Toeplitz Operators, Quantum Mechanicsand Mean Oscillation

Coburn教授将继续研究与Berezin-Toeplitz量子化相关的符号演算。一种特定的设置是具有高斯测度的复n-空间，并且具有关于符号奇点的适度增长和振荡条件。Coburn将寻求在这种情况下用符号的平均振荡来刻画有界的Hankel算子。球和多圆盘的Hardy空间的相应有界性问题也将受到攻击。Coburn希望在解决这些问题时开发的技术在各种自然生成的代数上的导子的研究中有用。这个数学研究的一般领域的推动力来自量子力学。在经典的、量子之前的世界概观中，物理系统的可观性是数字，或者，如果你想跟踪这些数字对其他一些量的依赖关系，那么就是函数。这是常识，但在微观层面上却行不通。人们需要用算子来代替函数，这是一个通常是无限维的空间的线性变换。这种替代就是数学物理学家所说的量子化。一般的想法是，运算符的行为几乎与它们产生的函数相似，只是校正项后来被证明包含重要信息。在这方面的努力中，物理洞察力告诉数学的频率与反过来一样高。

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.