Operator Algebra Theory in Applications

秩序很重要。在现实生活中，很容易观察到，某些操作的顺序会极大地改变最终结果。举个例子，你可能会想到烧水和加油。在经历了近一个世纪后，数学家们终于接受了量子力学的挑战，开发出了一种可以研究非交换运算的理论。该项目资助的工作将从字面上接受这一挑战，研究微积分中的经典运算，并引入空间、时间以及在空间和时间中执行的所有运算的顺序依赖关系。理论工作的特定应用范围将在量子信息理论中获得新的见解。量子信息理论修正了香农的理论，即通过噪声通道将可靠的信息发送到量子设备，以期利用“隐形传态”或其他非传统的“幽灵行为”。更具体地说，这项拟议的工作旨在研究非对易几何和非对易调和分析之间的联系。我们的目标是改编非对易几何中的一些术语，例如曲率，并证明它可以用于非对易空间上涉及Laplace和Dirac算子的估计。量子信息论的工作将发展与熵相关的表达式的不等式，并将它们应用于容量、不对称性和达到平衡的时间的概念。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Order matters. In real life it is easy to observe that the order of certain operations greatly change the final outcome. As an example one may think of heating water and adding oil. After almost a century mathematicians have finally embraced the challenge from quantum mechanics and developed a theory which allows to study noncommuting operations. The work funded in this project will take this challenge literally and study classical operations from calculus and introduce an order dependence for space, time and all the operations performed in space and time. A particular range of applications of the theoretical work will gain new insights in quantum information theory. Quantum information theory modifies Shannon's theory of sending reliable information through noisy channels to quantum devices in the hope of taking advantage of "teleportation" or other non-traditional "spooky behaviour."More concretely, the proposed work aims to study the connection between noncommutative geometry and noncommutative harmonic analysis. The goal is to adapt some terminology from noncommutative geometry, for example curvature, and show that it an be used for estimates involving Laplace and Dirac operators on noncommutative spaces. The work in quantum information theory will develop inequalities for entropy related expressions and apply them to notions of capacity, asymmetry and time to equilibrium.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.