Operator Algebra Theory in Applications

顺序很重要。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。作为一个例子，人们可能会想到加热水和加入油。经过近一个世纪的努力，数学家们终于接受了量子力学的挑战，并发展了一种可以研究非交换运算的理论。本项目资助的工作将从字面上理解这一挑战，从微积分中研究经典运算，并引入空间、时间和所有在空间和时间中执行的运算的顺序依赖性。该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。量子信息理论修改了香农的理论，即通过噪声信道向量子设备发送可靠信息，希望利用“隐形传态”或其他非传统的“怪异行为”。更具体地说，提出的工作旨在研究非交换几何和非交换谐波分析之间的联系。目标是采用非交换几何中的一些术语，例如曲率，并表明它可以用于非交换空间上涉及拉普拉斯和狄拉克算子的估计。量子信息理论的工作将发展熵相关表达式的不等式，并将其应用于容量、不对称和平衡时间的概念。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.