Approximate Commutators and K-theory

在经典物理学中，顺序并不重要：例如，如果你测量动量，然后测量位置，你得到的答案和你以相反的顺序测量的结果是一样的。在量子力学中，这是错误的：如果你在位置之前测量动量，你可能会得到不同的答案，而不是反过来测量动量；这是著名的海森堡测不准原理的一种表现。数学家称这种现象——顺序很重要——为“非交换性”。非交换系统比经典系统更难理解。这个项目的主要目的是更好地理解非交换系统，涉及所谓的“泛系数定理问题”。在一个非常粗略的层面上，它询问是否任何（合理的）非交换系统都可以被经典系统所取代，从而在经典系统中进行计算得到相同的结果。该项目的其他部分涉及将相同的思想直接应用于几何和对称研究中的近似问题。该项目还将涉及本科生和研究生的研究。该项目打算使用近似方法来研究所谓的普适系数定理（UCT）问题，该问题询问是否任何可分离核算子代数（可能高度非对易的）在k理论水平上等价于对易代数。最近，Elliott计划在分类简单核C*-代数方面取得了令人印象深刻的成功，这使这个问题得到了深刻的认识：它现在是获得最佳分类结果所需要的唯一额外因素。这些近似方法在其他领域也很有用：基于这些思想的两个项目研究了字双曲群的近似性质和流形上椭圆微分算子的有限维模型。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。

In classical physics, order does not matter: for example, if you measure momentum and then measure position, you get the same answer as if you do the measurements in the opposite order. In quantum mechanics, this is wrong: if you measure momentum before position, you can get a different answer than if you do the measurements the other way around; this is a manifestation of the famous Heisenberg uncertainty principle. Mathematicians call this phenomenon – that order matters – “noncommutativity.” Noncommutative systems are much harder to understand than classical ones. The bulk of this project, which aims to better understand noncommutative systems, concerns the so-called “universal coefficient theorem problem.” On a very rough level, it asks whether it is true that any (reasonable) noncommutative system can be replaced by a classical one, such that doing computations in the classical system gets the same results. Other parts of the project concern direct applications of the same ideas to approximation problems in the study of geometry and symmetry. The project will also involve both undergraduate and graduate students in research.The project intends to use approximation methods to study the so-called universal coefficient theorem (UCT) problem, which asks if any separable nuclear operator algebra (potentially highly noncommutative) is equivalent on the level of K-theory to a commutative one. The recent impressive successes in the Elliott program to classify simple nuclear C*-algebras have thrown this problem into sharp relief: it is now the only extra ingredient needed to get the best possible classification result. The approximation methods should be useful in other areas: two projects building on these ideas study approximation properties of word hyperbolic groups and finite-dimensional models for elliptic differential operators on manifolds.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.