Approximation and K-theory

传统上，数学研究精确的解：例如，一个人有一个未知量为x的方程，他想准确地知道x的哪个值(S)让它成为真的。通常在现实世界中，好的近似解总是非常接近精确解，因此也同样适用于实际目的。然而，有时一个问题的非常好的近似解可能存在，而根本没有任何实际的解决方案：例如，这是半导体物理中与本项目中使用的K理论方法密切相关的部分的基本现象。找到问题的近似解决方案，并看看可以用它们做什么，而不是精确解决方案的想法渗透到这些研究项目中。PI将继续他自己在这些方向上的研究，以及指导和建议初级研究人员，向更广泛的社区传播这一结果。在这个项目中使用的方法和研究的问题来自算子代数(量子物理推动的数学的一个分支)、群论(对称性研究)和K-理论(一种测量几何中“扭曲”现象的方法)。PI将通过首先求解近似形式来研究算子代数理论中的基本分类问题；正如其他研究人员最近所揭示的那样，这将在这类对象的结构中使用内在的低维现象。相反，PI打算研究高维障碍对对称群支配的矩阵方程组的近似解的存在的影响。最后，PI打算研究围绕精确度失败而出现的令人惊讶的分析现象，以及这一现象与几何的联系。这一奖项反映了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。

Traditionally, mathematics studies precise solutions: one has for example an equation with an unknown quantity 'x', and wants to know precisely which value(s) of x make it true. Often in the real world, good approximate solutions will always be very close to precise solutions, and are therefore just as good for practical purposes. However, sometimes very good approximate solutions to a problem can exist without there being any actual solution at all: for example, this is a fundamental phenomenon in parts of semi-conductor physics that are closely related to the 'K-theoretic' methods used in this project. The ideas of finding approximate solutions to problems and seeing what can be done with them versus precise solutions permeate these research projects. The PI will continue his own research in these directions, as well as mentor and advise junior researchers, disseminate the results of this to the broader community.The methods used, and problems studied, in this project come from operator algebras (a branch of mathematics motivated by quantum physics), group theory (the study of symmetry), and K-theory (a way of measuring "twistedness" phenomena in geometry). The PI will study fundamental classification problems in the theory of operator algebras by first solving approximate versions; this uses intrinsically low-dimensional phenomena in the structure of such objects, as recently revealed by other researchers. In contrast, the PI intends to study high-dimensional obstructions to the existence of approximate solutions to systems of matrix-equations governed by symmetry groups. Finally, the PI intends to study the surprising analytic phenomena that come up around the failure of exactness, and the connections of this to geometry.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.