Approximation and K-theory

传统上，数学研究精确解：例如，一个方程有一个未知量“x”，并希望精确地知道x的哪些值使其为真。通常在真实的世界中，好的近似解总是非常接近精确解，因此对于实际目的来说也一样好。 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解：例如，这是半导体物理学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。 寻找问题的近似解决方案，看看能用它们做些什么，而不是精确的解决方案，这些想法渗透到这些研究项目中。 PI将继续在这些方向上进行自己的研究，并指导和建议初级研究人员，向更广泛的社区传播其结果。该项目中使用的方法和研究的问题来自算子代数（受量子物理学启发的数学分支），群论（对称性研究）和K理论（测量几何中“扭曲”现象的方法）。 PI将通过首先解决近似版本来研究算子代数理论中的基本分类问题;这在这些对象的结构中使用了固有的低维现象，正如最近其他研究人员所揭示的那样。 相比之下，PI打算研究高维障碍的存在近似解的矩阵方程组由对称群。 最后，PI计划研究围绕精确性失败出现的令人惊讶的分析现象，以及这种现象与几何学的联系。该奖项反映了NSF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。

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.