Approximation and K-theory

University Of Hawaii, Honolulu

Investigators

Abstract

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.