The investigator will study a number of problems which, roughly speaking, belong to representation theory. The questions, however, differ from the classical theory (e.g. classification of irreducible representations). The goal is to a) better understand finite factor representations of certain quasidiagonal C*-algebras, b) study approximation properties of traces and c) work on applications of the previous two topics to various open questions in operator algebras. At this point, our hope is that this work will shed light on Elliott's classification program and Connes' embedding problem. However, there may be more as we have recently noticed that these ideas have K-homological implications, can be used to prove a general existence result for the finite section method (from numerical analysis) and we would not be surprised if connections with geometric group theory were soon worked out. Though these ideas are certainly in their infancy, they have solid historical foundations (e.g. Connes' uniqueness theorem for finite injective factors) and we believe the theory shows promise.
One very successful idea in mathematics is that problems about complicated objects can sometimes be solved using approximations by simpler objects. For example, in calculus we teach students that to compute the area under a curve one should first approximate by rectangles since the area of a rectangle is easy to compute. Operator algebras are (usually) infinite dimensional objects which provide the natural framework for many questions in quantum physics. Moreover, deep and unexpected connections with other areas of mathematics such as geometry, topology and probability were discovered over the years. As such, a solid understanding of the structure of operator algebras is important. The general philosophy of using approximations by simpler objects becomes especially relevant here since the objects of interest are infinite dimensional. The investigator will continue an established tradition of trying to use finite dimensional approximations to better understand some fundamental infinite dimensional objects.
