The primary thread of research will focus on emerging analogies between the structure of II_1 factors and simple C*-algebras. For example, there are now C*-analogues of McDuff's Theorem and Connes's "Injective implies McDuff" result (which played a fundamental role in classifying injective factors). The investigator hopes to explore these analogies with the ultimate goal of classifying simple C*-algebras of so-called finite nuclear dimension. In another direction, the investigator will continue working on C*-algebras associated to groups and related spaces that were recently introduced in joint work with Erik Guentner. Finally, we will continue our study of dynamical systems associated to II_1 factors which arise from certain homomorphisms.
One very successful idea in mathematics is that we can learn about complicated objects by approximating with simpler objects, then passing to a limit. For example, in calculus we compute the area under a curve using rectangular approximations, then refining the approximations over and over. Operator algebras are (usually) infinite dimensional objects which provide the natural framework for quantum mechanics, for example. 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.
