Rings of bounded operators on Hilbert space were first studied by Murray and von Neumann in the 1930s. These rings, now called von Neumann algebras, arose in part from quantum physics, and have the flavor of measure theory. Their topological analogs, C*-algebras, were introduced by Gelfand and Naimark in the 1940s. These operator algebras interact with each other, and with many branches of mathematics; enduring interest in them is largely due to the fact that they can encode many other mathematical structures, such as symmetries, time-evolving systems, graphs, number fields, etcetera.

Connes’ Fields Medal-winning work on the structure and classification of amenable von Neumann algebras in the 1970s was a pivotal moment in the theory. Topological (i.e., C*-algebraic) analogs of these breakthroughs have been several decades in the making, beginning in earnest in the early 90s with Elliott’s classification program. After a brief overview of this large-scale, worldwide endeavor, I will discuss my recent joint work with Gabe, Schafhauser, Tikuisis and White leading up to a proof of the capstone result in the program: the classification of simple regular C*-algebras of finite non-commutative covering dimension (modulo the universal coefficient theorem).