Becky Armstrong

Groupoids in Operator Algebra and Abstract Algebra

Abstract: Many abstract mathematical concepts have well known applications to popular puzzles and games, with one famous example being the connection between Group Theory and Rubik's cubes. The Fifteen Puzzle consists of a 4-by-4 grid of sliding tiles numbered 1 to 15, and is considered solved when these tiles are arranged consecutively. In this two-part lecture series, I will introduce a generalisation of groups, called groupoids, which can be exploited to solve the Fifteen Puzzle. I will then explain the major role that groupoids play in my research areas of Operator Algebra and Abstract Algebra.

Operator Algebra

C*-algebras are a class of mathematical objects that were first introduced to model quantum-mechanical phenomena. Much of the current research of C*-algebraists involves constructing interesting classes of C*-algebras from various mathematical objects, such as directed graphs, groups, and groupoids. In my lectures I will introduce groupoid C*-algebras, and I will discuss the role the underlying groupoid plays in understanding the structure of the associated C*-algebra.

Abstract Algebra

The quantum-mechanical applications of C*-algebras require that they are analytic in nature, in the sense that they are normed algebras (so we can make sense of limits and continuity). However, a significant portion of my C*-algebraic research involves overcoming analytic technicalities by working with purely algebraic (non-C*-algebraic) dense subalgebras of groupoid C*-algebras, called Steinberg algebras. Steinberg algebras are not just a useful operator-algebraic tool---they are also of independent interest to abstract-algebraists. In my lectures I will discuss Steinberg algebras and the role they have played in both Operator Algebra and Abstract Algebra.