I study von Neumann algebras, specifically the Jones theory of subfactors and planar algebras. I received my PhD in 2006 at UC Berkeley; my adviser was Vaughan Jones.

Algebras of operators on Hilbert space were introduced by von Neumann in the early twentieth century to provide a mathematical framework for quantum mechanics. In the early 1980's Vaughan Jones examined symmetries associated to certain inclusions of von Neumann algebras (subfactors) and found a powerful new knot invariant. This astonishing discovery introduced a new geometric dimension to the theory and revealed deep connections to low dimensional topology, quantum groups, and statistical mechanics.

A subfactor can be viewed as a quantum analogue of a group and is described by a standard invariant, which  is a tensor category of bimodule representations. The standard invariant has a rich algebraic and combinatorial structure, but also a strong geometric component: the standard invariant has the structure of a planar algebra, which is an algebra over the operad of planar tangles.

Subfactor theory is often thought of as a noncommutative Galois theory, and so determining the possible configurations of intermediate subfactors (algebras sandwiched by a subfactor) is a basic and central question. My research has focused on classifying noncommuting pairs of intermediate subfactors, using planar algebra techniques along with algebraic methods of bimodule calculus in tensor categories. This work has led to some strong and unexpected rigidity results. More generally, I am interested in exploring the planar algebra approach to subfactors and its connections to topology and mathematical physics.

Publications:

Intermediate Subfactors with No Extra Structure, with Vaughan F. R. Jones,
Journal of the AMS, 20 (2007), 219-265

Forked Temperley-Lieb Algebras and Intermediate Subfactors, Journal of Functional
Analysis, 247 (2007), 477-491

Classification of Noncommuting Quadrilaterals of Factors, with Masaki Izumi, 19 (2008), 557-643