Research Interests: Von Neumann Algebras, Subfactors, Quantum Information Theory, Quantum Physics.
Motivated by quantum mechanics and group representation theory, John von Neumann introduced in the early 30's certain algebras of bounded operators on a Hilbert space, the so-called von Neumann algebras. In his bicommutant theorem he showed that these algebras can be characterized either in purely topological or in purely algebraic terms, a fact, that has numerous beautiful and deep consequences. Today, the theory of operator algebras has many fruitful connections to other areas of mathematics and theoretical physics, in particular to low dimensional topology, statistical mechanics and quantum field theory. For instance, Alain Connes' noncommutative geometry provides a frame-work for the standard model in modern particle physics and implies predictions of properties of elementary particles. Vaughan Jones' theory of subfactors led to the discovery of the Jones polynomial, an invariant for knots and links, and had far reaching applications to low dimensional topology (e.g. new invariants for links and three manifolds, Atiyah-Witten topological quantum field theories etc.). Dan Voiculescu's free probability theory introduced probabilistic methods to the analysis of von Neumann algebras associated to free groups and his new concept of free entropy can be viewed as a measure of freeness in this context. K-theory, Kasporov's KK-theory and Connes' cyclic (co-) homology are used to study and classify C*-algebras, with applications to geometry such as the Novikov conjecture and various far reaching generalizations of the Atiyah-Singer index theorem.
My current research is mostly in the theory of subfactors, the study of inclusions of von Neumann algebras with trivial center. A subfactor can be viewed as a mathematical object encoding symmetry of a mathematical or physical problem, much like a group does. However, a subfactor is an infinite dimensional, highly noncommutative object and the symmetry it represents is more general than group symmetry. Operator algebra methods can be used to decode this symmetry and one obtains finite dimensional data in this process, which can be described combinatorially and computed numerically. The mathematical object associated to a subfactor that captures all this information is called the standard invariant or planar algebra of the subfactor. It has a simple description in terms of vector spaces of planar diagrams and actions of certain planar tangles on these vector spaces. Beautiful new structures emerge and planar algebra technology can be utilized to prove deep structural results for subfactors. There are numerous fruitful connections of the theory of subfactors to statistical mechanics, random matrices, algebraic quantum field theory, quantum computing, low dimensional topology and other areas of mathematics and physics.
I am also interested in quantum information theory and quantum computing. I am particularly interested Michael Freedman's program of building a topological quantum computer. The theory of subfactors seems to provide examples of structures that can be used to construct a (theoretical) topological quantum computer. Moreover, I am interested in the study of measures of entanglement, which turn out to be closely related to non-commutative entropies in operator algebras.