My research program lies at the confluence of three areas within modern functional analysis and mathematical physics: operator algebras, abstract harmonic analysis, and quantum information theory. In addition to the theoretical development of the individual areas, I am interested in the various interactions between them.
Motivated by representation theory and the mathematical structure of quantum mechanics, Murray and von Neumann laid the groundwork for operator algebras in the 1930's. The area has since blossomed into several research communities, but the common basic structure is a *-subalgebra of the algebra of bounded linear operators on a Hilbert space. This setting encapsulates infinite-dimensional spectral theory and provides a beautiful platform in which to study non-commutative generalizations of classical theory, including topology (C*-algebras), measure theory (von Neumann algebras), geometry (non-commutative geometry), probability (free probability) and information theory (quantum information theory).
Abstract harmonic analysis originated in the 1930's through the pioneering work of Pontryagin, van Kampen, and Weil, who generalized classical Fourier analysis (i.e., Fourier series, Fourier transforms, etc.) to the level of locally compact abelian groups. The primary feature of this theory is duality, where every locally compact abelian group G is assigned a dual group G^, which becomes a signifant tool in studying G and various function spaces associated to it. The attempt to generalize this duality to the non-abelian setting led to the theory of quantum groups, a mathematically rich subject at the interface between harmonic analysis, operator algebras, and mathematical physics.
In the course of the past two decades, it has been shown, through a number of different applications, that one can systematically harness the laws of quantum mechanics to perform information processing tasks that lie far beyond the realm of existing technologies. The resulting area of quantum information science continues to rapidly evolve, and lies at the confluence of mathematics, physics, and computer science. Both operator algebras and harmonic analysis have provided valuable tools for the area, and it is of great interest to explore further connections between them.