My research program lies at the confluence of three areas within modern functional analysis and mathematical physics: operator algebras, harmonic analysis on quantum groups, and quantum information. 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 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 mathematics, including topology (C*-algebras), measure theory (von Neumann algebras), geometry (non-commutative geometry), probability (free probability) and information theory (quantum information theory).
The past decade has witnessed a burst of interactions between operator algebras and quantum information theory, including quantum error correction, entanglement distillation, entropy theory, non-local games and self-testing, entanglement in quantum field theory, and most notably, connections with, and recent negative solution to, Connes' Embedding Problem -- a major problem in operator algebras dating back to the 1970's.
My current research within this area includes holographic quantum error correction in quantum field theory, quantum no-signalling correlations, and self-testing.
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.
My current research in this area includes operator module theoretic characterizations of quantum group approximation properties.