bischmath.ucsb.edu

Professor of Mathematics, Department of Mathematics, University of California, Santa Barbara, USA.

Operator Algebras, Quantum Physics, Quantum Information Theory.

His current research is mostly in the theory of subfactors, the study of inclusions of von Neumann algebras. 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. For instance, certain weighted bipartite graphs appear as basic structural ingredients and commuting squares (certain inclusions of four finite dimensional algebras) play a key role in this analysis. There are numerous fruitful connections of the theory of subfactors to statistical mechanics, algebraic quantum field theory, low dimensional topology and other areas of mathematics and physics.

Bimodules, higher relative commutants and the fusion algebra associated to a subfactor, Operator algebras and their applications (Waterloo, ON, 1994/1995), 13--63, Fields Inst. Commun., 13, Amer. Math. Soc., Providence, RI, 1997.

(with V.F.R. Jones) A note on free composition of subfactors, Geometry and physics (Aarhus, 1995), 339-361, Lecture Notes in Pure and Appl. Math., 184, Dekker, New York, 1997.

(with V.F.R. Jones) Algebras associated to intermediate subfactors, Invent. Math. 128, 89-157 (1997).

Principal graphs of subfactors with small Jones index, Math. Ann. 311 (1998) 2, 223-231.

(with S. Popa) Examples of subfactors with property T standard invariant, Geom. Funct. Anal. 9 (1999), no. 2, 215--225.

(with V.F.R. Jones) Singly generated planar algebras of small dimension, Duke Math. Journal 101 (2000), no. 1, 41-75.

(with V.F.R. Jones) Singly generated planar algebras of small dimension, Part II, to appear in Advances in Math..

Subfactors and Planar Algebras, appeared in the Proceedings of the ICM, Beijing 2002.

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