The Cuntz Semigroup and the Classification of C*-algebras

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This award provides support to defray expenses of US participants to visit the Centre de Recerca Matematica (CRM) in Barcelona during the program "The Cuntz Semigroup on the Classification of C*-algebras" which is being held in the period January through July 2011. The vast majority of the award funds will provide the opportunity for young US mathematicians, including graduate students and postdoctoral fellows, to attend significant portions of the CRM program. In particular, the award will support the attendance of these young mathematicians at an advanced course entitled "Dynamical Systems, C*-algebras and Classification," which will be held during June 14-23, 2011. The award is co-funded by the Office of International Science and Engineering.The classification of C*-algebras via K-theoretic invariants has recently been invigorated by the discovery that the Cuntz semigroup - roughly, a semigroup of isomorphism classes of countably generated Banach modules over a C*-algebra - contains an enormous amount of information. Explicit calculation of this semigroup has led to classification theorems for C*-algebras of minimal dynamical systems, among others. The connections between classification, C*-algebras, and dynamics have been further strengthened by progress on proving the hyperfiniteness of amenable group actions on the Cantor set. Another new front has opened at the interface of descriptive set theory and operator algebras, where the Borel complexity of various equivalence relations (including, naturally, isomorphism) allows one to quantify the complexity of various classification problems in C*-algebra theory. Thus, the Cuntz semigroup and classification problems in C*-algebra theory form an active and rapidly developing branch of functional analysis, one which will afford many research opportunities. The award provides a very good opportunity for young mathematicians with research interests in the area of C*-algebras to be exposed to the latest developments in the field, to interact with researchers from other countries, and to possibly initiate future collaborations. Among other things, this award will contribute to the future vitality of the field, and hopefully lead to further applications and connections with other areas of mathematics.