KK-theory, quantum groups, and quaternions

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This is a project in pure mathematics. Mathematics is about connections and developing provably correct deductions from available information we have about the world. Historically, it was synonymous with higher education itself. Progress in pure mathematics slowly influences the applied sciences; it is a fact that once esoteric concepts in pure mathematics now appear in engineering and physics. For the student or the HQP, pure mathematics teaches advanced logical thinking about complex topics, with possible applications to complicated scientific problems. Thus the benefits of this research program exist at the frontiers of today's science and technology, and may affect tomorrow's science and technology. The basic topic of the program is norm-closed self-adjoint algebras of operators on Hilbert space, called C*-algebras. They have remarkable properties, and connect disparate areas such as functional analysis, algebra, topology, geometry, and dynamical systems. Perhaps the most important single endeavour in this field is to classify such algebras by K-theoretical data (the Elliott program). The time is right to investigate branching the program out to new directions. C*-algebras can be viewed as noncommutative generalizations of topological spaces, and if we look for generalizations of topological groups rather than topological spaces, we obtain C*-algebraic quantum groups (QGs), which may have co-actions on C*-algebras. Our main objective is to generalize the Elliott program to suitable classes of QGs and co-actions. An Elliott style classification of QGs works very differently from existing algebraic classifications, and there is strong potential that this innovative approach will be adopted by an international community of researchers. HQP who obtain an early start in the proposed program will be well positioned for a productive research career. One of the benefits of the above program is that it lets classical algebraic topology be applied to QGs. Algebraic topology uses functors to prove that certain things cannot happen. For C*-algebras, there exists a Cuntz semigroup functor (CSF). We found important structural properties of the CSF of many QGs and some of their co-actions. This discovery may make the CSF a useful tool for studying the algebraic topology of quantum groups. For extending classical algebraic topology to a quantum dynamical framework, it seems desirable to further develop algebraic topological tools for QG and their co-actions. For example, we can construct quantum group isomorphisms from isomorphisms at the CSF level. This interesting phenomenon has no precise counterpart in classical algebraic topology, and is an explicitly `quantum' phenomenon. Jointly with G.A. Elliott and C. Ivanescu, we are studying the Cuntz semigroup of Villadsen algebras. We aim to classify Villadsen algebras through their Cuntz semigroup. This is an exciting, timely, and more importantly, feasible proposal.