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The University of Southampton
Mathematical Sciences

Research Group: Noncommutative Geometry

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The fundamental idea of noncommutative geometry is to study spaces through algebras of functions on them, using functional-analytic tools. Indeed, Gelfand's Theorem implies that all the information about a (compact Hausdorff, second countable topological) space is encoded in the algebra of (complex-valued, continuous) functions on the said space. This algebra can be represented as a commutative algebra of operators on a Hilbert space. By dropping the "commutative" requirement, we are entering the realm of operator algebras, which are instrumental in studying spaces which are "badly behaved" in the traditional sense, for example the space of representations of a discrete group; or a quotient of a space by a minimal action of a group.

Our research is invariably connected to operator algebras and variants of the Baum-Connes conjecture. Roughly speaking, this conjecture asserts that the algebraic topology (K-theory) of the space of tempered representations of a group (encoded as group C*-algebra) is computable in terms of algebraic topology (equivariant K-homology) of the classifying space of proper actions of the said group.

Currently, we focus mainly on two topics:

  • Coarse geometry and Roe algebras: Roe algebras are C*-algebras associated to metric spaces, encoding their large-scale structure. We study the interplay between the coarse properties of the metric spaces and C*-algebraic and K-theoretic properties of their Roe algebras; as well as the connections to the coarse Baum-Connes conjecture.
  • Relation between Langlands duality and the Baum-Connes conjecture: Langlands duality is a duality between Lie groups which arises by interchanging roots and coroots, and this duality induces a duality at the level of the affine and extended affine Weyl groups. We showed that this duality is also related to the Baum-Connes isomorphism in K-theory, and in this project we are carrying out the explicit calculations necessary to understand this isomorphism, providing a whole new family of examples to illuminate the mysterious assembly map.
Noncommutative Geometry

The group welcomes applications for postgraduate studies. Please contact group members (email, telephone, or in person) for more information; and when ready, please apply through the Graduate School application page.

List of related projects to Noncommutative Geometry
Related ProjectsStatus
K-theory and cyclic homology of affine Weyl groupsActive
Coarse geometry of non-positively curved spacesActive
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