My main research interest is algebraic topology. A topology on a space allows us to talk about closeness or neighbourhoods of points without actually having to talk about distances between them. This enables us to define continuity of maps as well. Many geometric objects, like surfaces or higher dimensional manifolds, are in particular topological spaces. The goal of algebraic topology is to associate algebraic invariants, often groups, rings or modules, to these spaces in a natural way to distinguish them up to homotopy equivalence, i.e. up to continuous deformations.
An important example of such an invariant is topological K-theory. It is a generalised cohomology theory, which means that it satisfies axioms that allow its computation from a decomposition of the space into smaller parts. It can be defined as an invariant associated to the algebra of continuous functions on the space (vanishing at infinity) and as such it can be generalised to noncommutative algebras, which should be considered as "noncommutative topological spaces". These operator algebras not only play an important role in mathematical physics, they also can be used to describe highly singular topological spaces, like orbifolds or foliations.
K-theory has many different variants, like equivariant K-theory or twisted K-theory. The outcome of one of my research projects, which was joint work with Marius Dadarlat, was a description of twisted K-theory via operator algebras that includes the higher twists, a feature of the theory, which was formerly only accessible via homotopy theory.
I am interested in these and other interactions between algebraic topology, homotopy theory and operator algebras. Since I studied mathematical physics before my PhD, I am also interested in topological and conformal field theories and their connection with another generalised cohomology theory, called elliptic cohomology.
An important example of such an invariant is topological K-theory. It is a generalised cohomology theory, which means that it satisfies axioms that allow its computation from a decomposition of the space into smaller parts. It can be defined as an invariant associated to the algebra of continuous functions on the space (vanishing at infinity) and as such it can be generalised to noncommutative algebras, which should be considered as "noncommutative topological spaces". These operator algebras not only play an important role in mathematical physics, they also can be used to describe highly singular topological spaces, like orbifolds or foliations.
K-theory has many different variants, like equivariant K-theory or twisted K-theory. The outcome of one of my research projects, which was joint work with Marius Dadarlat, was a description of twisted K-theory via operator algebras that includes the higher twists, a feature of the theory, which was formerly only accessible via homotopy theory.
I am interested in these and other interactions between algebraic topology, homotopy theory and operator algebras. Since I studied mathematical physics before my PhD, I am also interested in topological and conformal field theories and their connection with another generalised cohomology theory, called elliptic cohomology.
