A brief overview
My work has been centered for many years on regularisation and renormalisation methods in
- differential geometry, e.g. to make sense of the Ricci curvature on an infinite dimensional manifold
- the theory of operator algebras to express the locality of trace anomalies, in particular the index of a Dirac type operator
- complex analysis, e.g. to study Laurent expansions of multivariable meromorphic functions with linear poles
- combinatorics, e.g. to count integer lattice points on cones
- number theory, e.g. to evaluate at poles multizeta functions and their generalisations
- stochastic analysis, in the context of Hairer's regularity structures
- analytic tools such as pseudodifferential operators and symbols
- differential geometric tools such as the Dirac type operators, Chern-Weil forms and determinant bundles
- algebro-geometric tools such as groupoids
- algebraic tools such as Hopf algebras, trees, ProPs
and borrowing ideas and methods from both physics and mathematics, namely from
- perturbative quantum field theory,
- noncommutative geometry, index theory, lattice theory, and the geometry of cones.
Here is a link to a lecture I held on 19.09.2014 within the framework of the Hausdorff Trimester Program Non-commutative Geometry and its Applications in Bonn.
Traces on the noncommutative torus
Abstract:
The global symbol calculus for pseudodifferential operators on tori can be generalised to noncommutative tori. In this global approach, the quantisation map is invertible and traces are discrete sums. On the noncommutative torus, Fathizadeh and Wong had characterised the Wodzicki residue as the unique trace which vanishes on trace-class operators. In contrast, we build and characterise the canonical trace on classical pseudodifferential operators on a noncommutative torus, which extends the ordinary trace on trace-class operators. It can be written as a canonical discrete sum on the underlying toroidal symbols. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace which is thereby identified with a discrete sum. By means of the canonical trace, we derive defect formulae for regularised traces on noncommutative toris. The conformal invariance of the zeta-function at zero of the Laplacian on the noncommutative torus is then a straightforward consequence. This is based on joint work with Cyril Lévy and Carolina Neira Jiménez.
This video was created and edited with kind support from eCampus Bonn and is also available at https://mediaserver.uni-bonn.de.