Research projects

Here you may find further descriptions of some of the projects that I work on. The links below are to the arXiv versions, links to the published versions can be found here.

Index theory of hypoelliptic operators

The problem lying at the heart of index theory is to compute the Fredholm index. The Fredholm index of an operator T:V->W is defined as index(T)=dim(ker(T))-dim(W/TV), so it jointly measures the failure of uniqueness (the kernel) and the failure of solvability (the cokernel) of the equation Tu=f. For operators that are elliptic, the index theory is well understood since work of Atiyah-Singer in the '60s. More generally, one could ask for an Atiyah-Singer index theorem for hypoelliptic operators. The first such index problem was solved by Baum-van Erp for hypoelliptic operators in the 2010s for operators in the Heisenberg calculus on contact manifolds.

Continuing in the direction of Baum-van Erp, Alexey Kuzmin and I proved an index theorem for hypoelliptic operators in the Heisenberg calculus of a large class of Carnot manifolds. The index formula has similarities to Atiayah-Singer’s index formula but involves the space of representations of the osculating Lie groupoid of the Carnot manifold. While this work was first motivated by some questions for the Bernstein-Gelfand-Gelfand complex, those remain open…

Magnitude

The magnitude of a compact metric space is a recent construction due to Leinster and studied further in work by Willerton, Meckes and Barcelo-Carbery. The notion is only about a decade old, and has been heavily discussed in the blog the n-category café. One often considers the magnitude as a function of a parameter R>0 rescaling the metric. The magnitude function is an invariant that in important special cases captures several important geometric features, and these ideas have also been applied to new notions of diversities in biology. A conjecture that served as an important driving force in this field was the Leinster-Willerton conjecture, a conjecture Heiko Gimperlein and I disproved.

In work with Heiko Gimperlein, we studied the magnitude function of odd-dimensional euclidean domains. More recently, we generalized this work jointly with Heiko Gimperlein and Nikoletta Louca in two papers: one studying the relevant integral equation and another studying the geometric ramifications. We proved that the magnitude function admits a meromorphic extension to the complex plane and as R goes to infinity, it admits an asymptotic expansion whose coefficients are determined by the volume of the domain and integrals of the second fundamental form of the boundary and its derivatives. The first three coefficients were computed. It remains open problems to understand the pole structure of the magnitude function and to compute the lower order terms in the asymptotic expansion.

Higher index theory and secondary invariants

In higher index theory, one studies the topology and geometry of manifolds by means of K-theoretical indices of operators encoding further symmetries, i.e. its fundamental group. Properties of such indices relate to several important conjectures, e.g. the Novikov conjecture and the Baum-Connes conjecture. I am interested in the geometric consequences of such conjectures and ideas. For instance, how to describe and compute them in Baum-Douglas' geometric model? These questions are studied in this series of papers with Robin Deeley: paper I, paper II and paper III. Motivated by questions about rigidity and obstructions to PSC-metrics on manifolds with boundary, we also considered the relative situation in two papers: part I and part II.

In recent years, the consequences on secondary invariants and structures of spaces and PSC-metrics and homotopy structures have been studied. The secondary invariants are often of a spectral nature, or Atiyah-Patodi-Singer type indices. The problems surrounding this area, both computational and theoretical, interest me greatly.

Bivariant K-theory

I am interested in the unbounded model for bivariant K-theory and its geometric applications. These endeavours are motivated by problems from higher index theory. Building on Hilsum's notion of bordisms in the unbounded model for Kasparov's KK-theory, one can define KK-bordism groups - bivariant K-theory groups - whose cycles are unbounded Kasparov modules as Robin Deeley, Bram Mesland and I did in this paper. The properties of the KK-bordism groups, especially in relation to KK-theory, and its applications are currently being explored.

Foundations of Non-Commutative Geometry

Non-Commutative Geometry describes a geometry by means of its energy content and how it interacts with the particles thereon. It is a "non-theory", in the sense that it is to a large extent built on examples. Providing new examples and non-examples interest me greatly with the purpose of solidifying an underlying theory.

A project along these lines that I am working on is that of doing "differential geometry" in the absence of differentiability using Hölder continuous functions. This work is motivated by some previous papers on the spectral asymptotics of commutators between certain geometric operators and Hölder continuous functions. In this recent paper with Heiko Gimperlein we computed the spectral asymptotics in an averaged sense of singular traces and constructed new cohomology classes on Hölder geometries that pair non-trivially with algebraic K-theory and vanishes on smooth functions, building on some ideas from another paper with Heiko. These results opens up several problems, e.g. concerning computations and extensions to higher dimensions. This work also relates to this paper with Alex Usachev on singular traces. We developed these ideas even further in this paper with Ryszard Nest.

Other projects in this direction that I am involved in concerns constructions of new non-commutative geometries, i.e. spectral triples or unbounded Kasparov cycles, on exotic C*-algebras (such as Smale space C*-algebras, the Cuntz algebra, Cuntz-Krieger algebras, or Cuntz-Pimsner algebras). In some follow up work, Bram Mesland, Adam Rennie and I looked at twisted spectral triples and how to obtain ordinary spectral triples from them (at the expense of loosing potential finite dimensionality), and Adam Rennie, Alex Usached and I looked at the KMS-states associated with infinite dimensional spectral triples. The questions studied in this direction relates to possible dimensional- and regularity properties of spectral triples for non-commutative geometries beyond the classical situation of a manifold. Understanding such issues would improve the computational possibilities in higher index theory.

Topological phases

Topological effects are important in solid state physics (earning Thouless, Haldane and Kosterlitz a nobel prize in 2016). This area has lately been a popular application of index theory and KK-theory. I have not actively worked in this direction, but some of my papers (this one, also this and that one) are related to such problems and it is an area that I might continue in.