Research

My research lies at the interface of noncommutative geometry, operator algebras, index theory, spectral theory and partial differential equations. A recurring theme is the interaction between analysis and topology: how analytic properties of differential or pseudodifferential operators encode geometric and topological information, and conversely how K-theory and index theory can be used to understand analytic problems.

Much of my work can be grouped into the following interconnected themes. For all of my publications, see arXiv, or Zentralblatt or GoogleScholar.

Hypoelliptic operators, sub-Riemannian geometry and parabolic geometry

Elliptic operators provide one of the classical meeting points of analysis, geometry and topology. A central direction in my recent work is to understand how much of this picture survives for hypoelliptic operators arising from filtered, contact, Carnot and parabolic geometries.

The natural replacement for ordinary pseudodifferential analysis in this setting is the Heisenberg calculus and its generalizations. On the topological side, one encounters the representation theory of the osculating nilpotent groups rather than simply the cotangent bundle. This makes even the basic index problem considerably subtler than in elliptic theory.

With Alexey Kuzmin I developed a geometric approach to the index problem for hypoelliptic operators on regular Carnot manifolds, extending the contact-manifold index theory of Baum and van Erp to substantially more general settings. Subsequent work asks how far this phenomenon persists: with Bernard Helffer we find strong restrictions on the index theory of sub-Laplacians in higher-step geometries. My work on BGG sequences studies the same circle of questions in parabolic geometry, while joint work with Magnus Fries and Ada Masters develops a spectral noncommutative geometry adapted to the anisotropic nature of these spaces through the notion of tangled spectral triples.

Earlier work with Heiko Gimperlein on contact manifolds provides an analytic precursor to this programme, establishing sharp commutator estimates in the Heisenberg calculus and constructing geometric invariants from operators with nonclassical spectral behaviour.

There are several directions I would particularly like to pursue further: understanding which filtered geometries admit genuinely nontrivial hypoelliptic index theories; developing boundary value problems for hypoelliptic operators; finding local index formulas; and developing spectral geometry and noncommutative geometry adapted to general parabolic structures.

Papers and preprints

The index of sub-laplacians: beyond contact manifolds, with Bernard Helffer. Preprint. arXiv
Index theory of hypoelliptic operators on Carnot manifolds, with Alexey Kuzmin. Preprint. arXiv
Solving the index problem for (curved) Bernstein–Gelfand–Gelfand sequences. Accepted in Annals of Global Analysis and Geometry. arXiv
Parabolic noncommutative geometry, with Magnus Fries and Ada Masters. Advances in Mathematics, Volume 499 (2026). Journal · arXiv
Commutator estimates on contact manifolds and applications, with Heiko Gimperlein. Journal of Noncommutative Geometry 13 (2019). Journal · arXiv
Nonclassical spectral asymptotics and Dixmier traces: from circles to contact manifolds, with Heiko Gimperlein. Forum of Mathematics, Sigma 5 (2017), e3. Journal · arXiv

Spectral noncommutative geometry and singular traces

One of the central ideas of noncommutative geometry is that geometric information can be encoded spectrally by operators. I have been particularly interested in what happens when one moves outside the classical smooth setting: to Hölder and Lipschitz geometry, higher-order and twisted spectral triples, singular traces, non-measurable operators and geometries whose natural spectral asymptotics differ from the classical Weyl law.

A recurring problem is to understand how much geometry remains visible under weak regularity. My early work on topological degree showed that index theory continues to give explicit formulas for topological invariants of Hölder mappings. Work with Heiko Gimperlein subsequently uncovered nonclassical spectral asymptotics for commutators with Hölder functions. With Alexandr Usachev I developed techniques for computing and estimating Dixmier traces far beyond the classical pseudodifferential setting, while work with Ryszard Nest shows that Lipschitz and Hölder algebras carry cyclic-cohomological information that disappears completely on their smooth subalgebras.

A second direction concerns the flexibility of the spectral-triple formalism itself. Work with Bram Mesland and Adam Rennie shows how twisted and higher-order spectral triples can be transformed into ordinary spectral triples without losing their index-theoretic content. Work with Rennie and Usachev connects infinite-dimensional spectral triples to KMS states and hence to equilibrium states in operator algebras.

I am interested in continuing to develop a spectral geometry in which low regularity, anisotropic differential structures, singular traces and nonclassical eigenvalue asymptotics are features rather than pathologies.

Papers and preprints

Exotic cyclic cohomology classes and Lipschitz algebras, with Ryszard Nest. Annals of K-Theory 8 (2023), 221–243. Journal · arXiv
Estimating Dixmier traces of Hankel operators in Lorentz ideals, with Alexandr Usachev. Journal of Functional Analysis 279 (2020). Journal · arXiv
Dixmier traces and residues on weak operator ideals, with Alexandr Usachev. Journal of Mathematical Analysis and Applications 488 (2020). Journal · arXiv
Untwisting twisted spectral triples, with Bram Mesland and Adam Rennie. International Journal of Mathematics 30 (2019). Journal · arXiv
Constructing KMS states from infinite-dimensional spectral triples, with Adam Rennie and Alexandr Usachev. Journal of Geometry and Physics 143 (2019), 107–149. Journal · arXiv
Analytic formulas for topological degree of non-smooth mappings: the even-dimensional case. Journal of Pseudo-Differential Operators and Applications 4 (2013). Journal · arXiv
Analytic formulas for topological degree of non-smooth mappings: the odd-dimensional case. Advances in Mathematics 231 (2012). Journal · arXiv
Функции Вейерштрасса разделяют все следы (Weierstrass functions separate all traces), with Alexandr Usachev. Mathematical Notes 110 (2021). Journal

Noncommutative geometry of dynamics and Cuntz–Pimsner algebras

Symbolic and hyperbolic dynamical systems provide natural examples of genuinely noncommutative spaces. Their C*-algebras retain information about the underlying dynamics, while K-theory, spectral triples, equilibrium states and spectral invariants offer different ways of extracting it.

With Bram Mesland I constructed finitely summable Fredholm modules and spectral triples on Cuntz–Krieger algebras representing their K-homology. Subsequent work extended this picture to Cuntz–Pimsner algebras and to C*-algebras arising from Wieler solenoids and Smale spaces.

More recently, with Dimitris Gerontogiannis and Bram Mesland, I have returned to Cuntz–Krieger algebras from a more genuinely spectral-geometric viewpoint. We constructed natural heat operators and spectral triples whose isometry groups recover symmetries of the underlying dynamics. With Gerontogiannis I subsequently studied their eigenvalue and heat-trace asymptotics. A striking feature is that the KMS state of the underlying dynamical system plays the role normally played by Riemannian volume: it appears in local Weyl laws, heat asymptotics and a noncommutative analogue of Connes' trace theorem.

Together these results are moving toward a rather complete noncommutative geometry of Cuntz–Krieger algebras, incorporating topology and index theory, measure/KMS theory, symmetry and spectral geometry. Natural next questions concern quantum metric geometry, finer spectral invariants, rigidity, and extensions to broader classes of Cuntz–Pimsner and groupoid C*-algebras.

Papers and preprints

Five shades of KMS: Statistical properties in the spectral geometry of Cuntz–Krieger algebras, with Dimitris Michail Gerontogiannis. Preprint. arXiv
Wieler solenoids: non-Hausdorff expansiveness, Cuntz–Pimsner models, and functorial properties, with Robin J. Deeley, Menevse Eryüzlü and Allan Yashinski. Transactions of the American Mathematical Society 378 (2025), 7033–7074. Journal · arXiv
Heat operators and isometry groups of Cuntz–Krieger algebras, with Dimitris Gerontogiannis and Bram Mesland. International Mathematics Research Notices (2025). Journal · arXiv
Fell algebras, groupoids, and projections, with Robin Deeley and Allan Yashinski. Proceedings of the American Mathematical Society 150 (2022), 4891–4907. Journal · arXiv
Smale space C*-algebras have non-zero projections, with Robin Deeley and Allan Yashinski. Proceedings of the American Mathematical Society 148 (2020). Journal · arXiv
Wieler solenoids, Cuntz–Pimsner algebras and K-theory, with Robin Deeley, Bram Mesland and Michael Whittaker. Ergodic Theory and Dynamical Systems 38 (2018), 2942–2988. Journal · arXiv
Shift tail equivalence and an unbounded representative of the Cuntz–Pimsner extension, with Bram Mesland and Adam Rennie. Ergodic Theory and Dynamical Systems 38 (2018). Journal · arXiv
Spectral triples and finite summability on Cuntz–Krieger algebras, with Bram Mesland. Documenta Mathematica 20 (2015), 89–170. Journal · arXiv
Spectral triples on ON, with Bram Mesland. MATRIX proceedings (2018). Published version · arXiv

Geometric K-homology, unbounded KK-theory and secondary invariants

K-homology and Kasparov's KK-theory provide a common language for index theory, topology and operator algebras. A longstanding interest of mine has been to make these theories more geometric and more directly calculable.

A substantial part of my work with Robin Deeley develops geometric models for analytic objects. This includes twisted K-homology and fractional indices, relative assembly maps, and a geometric realization of the Higson–Roe analytic surgery exact sequence. The latter gives geometric interpretations of rho- and eta-type secondary invariants and connects index theory to rigidity and positive scalar curvature.

A related programme with Robin Deeley and Bram Mesland concerns unbounded KK-theory. Unbounded cycles contain much more geometric information than their bounded transforms, but the equivalence relation defining KK-theory is difficult to see directly at the unbounded level. We introduced bordism as a geometric relation between unbounded KK-cycles and are now developing this into a systematic model for KK-theory. The aim is to combine the computational power of Kasparov theory with geometric constructions familiar from manifolds and bordism.

Boundary phenomena form another part of this picture. Joint work with Iain Forsyth, Bram Mesland and Adam Rennie introduced relative spectral triples and a noncommutative version of the principle that “the boundary of Dirac is Dirac on the boundary.”

I am interested in pushing these ideas toward a genuinely geometric calculus for unbounded KK-theory, including secondary invariants, manifolds and noncommutative spaces with boundary, and applications to assembly maps and index theory.

Papers and preprints

Bordisms and unbounded KK-theory, with Robin Deeley and Bram Mesland. Preprint/monograph. arXiv
Boundaries, spectral triples and K-homology, with Iain Forsyth, Bram Mesland and Adam Rennie. Journal of Noncommutative Geometry 13 (2019), 407–472. Journal · arXiv
Relative geometric assembly and mapping cones, Part II: Chern characters and the Novikov property, with Robin Deeley. Münster Journal of Mathematics 12 (2019), 57–92. Journal · arXiv
Relative geometric assembly and mapping cones, Part I: The geometric model and applications, with Robin Deeley. Journal of Topology 11 (2018). Journal · arXiv
The bordism group of unbounded KK-cycles, with Robin Deeley and Bram Mesland. Journal of Topology and Analysis 10 (2018). Journal · arXiv
Applying geometric K-cycles to fractional indices, with Robin Deeley. Mathematische Nachrichten 290 (2017), 2207–2233. Journal · arXiv
Realizing the analytic surgery group of Higson and Roe geometrically. Part I: The geometric model, with Robin Deeley. Journal of Homotopy and Related Structures 12 (2017), 109–142. Journal · arXiv
Realizing the analytic surgery group of Higson and Roe geometrically. Part II: Relative η-invariants, with Robin Deeley. Mathematische Annalen 366 (2016), 1319–1363. Journal · arXiv
Realizing the analytic surgery group of Higson and Roe geometrically. Part III: Higher invariants, with Robin Deeley. Mathematische Annalen 366 (2016), 1513–1559. Journal · arXiv
Equivariant extensions of *-algebras. New York Journal of Mathematics 16 (2010). Journal · arXiv

Spectral theory, boundary value problems and inverse problems

Another strand of my work concerns spectral problems for differential and pseudodifferential operators, particularly when boundaries play an essential role.

My earlier work in magnetic spectral theory studied Landau Hamiltonians and magnetic Robin problems. With Ayman Kachmar and Mikael Persson Sundqvist I studied eigenvalue clusters around Landau levels, and with Elmar Schrohe I used spectral flow to analyze the dependence of exterior Landau–Robin Hamiltonians on their boundary conditions. More recently, with Søren Fournais, Rupert Frank, Ayman Kachmar and Mikael Sundqvist, index-theoretic ideas led to sharp estimates for the number of negative eigenvalues of magnetic Pauli operators and new semiclassical information about the magnetic Neumann Laplacian.

Boundary value problems themselves have become an increasingly important part of my work. With Lashi Bandara and Hemanth Saratchandran I developed a general framework for realizations of elliptic differential operators of arbitrary order on manifolds with boundary. With Magnus Fries and Germán Miranda, differential complexes provide a new route to eigenvalue inequalities between Dirichlet and Neumann problems.

Most recently, with Heiko Gimperlein, I have begun studying inverse boundary problems for fractional powers of Schrödinger operators. Here pseudodifferential boundary calculus allows one to recover the full Taylor expansion of the potential at the boundary from a local Dirichlet-to-Neumann map.

I am particularly interested in interactions between boundary microlocal analysis, inverse problems, spectral asymptotics and index theory, as well as extensions of boundary-value techniques from elliptic to hypoelliptic operators.

Papers and preprints

Boundary determination from local boundary data for a fractional Calderón problem, with Heiko Gimperlein. Preprint. arXiv
Reproving Friedlander's inequality with the de Rham complex, with Magnus Fries and Germán Miranda. Journal of Spectral Theory 15 (2025), 1593–1613. Journal · arXiv
Counting Negative Eigenvalues for the Magnetic Pauli Operator, with Søren Fournais, Rupert L. Frank, Ayman Kachmar and Mikael Sundqvist. Duke Mathematical Journal 174 (2025), 313–353. Journal · arXiv
Realizations of elliptic operators on compact manifolds with boundary, with Lashi Bandara and Hemanth Saratchandran. Advances in Mathematics 420 (2023). Journal · arXiv
Spectral flow of exterior Landau–Robin Hamiltonians, with Elmar Schrohe. Journal of Spectral Theory 7 (2017). Journal · arXiv
Clusters of eigenvalues for the magnetic Laplacian with Robin condition, with Ayman Kachmar and Mikael Persson Sundqvist. Journal of Mathematical Physics 57 (2016). Journal · arXiv
Noncommutative deformation of the Ward metric, with Olaf Lechtenfeld. Conference proceedings (2011). Published version · arXiv
Index formulas and charge deficiencies on the Landau levels. Journal of Mathematical Physics 51 (2010). Journal · arXiv

Magnitude and spectral geometry

Magnitude originated as an invariant of finite metric spaces, inspired by category theory, before being extended to general metric spaces. A surprising discovery is that for smooth domains and manifolds it behaves very much like a spectral invariant.

With Heiko Gimperlein, and subsequently with Nikoletta Louca, I have developed an analytic approach to magnitude based on pseudodifferential and semiclassical analysis. For Euclidean domains, magnitude admits a meromorphic continuation whose poles behave like scattering resonances, while its large-scale asymptotic expansion contains classical geometric quantities such as volume, surface area, total mean curvature and the Willmore energy.

The underlying operator is nonlocal, and its analysis leads naturally to boundary pseudodifferential problems and semiclassical analysis. More recent work extends the theory from odd-dimensional Euclidean domains to manifolds of arbitrary dimension and relates magnitude to Brylinski's beta function and Riesz energies.

This suggests a broader spectral geometry of magnitude. Among the natural questions are inverse problems—how much geometry does the magnitude function determine?—the geometric meaning of its poles and lower-order coefficients, analogues of isospectrality, and extensions to singular, sub-Riemannian and noncommutative spaces.

Papers and preprints

The magnitude and spectral geometry, with Heiko Gimperlein and Nikoletta Louca. Accepted in American Journal of Mathematics. arXiv
Riesz energies and the magnitude of manifolds, with Heiko Gimperlein. Proceedings of the American Mathematical Society 153 (2025), 3173–3184. Journal · arXiv
Semiclassical analysis of a nonlocal boundary value problem related to magnitude, with Heiko Gimperlein and Nikoletta Louca. Journal d'Analyse Mathématique 153 (2024), 401–487. Journal · arXiv
The Willmore energy and the magnitude of Euclidean domains, with Heiko Gimperlein. Proceedings of the American Mathematical Society 151 (2023), 897–906. Journal · arXiv
On the magnitude function of domains in Euclidean space, with Heiko Gimperlein. American Journal of Mathematics 143 (2021). Journal · arXiv

Representation theory, group C*-algebras and the Baum–Connes conjecture

A more recent direction in my research uses operator algebras and noncommutative geometry to study problems in representation theory.

With Bram Mesland and Haluk Sengün, I have been developing C*-algebraic formulations of induction and correspondence in representation theory. Restricted tensor products of C*-correspondences give an operator-algebraic realization of adelic parabolic induction. In our work on theta correspondence, Li's construction of local theta lifts is realized through group C*-algebras and Hilbert C*-modules, and the same framework connects local and global theta correspondence.

With Heiko Gimperlein I have approached the Baum–Connes conjecture for rank-one semisimple Lie groups through analysis of Poisson transforms. Sharp Sobolev mapping properties of the Poisson transform, formulated using Heisenberg analysis on the boundary, yield the compactness property required in Julg's programme for the Baum–Connes conjecture. This is an example of a theme that interests me more broadly: using fine analytic properties of representation-theoretic operators to construct K-homology and KK-theory classes.

Earlier work on equivariant KK-theory and quantum groups also belongs to this circle of ideas. Natural directions forward include higher-rank groups, the interaction between automorphic representation theory and operator K-theory, and C*-algebraic formulations of other representation-theoretic correspondences.

Papers and preprints

Sharp mapping properties of Poisson transforms and the Baum–Connes conjecture, with Heiko Gimperlein. Preprint. arXiv
Theta correspondence via group C*-algebras, with Bram Mesland and Haluk Sengün. Preprint. arXiv
Adelic C*-correspondences and parabolic induction, with Bram Mesland and Haluk Sengün. Journal of Functional Analysis 290 (2026). Journal · arXiv
The Pimsner–Voiculescu sequence for coactions of compact Lie groups. Mathematica Scandinavica 110 (2012). Journal · arXiv
A remark on twists and the notion of torsion-free discrete quantum groups. Algebras and Representation Theory 15 (2012). Journal · arXiv

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