arXiv subjects

Magnus Goffeng

Publications and source records attributed to Magnus Goffeng.

At least 19 recordsLinked to original sources

arXiv2026

Boundary determination from local boundary data for a fractional Calder\'{o}n problem

We introduce a new Calder\'{o}n-type problem for fractional powers of Schr\"{o}dinger operators, with local boundary conditions. The associated Dirichlet-to-Neumann operator maps Dirichlet data to Neumann data on the boundary. We show that this operator determines the Taylor series of the potential at the boundary. In particular, analytic potentials are uniquely determined. These are the first results for a fractional Calder\'{o}n problem with sources and measurements on the boundary. Our proof builds on recent advances for pseudodifferential boundary value problems to compute the complete symbol of the Dirichlet-to-Neumann operator.

math.AP↗

arXiv2026

Five shades of KMS: Statistical properties in the spectral geometry of Cuntz--Krieger algebras

We study spectral invariants arising in the noncommutative geometry of topological Markov chains and Cuntz--Krieger algebras. Their noncommutative geometry is described by spectral triples built from log-Laplacians, which are known to have non-trivial index theory and exotic quantum symmetries. We prove statistical eigenvalue asymptotics, local heat trace asymptotics, local Weyl laws, and an analogue of Connes' trace theorem. In all cases the local asymptotics are governed by the Kubo--Martin--Schwinger state of the gauge action on the Cuntz--Krieger algebra.

math.OA↗

arXiv2026

Bordisms and unbounded $KK$-theory

This monograph studies $KK$-theory in its unbounded model. The central object is the $KK$-bordism group obtained by imposing the $KK$-bordism relation on unbounded $KK$-cycles. In the paradigm of noncommutative geometry, an unbounded $KK$-cycle is a noncommutative geometry in its own right and our approach allow for the study of mildly noncommutative geometries (orbifolds, foliations et cetera) as if they were closed manifolds. The techniques we introduce enable us to directly import manifold techniques and arguments into the important yet technical field of unbounded $KK$-theory. Recent decades has seen a tremendous progress in the study of the unbounded model for $KK$ as well as secondary invariants, the first motivated by refining computational tools in Kasparov's $KK$-theory and the second by applications to geometry and topology. The aim of this work is to provide a common framework for these two areas: equipping unbounded $KK$-cycles with a geometrically motivated relation recovering Kasparov's $KK$-theory that is computationally tractable for working with secondary invariants.

math.KT↗

arXiv2025

Sharp mapping properties of Poisson transforms and the Baum-Connes conjecture

We prove a sharp, quantitative analogue of Helgason's conjecture at the level of distributions: For a semisimple Lie group $G$ of real rank one, Poisson transforms map a Sobolev space on $P\backslash G$ boundedly with closed range to an $L^2$-space on $K\backslash G$. The result is obtained for the Poisson transform studied by Knapp-Wallach under the name Szeg\"o map, and the appropriate Sobolev spaces are defined using van Erp-Yuncken's Heisenberg calculus. The proof generalizes to show that commutators of this Poisson transform with smooth functions on the Furstenberg compactification are compact. This proves the remaining open conjecture in Julg's seminal program to establish the Baum-Connes conjecture for closed subgroups of semisimple Lie groups of real rank one.

math.KT↗

arXiv2025

Parabolic noncommutative geometry

We introduce to spectral noncommutative geometry the notion of tangled spectral triple, which encompasses the anisotropies arising in parabolic geometry as well as the parabolic commutator bounds arising in so-called "bad Kasparov products". Tangled spectral triples incorporate anisotropy by replacing the unbounded operator in a spectral triple that mimics a Dirac operator with several unbounded operators mimicking directional Dirac operators. We allow for varying and dependent orders in different directions, controlled by using the tools of tropical combinatorics. We study the conformal equivariance of tangled spectral triples as well as how they fit into $K$-homology by means of producing higher order spectral triples. Our main examples are hypoelliptic spectral triples constructed from Rockland complexes on parabolic geometries; we also build spectral triples on nilpotent group $C^*$-algebras from the dual Dirac element and crossed product spectral triples for parabolic dynamical systems.

math.OA↗

arXiv2024

Theta correspondence via group C*-algebras

We prove that the well-known explicit construction of the local theta correspondence by Li has a simple interpretation in terms of group C*-algebras. In particular, we deduce that in two standard cases where Li's method work, local theta correspondence arises from a continuous functor. Moreover, using results from a companion paper, we treat global theta correspondence using C*-algebraic methods. As a byproduct, we exhibit that Rallis inner product formula can be interpreted as a certain natural inclusion being an isometry.

math.OA↗

arXiv2024

Friedlander's inequality and the de Rham complex

Inequalities between Dirichlet and Neumann eigenvalues of the Laplacian and of other differential operators have been intensively studied in the past decades. The aim of this paper is to introduce differential forms and the de Rham complex in the study of such inequalities. We show how differential forms lie hidden at the heart of the work of Rohleder on inequalities between Dirichlet and Neumann eigenvalues for the Laplacian on planar domains.

math.SP↗

arXiv2024

Adelic C*-correspondences and parabolic induction

In analogy with the construction of representations of adelic groups as restricted products of representations of local groups, we study restricted tensor products of Hilbert C*-modules and of C*-correspondences. The construction produces global C*-correspondences from compatible collections of local C*-correspondences. When applied to the collection of C*-correspondences capturing local parabolic induction, the construction produces a global C*-correspondence that captures adelic parabolic induction.

math.OA↗

arXiv2024

Riesz energies and the magnitude of manifolds

We study the geometric significance of Leinster's magnitude invariant. For closed manifolds we find a precise relation with Brylinski's beta function and therefore with classical invariants of knots and submanifolds. In the special case of compact homogeneous spaces we obtain an elementary proof that the residues of the beta function contain the same geometric information as the asymptotic expansion of the magnitude function. For general closed manifolds we use the recent pseudodifferential analysis of the magnitude operator to relate these via an interpolating polynomial family. Beyond manifolds, the relation with the Brylinski beta function allows to deduce unexpected properties of the magnitude function for the $p$-adic integers.

math.DG↗

arXiv2024

The index of sub-laplacians: beyond contact manifolds

In this paper we study the following question: do sub-Laplacian type operators have non-trivial index theory on Carnot manifolds in higher degree of nilpotency? The problem relates to characterizing the structure of the space of hypoelliptic sub-Laplacian type operators, and results going back to Rothschild-Stein and Helffer-Nourrigat. In two degrees of nilpotency, there is a rich index theory by work of van Erp-Baum on contact manifolds, that was later extended to polycontact manifolds by Goffeng-Kuzmin. We provide a plethora of examples in higher degree of nilpotency where the index theory is trivial.

math.AP↗

arXiv2024

Solving the index problem for (curved) Bernstein-Gelfand-Gelfand sequences

We study the index theory of curved Bernstein-Gelfand-Gelfand (BGG) sequences in parabolic geometry and their role in $K$-homology and noncommutative geometry. The BGG-sequences fit into $K$-homology, and we solve their index problem. We provide a condition for when the BGG-complex on the flat parabolic geometry $G/P$ of a semisimple Lie group $G$ fits into $G$-equivariant $K$-homology by means of Heisenberg calculus. For higher rank Lie groups, we prove a no-go theorem showing that the approach fails.

math.KT↗

arXiv2024

Heat operators and isometry groups of Cuntz-Krieger algebras

This paper introduces heat semigroups of topological Markov chains and Cuntz-Krieger algebras by means of spectral noncommutative geometry. Using recent advances on the logarithmic Dirichlet Laplacian on Ahlfors regular metric-measure spaces, we construct spectral triples on Cuntz-Krieger algebras from singular integral operators. These spectral triples exhaust odd K-homology and for Cuntz algebras we can compute their heat operators explicitly as Riesz potential operators. We also describe their isometry group in terms of the automorphism group of the underlying directed graph and prove that the Voiculescu noncommutative topological entropy vanishes on isometries.

math.OA↗

arXiv2023

Wieler solenoids: non-Hausdorff expansiveness, Cuntz-Pimsner models, and functorial properties

Building on work of Williams, Wieler proved that every irreducible Smale space with totally disconnected stable sets can be realized via a stationary inverse limit. Using this result, the first and fourth listed authors of the present paper showed that the stable $C^*$-algebra associated to such a Smale space can be obtained from a stationary inductive limit of a Fell algebra. Its spectrum is typically non-Hausdorff and admits a self-map related to the stationary inverse limit. With the goal of understanding the fine structure of the stable algebra and the stable Ruelle algebra, we study said self-map on the spectrum of the Fell algebra as a dynamical system in its own right. Our results can be summarized into the statement that this dynamical system is an expansive, surjective, local homeomorphism of a compact, locally Hausdorff space and from its $K$-theory we can compute $K$-theoretical invariants of the stable and unstable Ruelle algebra of a Smale space with totally disconnected stable sets.

math.OA↗

arXiv2023

Counting Negative Eigenvalues for the Magnetic Pauli Operator

We study the Pauli operator in a two-dimensional, connected domain with Neumann or Robin boundary condition. We prove a sharp lower bound on the number of negative eigenvalues reminiscent of the Aharonov-Casher formula. We apply this lower bound to obtain a new formula on the number of eigenvalues of the magnetic Neumann Laplacian in the semi-classical limit. Our approach relies on reduction to a boundary Dirac operator. We analyze this boundary operator in two different ways. The first approach uses Atiyah-Patodi-Singer index theory. The second approach relies on a conservation law for the Benjamin-Ono equation.

math.SP↗

arXiv2022

The Willmore energy and the magnitude of Euclidean domains

We study the geometric significance of Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain in an odd-dimensional Euclidean space, we show that the asymptotic expansion of the magnitude function at infinity determines the Willmore energy of the boundary. This disproves the Leinster-Willerton conjecture for a compact convex body in all odd dimensions.

math.MG↗

arXiv2022

Exotic cyclic cohomology classes and Lipschitz algebras

We study the noncommutative geometry of algebras of Lipschitz continuous and Hölder continuous functions where non-classical and novel differential geometric invariants arise. Indeed, we introduce a new class of Hochschild and cyclic cohomology classes that pair non-trivially with higher algebraic $K$-theory yet vanish when restricted to the algebra of smooth functions. Said cohomology classes provide additional methods to extract numerical invariants from Connes-Karoubi's relative sequence in $K$-theory.

math.KT↗

arXiv2022

Index theory of hypoelliptic operators on Carnot manifolds

We study the index theory of hypoelliptic operators on Carnot manifolds -- manifolds whose Lie algebra of vector fields is equipped with a filtration induced from sub-bundles of the tangent bundle. A Heisenberg pseudodifferential operator, elliptic in the calculus of van Erp-Yuncken, is hypoelliptic and Fredholm. Under some geometric conditions, we compute its Fredholm index by means of operator $K$-theory. These results extend the work of Baum-van Erp (Acta Mathematica '2014) for co-oriented contact manifolds to a methodology for solving this index problem geometrically on Carnot manifolds. Under the assumption that the Carnot manifold is regular, i.e. has isomorphic osculating Lie algebras in all fibres, and admits a flat coadjoint orbit, the methodology derived from Baum-van Erp's work is developed in full detail. In this case, we develope $K$-theoretical dualities computing the Fredholm index by means of geometric $K$-homology a la Baum-Douglas. The duality involves a Hilbert space bundle of flat orbit representations. Explicit solutions to the index problem for Toeplitz operators and operators of the form "$\Delta_H+\gamma T$" are computed in geometric $K$-homology, extending results of Boutet de Monvel and Baum-van Erp, respectively, from co-oriented contact manifolds to regular polycontact manifolds.

math.DG↗

arXiv2022

Fell algebras, groupoids, and projections

Examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant are constructed to illustrate differences with the case of continuous trace $C^*$-algebras. At the level of the spectrum, this translates to only assuming the spectrum is locally Hausdorff (rather than Hausdorff). The existence of (full) projections is the fundamental question considered. The class of Fell algebras studied here arise naturally in the study of Wieler solenoids and applications to dynamical systems will be discussed in a separate paper.

math.OA↗