arXiv subjects

Magnus Goffeng

Publications and source records attributed to Magnus Goffeng.

51 records · Page 3Linked to original sources

arXiv2016

Realizing the analytic surgery group of Higson and Roe geometrically, Part III: Higher invariants

We construct an isomorphism between the geometric model and Higson-Roe's analytic surgery group, reconciling the constructions in the previous papers in the series on "Realizing the analytic surgery group of Higson and Roe geometrically" with their analytic counterparts. Following work of Lott and Wahl, we construct a Chern character on the geometric model for the surgery group; it is a "delocalized Chern character", from which Lott's higher delocalized $ρ$-invariants can be retrieved. Following work of Piazza and Schick, we construct a geometric map from Stolz' positive scalar curvature sequence to the geometric model of Higson-Roe's analytic surgery exact sequence.

math.KT↗

arXiv2015

Commutator estimates on contact manifolds and applications

This article studies sharp norm estimates for the commutator of pseudo-differential operators with multiplication operators on closed Heisenberg manifolds. In particular, we obtain a Calderon commutator estimate: If $D$ is a first-order operator in the Heisenberg calculus and $f$ is Lipschitz in the Carnot-Caratheodory metric, then $[D,f]$ extends to an $L^2$-bounded operator. Using interpolation, it implies sharp weak--Schatten class properties for the commutator between zeroth order operators and Hölder continuous functions. We present applications to sub-Riemannian spectral triples on Heisenberg manifolds as well as to the regularization of a functional studied by Englis-Guo-Zhang.

math.SP↗

arXiv2015

Spectral flow of exterior Landau-Robin hamiltonians

We study the spectral flow of Landau-Robin hamiltonians in the exterior of a compact domain with smooth boundary. This provides a method to study the spectrum of the exterior Landau-Robin hamiltonian's dependence on the choice of Robin data, even explaining the heuristics of how the spectrum of the Robin problem asymptotically tends to the spectrum of the Dirichlet problem. The main technical result concerns the continuous dependence of Landau-Robin hamiltonians on the Robin data in the gap topology. The problem can be localized to the compact boundary where the asymptotic behavior of the spectral flow in some special cases can be described.

math.SP↗

arXiv2015

Spectral triples and finite summability on Cuntz-Krieger algebras

We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz-Krieger algebras by means of realizing these algebras as "the algebra of functions on a non-commutative space" coming from a sub shift of finite type. We show that any odd $K$-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are $θ$-summable although their bounded transform, when constructed using the sign-function, will already on the level of analytic $K$-cycles be finitely summable bounded Fredholm modules. Using the unbounded Kasparov product, we exhibit a family of generalized spectral triples, possessing mildly unbounded commutators, whilst still giving well defined $K$-homology classes.

math.KT↗

arXiv2014

Clusters of eigenvalues for the magnetic Laplacian with Robin condition

We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in euclidean space. Functions in the domain of the operator are subject to a boundary condition of the third type (a magnetic Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which is independent of the boundary condition.

math.SP↗

arXiv2014

Applying geometric K-cycles to fractional indices

A geometric model for twisted $K$-homology is introduced. It is modeled after the Mathai-Melrose-Singer fractional analytic index theorem in the same way as the Baum-Douglas model of $K$-homology was modeled after the Atiyah-Singer index theorem. A natural transformation from twisted geometric $K$-homology to the new geometric model is constructed. The analytic assembly mapping to analytic twisted $K$-homology in this model is an isomorphism for torsion twists on a finite CW-complex. For a general twist on a smooth manifold the analytic assembly mapping is a surjection. Beyond the aforementioned fractional invariants, we study $T$-duality for geometric cycles.

math.KT↗

arXiv2014

Realizing the analytic surgery group of Higson and Roe geometrically, Part I: The geometric model

We construct a geometric analog of the analytic surgery group of Higson and Roe for the assembly mapping for free actions of a group with values in a Banach algebra completion of the group algebra. We prove that the geometrically defined group, in analogy with the analytic surgery group, fits into a six term exact sequence with the assembly mapping and also discuss mappings with domain the geometric group. In particular, given two finite dimensional unitary representations of the same rank, we define a map in the spirit of $η$-type invariants from the geometric group (with respect to assembly for the full group $C^*$-algebra) to the real numbers.

math.KT↗

arXiv2013

Analytic formulas for topological degree of non-smooth mappings: the even-dimensional case

Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölder continuous vector bundle. The index formula gives an analytic formula for the degree of a Hölder continuous mapping between even-dimensional manifolds. The paper is an independent continuation of the paper "Analytic formulas for topological degree of non-smooth mappings: the odd-dimensional case".

math.KT↗

arXiv2012

Noncommutative deformation of the Ward metric

The moduli-space metric in the static non-Abelian charge-two sector of the Moyal-deformed CP^1 sigma model in 1+2 dimensions is analyzed. After recalling the commutative results of Ward and Ruback and the zeta-regularized construction of the noncommutative K"ahler potential due to the second author, explicit expressions and asymptotics for it are presented and discussed in different regions of the moduli space. Along two curves in the moduli space the potential can be calculated analytically. In the region of solitons known as "ring-like", perturbation theory is used. In the region of "lump-like" solitons, both perturbation theory and the zeta-function approach are employed. While the strong noncommutativity limit is smooth and under control, the commutative limit in the two-lump region remains a semiclassical challenge.

hep-th↗

arXiv2011

Operator-valued pseudo-differential operators and the twisted index pairing

The notion of pseudo-differential operators with coefficients in a continuous trace algebra over a manifold are introduced and their index theory is studied. The algebra of principal symbols in this calculus provides an abstract Poincaré dual to the continuous trace algebra. Index formulas for pseudo-differential operators twisted by a bundle on the opposite continuous trace algebra are obtained in terms of an Atiyah-Singer type index formula describing the twisted index pairing as index theory for elliptic operators.

math.KT↗

arXiv2011

The Pimsner-Voiculescu sequence for coactions of compact Lie groups

The Pimsner-Voiculescu sequence is generalized to a Pimsner-Voiculescu tower describing the $KK$-category equivariant with respect to coactions of a compact Lie group satisfying the Hodgkin condition. A dual Pimsner-Voiculescu tower is used to show that coactions of a compact Hodgkin-Lie group satisfy the Baum-Connes property.

math.KT↗

arXiv2011

A remark on twists and the notion of torsion-free discrete quantum groups

In this paper twists of reduced locally compact quantum groups are studied. Twists of the dual coaction on a reduced crossed product are introduced and the twisted dual coactions are proved to satisfy a type of Takesaki-Takai duality. The twisted Takesaki-Takai duality implies that twists of discrete, torsion-free quantum groups are torsion-free. Cocycle twists of duals of semisimple, compact Lie are studied leading to a locally compact quantum group contained in the Drinfeld-Jimbo algebra which gives a dual notion of Woronowicz deformations for semisimple, compact Lie groups. These cocycle twists are proven to be torsion-free whenever the Lie group is simply connected.

math.QA↗

arXiv2010

Equivariant extensions of *-algebras

A bivariant functor is defined on a category of *-algebras and a category of operator ideals, both with actions of a second countable group $G$, into the category of abelian monoids. The element of the bivariant functor will be $G$-equivariant extensions of a *-algebra by an operator ideal under a suitable equivalence relation. The functor is related with the ordinary $Ext$-functor for $C^*$-algebras defined by Brown-Douglas-Fillmore. Invertibility in this monoid is studied and characterized in terms of Toeplitz operators with abstract symbol.

math.KT↗

arXiv2010

Analytic formulas for topological degree of non-smooth mappings: the odd-dimensional case

The notion of topological degree is studied for mappings from the boundary of a relatively compact strictly pseudo-convex domain in a Stein manifold into a manifold in terms of index theory of Toeplitz operators on the Hardy space. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a Toeplitz operator with Hölder continuous symbol. The index formula gives an analytic formula for the degree of a Hölder continuous mapping from the boundary of a strictly pseudo-convex domain.

math.KT↗

arXiv2010

Index formulas and charge deficiencies on the Landau levels

The notion of charge deficiency from Avron, Seiler, Simon (Charge deficiency, charge transport and comparison of dimensions, Comm. Math. Phys. 159) is studied from the view of $KK$-theory and is applied to the Landau levels in $\C^n$. We calculate the charge deficiencies of the higher Landau levels in $\C^n$ by means of an Atiyah-Singer type index theorem.

math-ph↗