arXiv subjects

Karsten Bohlen

Publications and source records attributed to Karsten Bohlen.

13 recordsLinked to original sources

arXiv2026

Reentrant value fields as delayed coupled reaction-diffusion systems on finite graphs

We describe a dynamical system in which a symbolic field is coupled to a geometric field via a bipartite Hilbert-Schmidt kernel. The system is fully described by a retarded functional differential equation (RFDE) on the history space, subject to Lipschitz and small gain conditions. We show that the RFDE is well-posed under constant input and that it admits a compact global attractor. The principal subsystem $(H_L, X_R, P)$, which is comprised of the two primary fields as well as an executive field, is shown to be globally stable independent of delay, provided that the interfield coupling satisfies $C_{\mathcal{K}}^2<\mu_L\mu_R$. In addition, we describe design specifications that fulfill the hypotheses of the main Theorem.

math.DS↗

arXiv2025

A Volterra Calculus for Lie Groupoids

A pseudodifferential Volterra calculus for inverting parabolic differential equations on Lie groupoids is introduced. This enables the study of fundamental solutions of various cases of heat flows on singular manifolds with corners with non-resonant boundary indicial symbols, such as the $b$-manifolds, as well as other geometric bisection covariant heat flows. We also establish the short time asymptotic expansion for the heat kernel of a positive, elliptic differential operator on a Lie groupoid that acts on suitable Sobolev Hilbert modules and is positive definite with respect to the appropriate $L^2$ inner product.

math.AP↗

arXiv2022

The Fredholm index for operators of tensor product type

We consider bisingular pseudodifferential operators which are pseudodifferential operators of tensor product type. These operators are defined on the product manifold $M_1 \times M_2$, for closed manifolds $M_1$ and $M_2$. We prove a topological index theorem of product type. In addition, we show that the Fredholm index of elliptic bisingular operators equals the topological index, whenever the operator takes the form of an external tensor product of pseudodifferential operators, up to equivalence. To this end we construct a suitable double deformation groupoid and a Poincaré duality type homomorphism.

math.FA↗

arXiv2022

A geometric approach to K-homology for Lie manifolds

We show that the computation of the Fredholm index of a fully elliptic pseudodifferential operator on an integrated Lie manifold can be reduced to the computation of the index of a Dirac operator, perturbed by a smoothing operator, canonically associated, via the so-called clutching map. To this end we adapt to our framework ideas coming from Baum-Douglas geometric $K$-homology and in particular we introduce a notion of geometric cycles, that can be categorized as a variant of the famous geometric $K$-homology groups, for the specific situation here. We also define a comparison map between this geometric $K$-homology theory and a relative $K$-theory group, directly associated to a fully elliptic pseudodifferential operator.

math.KT↗

arXiv2021

Internal absolute geometry I: desingularization

We introduce an axiomatization of Grothendieck sites with additional structure, and we describe sheaves that reconstruct groupoids which are internal to the site structure. This setting applies to various concrete situations, where a Nash blowup of a singular space results in an almost regular foliation. It also potentially applies to various types of moduli spaces. The sheaf can encode candidate holonomy groupoids that desingularize such spaces.

math.CT↗

arXiv2017

Quantization on manifolds with an embedded submanifold

We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic problems is achieved using the theory of Fourier integral operators on Lie groupoids. We show that our calculus is closed under composition and furnishes a so-called noncommutative completion of the given embedding. A representation of the algebra is defined and the continuity of the operators in the algebra on suitable Sobolev spaces is established.

math.DG↗

arXiv2017

Groupoids and singular manifolds

We describe how Lie groupoids are used in singular analysis, index theory and non-commutative geometry and give a brief overview of the theory. We also expose groupoid proofs of the Atiyah-Singer index theorem and discuss the Baum-Connes conjecture for Lie groupoids. With the help of the general framework of Lie groupoids and related structures we survey recent progress on problems which were outside the scope of the original work of Atiyah and Singer. This includes the Atiyah-Singer type index problem for many classes of non-compact manifolds (e.g. manifolds with a Lie structure at infinity). We also consider generalizations of the pseudodifferential calculus on Lie groupoids, e.g. for boundary value problems.

math.OA↗

arXiv2017

Getzler rescaling via adiabatic deformation and a renormalized local index formula

We prove a local index theorem of Atiyah-Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). With the help of a renormalized supertrace, defined on a suitable class of regularizing operators, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of a rescaled bundle over the adiabatic groupoid. Finally, we calculate the right coefficient in the heat kernel expansion by deforming the Dirac operator into a polynomial coefficient operator over the rescaled bundle and applying the Lichnerowicz theorem to the fibers of the groupoid and the Lie manifold.

math.OA↗

arXiv2017

Index formulae for mixed boundary conditions on manifolds with corners

We investigate the problem of calculating the Fredholm index of a geometric Dirac operator subject to local (e.g. Dirichlet and Neumann) and non-local (APS) boundary conditions posed on the strata of a manifold with corners. The boundary strata of the manifold with corners can intersect in higher codimension. To calculate the index we introduce a glueing construction and a corresponding Lie groupoid. We describe the Dirac operator subject to mixed boundary conditions via an equivariant family of Dirac operators on the fibers of the Lie groupoid. Using a heat kernel method with rescaling we derive a general index formula of the Atiyah-Singer type.

math.AP↗

arXiv2017

Boutet de Monvel operators on Lie manifolds with boundary

We introduce and study a general pseudodifferential calculus for boundary value problems on a class of non-compact manifolds with boundary (so-called Lie manifolds with boundary). This is accomplished by constructing a suitable generalization of the Boutet de Monvel calculus for boundary value problems. The data consists of a compact manifold with corners $M$ that is endowed with a Lie structure of vector fields $\mathcal{V}$, a so-called Lie manifold. The manifold $M$ is split into two equal parts $X_{+}$ and $X_{-}$ which intersect in an embedded hypersurface $Y \subset X_{\pm}$. Our goal is to describe a transmission Boutet de Monvel calculus for boundary value problems compatible with the structure of Lie manifolds. Starting with the example of $b$-vector fields, we show that there are two groupoids integrating the Lie structures on $M$ and on $Y$, respectively. These two groupoids form a bibundle (or a groupoid correspondence) and, under some mild assumptions, these groupoids are Morita equivalent. With the help of the bibundle structure and canonically defined manifolds with corners, which are blow-ups in particular cases, we define a class of Boutet de Monvel type operators. We then define the representation homomorphism for these operators and show closedness under composition with the help of a representation theorem. Finally, we consider appropriate Fredholm conditions and construct the parametrices for elliptic operators in the calculus.

math.AP↗

arXiv2016

On the $η$-function for bisingular pseudodifferential operators

In this work we consider the $η$-invariant for pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators. We study complex powers of classical bisingular operators. We prove the trace property for the Wodzicki residue of bisingular operators and show how the residues of the $η$-function can be expressed in terms of the Wodzicki trace of a projection operator. Then we calculate the $K$-theory of the algebra of $0$-order (global) bisingular operators. With these preparations we establish the regularity properties of the $η$-function at the origin for global bisingular operators which are self-adjoint, elliptic and of positive orders.

math.SP↗

arXiv2016

Boutet de Monvel operators on singular manifolds

We construct a Boutet de Monvel calculus for general pseudodifferential boundary value problems defined on a broad class of non-compact manifolds, the class of so-called Lie manifolds with boundary. It is known that this class of non-compact manifolds can be used to model many classes of singular manifolds.

math.AP↗

arXiv2015

The K-theory of bisingular pseudodifferential algebras

In this paper we calculate the K-theory of $C^{\ast}$-algebras given by the norm-closures of spaces of bisingular pseudodifferential operators. We obtain results for the \emph{global} bisingular calculus in the flat ($\Rr^{n_1 + n_2}$) case.

math.FA↗