Related papers: An Obstruction to Higher-Dimensional Kernel of Dir…
We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation…
We study the relation between spectral flow and index theory within the framework of (unbounded) KK-theory. In particular, we consider a generalised notion of 'Dirac-Schr\"odinger operators', consisting of a self-adjoint elliptic…
The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the…
In the old spirit of Kaluza-Klein, we consider a spacetime of the form $P = M_4 \times K$, where $K$ is the Lie group $\mathrm{SU}(3)$ equipped with a left-invariant metric that is not fully right-invariant. We observe that a complete…
We describe two topologies on the space of unbounded Fredholm operators and we explain their K-theoretic relevance. In the process we also prove a very general result concerning the continuity of families of first order, elliptic boundary…
We set up a fibred categorical theory of obstruction and classification of morphisms that specializes to the one of monoidal functors between categorical groups and also to the Schreier-Mac Lane theory of group extensions. Further…
We establish the factorization of the Dirac operator on an almost-regular fibration of spin$^c$ manifolds in unbounded KK-theory. As a first intermediate result we establish that any vertically elliptic and symmetric first-order…
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,…
We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture…
We carry the index theory for manifolds with boundary of B\"ar and Ballmann over to first order differential operators on metric graphs. This approach results in a short proof for the index of such operators. Then the self-adjoint…
The structured operators and corresponding operator identities, which appear in inverse problems for the self-adjoint and skew-self-adjoint Dirac systems with rectangular potentials, are studied in detail. In particular, it is shown that…
A Dirac operator on the plane with constant (positive) mass is a Chern insulator, sitting in class D of the Kitaev table. Despite its simplicity, this system is topologically ill-behaved: the non-compact Brillouin zone prevents definition…
Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. We compute the obstruction for a vector bundle $S$ over $(M,g)$ to admit a Dirac operator whose principal symbol induces on $S$ the structure of a bundle of irreducible real…
We analyze the topological properties of a chiral ${p}+i{p}$ superconductor for a two-dimensional metal/semimetal with four Dirac points. Such a system has been proposed to realize second-order topological superconductivity and host corner…
I derive the overlap Dirac operator starting from the overlap formalism, discuss the numerical hurdles in dealing with this operator and present ways to overcome them.
For this quarter of century, differential operators in a lower dimensional submanifold embedded or immersed in real $n$-dimensional euclidean space $\EE^n$ have been studied as quantum mechanical models, which are realized as restriction of…
We develop the method of similar operators to study the spectral properties of unbounded perturbed linear operators that can be represented by matrices of various kinds. The class of operators under consideration includes various…
We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by…
We describe a topological predual to differential forms constructed as an inductive limit of a sequence of Banach spaces. This subspace of currents has nice properties, in that Dirac chains and polyhedral chains are dense, and its operator…
In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by A.Vaintrob and two of the authors.…