In this work we propose a generalization of the Heat Kernel Signature (HKS). The HKS is a point signature derived from the heat kernel of the Laplace-Beltrami operator of a surface. In the theory of exterior calculus on a Riemannian manifold, the Laplace-Beltrami operator of a surface is a special case of the Hodge Laplacian which acts on r-forms, i. e. the Hodge Laplacian on 0-forms (functions...

The Hodge theory of the de Rham complex in the setting of the Sobolev topology is studied. As a result, a new elliptic boundaryvalue problem is obtained. Next, the Hodge theory of the @-Neumann problem in the Sobolev topology is studied. A new @-Neumann boundary condition is obtained, and the corresponding subelliptic estimate derived. The classical Hodge theory on a domain in R N+1 (or, more g...

In a survey paper [1], Agarwal and Ding summarized the advances achieved in the study of A-harmonic equations. Some recent results about A-harmonic equations can also be found in [2, 3, 5, 6]. The purpose of this note is to establish some estimates about Green’s operator and the Hodge codifferential operator d∗, which will enrich the existing literature in the field of A-harmonic equations. Let...

We study ∗-differential calculi over compact quantum groups in the sense of S.L. Woronowicz. Our principal results are the construction of a Hodge operator commuting with the Laplacian, the derivation of a corresponding Hodge decomposition of the calculus of forms, and, for Woronowicz’ first calculus, the calculation of the eigenvalues of the Laplacian.

of the Dissertation On the Algebra and Geometry of a Manifold’s Chains and Cochains by Scott Owen Wilson Doctor of Philosophy in Mathematics Stony Brook University 2005 This dissertation consists of two parts, each of which describes new algebraic and geometric structures defined on chain complexes associated to a manifold. In the first part we define, on the simplicial cochains of a triangulat...

The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several new features, includin...

The purpose of these notes is to show that the methods introduced by Bauer and Furuta, see [5, 6, 7], in order to refine the Seiberg-Witten invariants of smooth 4-dimensional manifolds can also be used to obtain stable homotopy classes from 2-dimensional manifolds, using the vortex equations on the latter. So far these notes contain barely more than the necessary analytic estimates to prove thi...

The differential-geometric and topological structure of Delsarte transmutation operators and associated with them Gelfand-Levitan-Marchenko type eqautions are studied making use of the De Rham-Hodge-Skrypnik differential complex. The relationships with spectral theory and special Berezansky type congruence properties of Delsarte transmuted operators are stated. Some applications to multidimensi...