Related papers: Helmholtzian Eigenmap: Topological feature discove…

Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians…

Current point cloud processing algorithms do not have the capability to automatically extract semantic information from the observed scenes, except in very specialized cases. Furthermore, existing mesh analysis paradigms cannot be directly…

In symplectic topology one uses elliptic methods to prove rigidity results about symplectic manifolds and solutions of Hamiltonian equations on them, where the most basic example is given by geodesics on Riemannian manifolds. Harmonic maps…

We first develop a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. This includes, among others, the graph Laplacian, the combinatorial Laplacian on simplicial complexes, the…

This paper gives a geometric description of functional spaces related to Domain Decomposition techniques for computing solutions of Laplace and Helmholtz equations. Understanding the geometric structure of these spaces leads to algorithms…

This paper presents a novel physics-inspired deep learning approach for point cloud processing motivated by the natural flow phenomena in fluid mechanics. Our learning architecture jointly defines data in an Eulerian world space, using a…

We establish conditions for which graph Laplacians $\Delta_{\lambda,\epsilon}$ on compact, boundaryless, smooth submanifolds $\mathcal{M}$ of Euclidean space are semiclassical pseudodifferential operators ($\Psi$DOs): essentially, that the…

Let $X$ be a compact polyhedral surface (a compact Riemann surface with flat conformal metric $\mathfrak{T}$ having conical singularities). The $\zeta$-function $\zeta_\Delta(s)$ of the Friedrichs Laplacian on $X$ is meromorphic in…

We propose a mathematical model for fluids in multiphase flows in order to establish a solid theoretical foundation for the study of their complex topology, large geometric deformations, and topological changes such as merging. Our modeling…

Let $(E,h)$ be a holomorphic Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of $E$. If $E$ is simple we obtain an…

Inspired by persistent homology in topological data analysis, we introduce the homological eigenvalues of the graph $p$-Laplacian $\Delta_p$, which allows us to analyse and classify non-variational eigenvalues. We show the stability of…

The concept of a Point Cloud has played an increasingly important role in many areas of Engineering, Science, and Mathematics. Examples are: LIDAR, 3D-Printing, Data Analysis, Computer Graphics, Machine Learning, Mathematical Visualization,…

The Laplace-Beltrami operator has established itself in the field of non-rigid shape analysis due to its many useful properties such as being invariant under isometric transformation, having a countable eigensystem forming an orthornormal…

Let H be a Tonelli Hamiltonian defined on the cotangent bundle of a compact and connected manifold and let u be a semi-concave function defined on M. If E (u) is the set of all the super-differentials of u and (\phi t) the Hamiltonian flow…

Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves…

We introduce the logarithmic analogue of the Laplace-Beltrami operator on Ahlfors regular metric-measure spaces. This operator is intrinsically defined with spectral properties analogous to those of elliptic pseudo-differential operators on…

Let $\mathscr{L}$ be either the Laplace--Beltrami operator, its shift without spectral gap, or the distinguished Laplacian on a symmetric space of noncompact type $\mathbb{X}$ of arbitrary rank. We consider the heat equation, the fractional…

We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation. We prove the appropriate generalizations…

Let $M$ be a Riemannian manifold with a smooth boundary. The main question we address in this article is: "When is the Laplace-Beltrami operator $\Delta\colon H^{k+1}(M)\cap H^1_0(M) \to H^{k-1}(M)$, $k\in \mathbb{N}_0$, invertible?" We…

The main objective of this dissertation is to analyse thoroughly the construction of self-adjoint extensions of the Laplace-Beltrami operator defined on a compact Riemannian manifold with boundary and the role that quadratic forms play to…