arXiv research

Authors prove an asymptotic expansion for spectral zeta functions on discrete tori.

Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We s…

We construct an infinite family of simply connected, pairwise nondiffeomorphic 4-manifolds, all homeomorphic to 3CP^2 blown up at 9 points.

Authors prove a formula relating the Gaussian curvature of polyhedral vertex stars to their Gauss images.

We study the Berezin-Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-pro…

The paper constructs quantizations for symplectic manifolds with specific Laplacian properties.

We use persistent homology along with the eigenfunctions of the Laplacian to study similarity amongst triangulated 2-manifolds. Our method relies on studying the lower-star filtration induced by the eigenfunctions of the Laplacian. This gives us a shape descriptor that inherits the rich information encoded in the eigen…

The paper proves a new discrete Laplacian for 3D meshes and shows its superiority over primal construction.

BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.

Discrete Laplacians defined for spherical and hyperbolic surfaces.

STAR framework reduces OPE variance by distilling complex problems into discrete ARPs.

Sharp Lipschitz bounds for flow-matching and diffusion models with optimal sampling rates.

We establish a uniform comparison between the spectrum of the rough Laplacian (acting on sections of a vector bundle of complex rank one or of harmonic curvature) with the spectrum of a discrete operator (a generalization of a discrete magnetic Laplacian added with a potential) acting on a finite dimensional space comi…

The paper analyzes Laplacian pyramids for extending and denoising discrete functions.

Let $f^{\star}$ be a function on $\mathbb{R}^d$ with an assumption of a spectral norm $v_{f^{\star}}$. For various noise settings, we show that $\mathbb{E}\|\hat{f} - f^{\star} \|^2 \leq \left(v^4_{f^{\star}}\frac{\log d}{n}\right)^{1/3}$, where $n$ is the sample size and $\hat{f}$ is either a penalized lea…

The paper models star dynamics using Ricci flow and Perelman entropy, revealing chaotic behavior.

The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on …

This paper uses the technology of weighted and regular triangulations to study discrete versions of the Laplacian on piecewise Euclidean manifolds. Regular triangulations are studied in some detail, including flip algorithms. The Laplacian is then studied as an operator on functions of the vertices as a generalized wei…

The paper analyzes spectral properties of connection Laplacian on tori, proving convergence to real torus.

We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of Eastwood, Leistner, Gover and Šilhan. In particular, conformally equivariant qu…

The spectral geometry of mesh matrices of graphs is explored, leading to new formulas and eigenvalue estimates.

It is shown that a (curved) projective structure on a smooth manifold determines on the Poisson algebra of smooth, fiberwise-polynomial functions on the cotangent bundle a one-parameter family of graded star products. For a particular value of the parameter (corresponding to half-densities) the star product is symmetri…

Universal inequalities for Laplacian eigenvalues on discrete groups.

Introduces Star-Shaped DDPMs for non-Gaussian distributions.

Study discrete analog of zeta-determinant maximization on triangulated surfaces.

Existence and uniqueness theorem for Ricci flow on weighted graphs proved.

Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representa…

Study on the spectrum of drift Laplacian on Ricci expanders.

The paper finds minimum Steklov eigenvalues on combinatorial graphs.

Fast linear transforms are ubiquitous in machine learning, including the discrete Fourier transform, discrete cosine transform, and other structured transformations such as convolutions. All of these transforms can be represented by dense matrix-vector multiplication, yet each has a specialized and highly efficient (su…

Graph Laplacian approximates manifold eigenvalues with controlled curvature bounds.

We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In…

To every Hermitian vector bundle with connection over a compact Riemannian manifold $M$ one can associate a corresponding connection Laplacian acting on the sections of the bundle. We define analogous combinatorial metric dependent Laplacians associated to triangulations of $M$ and prove that their spectra converge, as…

Study asymptotic expansion of graph Laplacian on discretized surfaces, relating spanning trees and cycle-rooted forests.

We consider the Laplacian with attractive Robin boundary conditions, \[ Q^Ω_αu=-Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on } \partialΩ, \] in a class of bounded smooth domains $Ω\in\mathbb{R}^ν$; here $n$ is the outward unit normal and $α>0$ is a constant. We show that for each $j\in\mathbb{N}$ and $α\to+\in…

We prove some estimates on the spectrum of the Laplacian of the total space of a Riemannian submersion in terms of the spectrum of the Laplacian of the base and the geometry of the fibers. When the fibers of the submersions are compact and minimal, we prove that the total space is discrete if and only if the base is di…

Semiparametric STAR model improves mental health data analysis.

We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a half-translation surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian with von Neumann boundary conditions. As an interesting byproduct of our study, w…

An adapted version of the proof (due to A. Weil) of the well-known de Rham Theorem allows us to compare uniformly the spectrum of the Hodge Laplacian acting on differential forms (on a compact Riemannian manifold) to the spectrum of the combinatorial Laplacian acting on cochains associated to an open cover (made of bal…

New methods improve solving linear systems and preconditioning with reduced complexity.

We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on finding harmonic cochains cohomologous to a given cocycle. Amongst other …

HLTF generates chemically valid 3D molecules with improved topology control.

Method detects trajectory outliers using Hodge Laplacian embeddings.

We derive a numerical method for Darcy flow, hence also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedr…

Study ancient solutions on graphs with unbounded Laplacians, generalizing previous results.

By introducing a shape manifold as a solution set to solve inverse obstacle scattering problems we allow the reconstruction of general, not necessarily star-shaped curves. The bending energy is used as a stabilizing term in Tikhonov regularization to gain independence of the parametrization. Moreover, we discuss how se…

We consider Toeplitz operators associated with the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold. We study the asymptotic behavior, in the semiclassical limit, of low-lying eigenvalues and the corresponding eigenfunctions of a self-adjoint Toeplitz opera…