arXiv research

Open and discrete maps with specific branch set images are equivalent to PL branched covers.

Develops TCD maps to relate discrete differential geometry and cluster algebras.

Riemannian Neural OT maps improve scalability on manifolds.

Stochastic optimization improves semi-discrete OT map estimation with a minimax rate.

We show that an entire branched cover of finite distortion cannot have a compact branch set if its distortion satisfies a certain asymptotic growth condition. We furthermore show that this bound is strict by constructing an entire, continuous, open and discrete mapping of finite distortion which is piecewise smooth, ha…

Let $H^n$ be the hyperbolic n-space with $n\geq 2$. Suppose that $Γ<Isom H^n$ is a discrete, torsion free subgroup and $a$ is a point in the domain of discontinuity $Ω(Γ)$. Let $p$ be the projection map from $H^n$ to the quotient manifold $M=H^n/Γ$. In this paper we prove that there exists an open neighborhood $U$ of $…

The paper proves the existence of minimal surfaces avoiding specific points.

New algorithm improves OT map estimation for semi-discrete settings.

New theorem proves convergence of various discrete conformal structures to conformal maps.

The paper studies stability of discretized Anosov flows.

The paper proves convergence of discrete maps to Riemann mappings for polyhedral surfaces.

Isothermic nets created from special maps for smooth surfaces.

No stable discrete maps into certain curved spaces exist.

Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete Riemannian manifold of non-positive sectional curvature or a locally finite tree). Isometric actions of G on M are (by definition) points in the space R := Hom(G, Isom(M)) with the compact open topology. Sample theorems: 1. The…

The paper studies convergence of discrete harmonic maps to smooth ones.

Discrete Flow Maps bypass sequential prediction limits for parallel text generation.

The paper explores heat flow and constants on graphs, proving properties and proposing new concepts.

Open problem: Establishing bounds for Cayley-table completion to discover discrete algorithmic axioms.

New representations for discrete surfaces derived from dual transforms.

The study of topological groups with compact open subgroups and their geometric properties.

Discrete conformal maps on surfaces with vertex decorations are studied.

Discrete exterior calculus shows natural properties of wedge product and averaging.

We prove that for each integer k of at least 2, there is an open neigborhood ν_k of the identity map of the 2-sphere S^2, in C^1-topology such that: if G is a nilpotent subgroup of Diff^1(S^2) with length k of nilpotency, generated by elements in ν_k, then the natural action on S^2 has non-empty fixed point set. Moreov…

Optimal discrete harmonic maps between hyperbolic surfaces are found via minimizing energy.

We establish a connection between recent developments in the study of vortices in the abelian Higgs models, and in the theory of structure-preserving discrete conformal maps. We explain how both are related via conformal mapping problems involving prescribed linear combinations of the curvature and volume form, and sho…

Analytic submanifolds of cocycles reveal discrete cohomology spaces.

The present chapter gives an overview on results for discrete knot energies. These discrete energies are designed to make swift numerical computations and thus open the field to computational methods. Additionally, they provide an independent, geometrically pleasing and consistent discrete model that behaves similarly …

Paper studies third order open mapping in sub-Riemannian geometry.

Minimal surfaces with symmetries exist under certain group actions.

We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by s…

Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume.

Effective methods compute equivariant harmonic maps from surfaces to nonpositively curved spaces.

Unified framework for various geometric constructions.

Discrete version of Liouville's theorem for simplicial complexes.

Overlays were introduced by R. H. Fox [6] as a subclass of covering maps. We offer a different view of overlays: it resembles the definition of paracompact spaces via star refinements of open covers. One introduces covering structures for covering maps and $p:X\to Y$ is an overlay if it has a covering structure that ha…

DRAG decreases regularization to accelerate semi-discrete OT convergence.

A discrete conformal map (DCM) maps the square lattice to the Riemann sphere such that the image of every irreducible square has the same cross-ratio. This paper shows that every periodic DCM can be determined from spectral data (a hyperelliptic compact Riemann surface, called the spectral curve, equipped with some mar…

New mechanics on non-associative octonions discovered.

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to\mathbb{R}^n$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the…

Uniformizes surfaces using discrete harmonic maps and hyperbolic metrics.

Let $(W,S)$ be a Coxeter system with Davis complex $Σ$. The polyhedral automorphism group $G$ of $Σ$ is a locally compact group under the compact-open topology. If $G$ is a discrete group (as characterised by Haglund--Paulin), then the set $\mathcal V_u(G)$ of uniform lattices in $G$ is discrete. Whether the converse i…

Brezis' open problem on harmonic maps resolved

Discrete approximation solves Björling's minimal surface problem.

Maps discrete manifolds to partitions to define new manifolds.

Discrete hyperbolic isometries proven via test maps.

Estimates discontinuous optimal transport maps between a discrete and continuous distribution.

Develops a new method for learning discrete distributions without embedding them in a continuous space.

Maps between Hadamard manifolds are quasi-isometric to harmonic maps.