arXiv research

CLAMP uses neural manifold packing to improve self-supervised learning.

Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.

Proves rigidity of circle packings in the plane, generalizing previous work.

The paper studies rigidity of sphere packings on 3D manifolds with boundary.

The paper studies circle packings using renormalization and subdivision rules.

Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.

Study generates infinite circle packings with a specific property.

Paper proves circle packings converge to Riemann mapping for Jordan domains.

Paper introduces new flows to find circle packings with specific curvature.

Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.

The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.

The paper solves the existence problem of sphere packings in higher dimensions.

Projective rigidity of circle packings on complex surfaces proved.

The paper studies rigid sphere packings on 3D manifolds with boundary.

Thurston's sphere packing on a 3-dimensional manifold is a generalization of Thusrton's circle packing on a surface, the rigidity of which has been open for many years. In this paper, we prove that Thurston's Euclidean sphere packing is locally determined by combinatorial scalar curvature up to scaling, which generaliz…

The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.

Study of rod packings in 3-torus using 3-manifold geometry.

Paper studies degenerated circle packings in hyperbolic geometry and finds conditions for their existence.

As neural networks are increasingly employed in machine learning practice, how to efficiently share limited training resources among a diverse set of model training tasks becomes a crucial issue. To achieve better utilization of the shared resources, we explore the idea of jointly training multiple neural network model…

Kernel sparsity ("dying ReLUs") and lack of diversity are commonly observed in CNN kernels, which decreases model capacity. Drawing inspiration from information theory and wireless communications, we demonstrate the intersection of coding theory and deep learning through the Grassmannian subspace packing problem in CNN…

Paper constructs hyperbolic metrics using circle packings and curvature parameters.

Study on packing links with geometric constraints.

Confirms unique eigenfunction in hyperbolic packing has maximal spectral gap.

Paper calculates ball number of links using Lorentz geometry and circle packing.

The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.

New theorem proves rigidity of circle packings in hyperbolic geometry.

The paper connects Apollonian packings to knot theory and improves link representations.

Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.

The paper solves circle packings on surfaces with boundaries.

The paper finds circle packings with specific curvatures in hyperbolic geometry.

From the geometric study of the elementary cell of hexagonal circle packings --- a flower of 7 circles --- the class of conformally symmetric circle packings is defined. Up to Moebius transformations, this class is a three parameter family, that contains the famous Doyle spirals as a special case. The solutions are giv…

The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on statistical minimax rates for the problem of estimating a probability distribution under W…

An extremal $k$-packing is a collection of $k$ mutually disjoint metric discs, embedded in a surface, whose radius is maximal for the given topology. We study compact non-orientable surfaces of genus $g\ge 3$ containing extremal $k$-packings.

We propose a new approach to the value distribution theory of entire holomorphic curves. We define a ``packing density'' of an entire holomorphic curve, and show that it has various non-trivial properties. We prove a ``gap theorem'' for holomorphic maps from elliptic curves to the complex projective space, and study th…

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity a…

The Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σ_g of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1, 0 or -1 on the surface depending on whether g=0, 1 or \ge 2 such that the surface with this metric admits a…

Recently, Freedman [arXiv:2301.00295] introduced the idea of packing a maximal number of links into a bounded region subject to geometric constraints, and produced upper bounds on the packing number in some cases, while commenting that these bounds seemed far too large. We show that the smallest of these "extravagantly…

We study circle packings with the combinatorics of a triangulated disk in the plane and parametrize deformations of circle packings in terms of vertex rotation and cross ratios. We show that there is a Weierstrass representation formula relating infinitesimal deformations of circle packings to discrete minimal surfaces…

This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive dist…

Pack hyperbolic surfaces with circles or horocycles, noting symmetries.

Solving polynomial equations finds circle packings on surfaces.

We discuss closed symplectic 4-manifolds which admit full symplectic packings by $N$ equal balls for large $N$'s. We give a homological criterion for recognizing such manifolds. As a corollary we prove that ${\Bbb C}P^2$ can be fully packed by $N$ equal balls for every $N\geq 9$.

The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.

We describe some problems, observations, and conjectures concerning thickness and packing density of knots and links in $\sp^3$ and $\R^3$. We prove the thickness of a nontrivial knot or link in $\sp^3$ is no more than $\fracπ{4}$, the thickness of a Hopf link. We also give arguments and evidence supporting the conject…

Inversive distance circle packing on surfaces was introduced by Bowers-Stephenson as a generalization of Thurston's circle packing and conjectured to be rigid. The infinitesimal and global rigidity of circle packing with nonnegative inversive distance were proved by Guo and Luo respectively. The author proved the globa…

In this paper, we generalize Chow-Luo's combinatorial Ricci flow to inversive distance circle packing setting. Although the solution to the generalized flow may develop singularities in finite time, we can always extend the solution so as it exists for all time and converges exponentially fast. Thus the generalized flo…

We show that the analog of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algo…

We use Bryant Representation to construct constant mean curvature one surfaces in hyperbolic space that desingularize a horosphere packing.