arXiv research

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

This is the second part of the works on Hausdorff dimensions of Schottky groups. It has been conjectured that the Hausdorff dimensions of nonclassical Schottky groups are strictly bounded from below. In this second part of our works we provide a resolution of this conjecture, we prove that there exists a universal posi…

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…

This document consists of the collection of handouts for a two-week summer workshop entitled 'Geometry and the Imagination', led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 17-28, 1991. The workshop was based on a course `Geometry and the Imagination' which we …

We introduce a new discretization of O'Hara's Möbius energy. In contrast to the known discretizations of Simon and Kim and Kusner it is invariant under Möbius transformations of the surrounding space. The starting point for this new discretization is the cosine formula of Doyle and Schramm. We then show $Γ$-convergence…

The purpose of this paper is to extend the Green-Naghdi-Rivlin balance of energy method to continua with microstructure. The key idea is to replace the group of Galilean transformations with the group of diffeomorphisms of the ambient space. A key advantage is that one obtains in a natural way all the needed balance la…

O'Hara introduced several functionals as knot energies. One of them is the Möbius energy. We know its Möbius invariance from Doyle-Schramm's cosine formula. It is also known that the Möbius energy was decomposed into three components keeping the Möbius invariance. The first component of decomposition represents the ext…

Energy functional for Legendrian knots in Heisenberg group, invariant under PU(2,1).

We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form $X/Γ$ where $X$ is a contractible Riemannian manifold and $Γ<\Isom(X)$ is a discrete subgroup, typically with infinite co-volume. The existence depends on the $L^2$-Betti numbers of $Γ$, its subgroups and of a uniform latt…

Polynomial chaos surrogates quantify epistemic uncertainty in AI-driven scientific models.

New metrics help predict Brownian motion on surfaces and higher dimensions.

Polyhedra volume conjecture supports Stoker conjecture weakly.

Survey on two non-Kähler geometry conjectures.

Numerical study confirms Brennan's conjecture for a counterexample to Thurston's $K=2$ conjecture.

The non-vanishing conjecture implies the abundance conjecture in certain cases.

Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…

Symmetry-breaking in three differential geometry conjectures.

We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…

Study proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.

New proof shows most thin knots satisfy Cabling Conjecture.

We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum-Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which tog…

Metric SYZ conjecture proved using non-archimedean geometry.

Paper confirms Chen's biharmonic conjecture for hypersurfaces in 5D.

Counterexample disproves recent Penrose conjecture variant.

In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture.

Proof outlined for 4D smooth Poincaré conjecture.

In order to give a unified generalization of the BW inequality and the DDVV inequality, Lu and Wenzel proposed three Conjectures 1, 2, 3 and an open Question 1 in 2016. In this paper we discuss further these conjectures and put forward several new conjectures which will be shown equivalent to Conjecture 2. In particula…

In this paper, we generalize the Cosmetic Surgery Conjecture to an $n$-cusped hyperbolic $3$-manifold and prove it under the assumption of another well-known conjecture in number theory, so called the Zilber-Pink Conjecture. For $n=1$ and $2$, we show them without the assumption.

Paper connects AJ conjecture and colored Jones polynomial potential function.

Akbulut and Kirby conjectured that two knots with the same $0$-surgery are concordant. In this paper, we prove that if the slice-ribbon conjecture is true, then the modified Akbulut-Kirby's conjecture is false. We also give a fibered potential counterexample to the slice-ribbon conjecture.

Counterexample disproves conjectures about log canonical thresholds.

Study confirms conjecture on Hermitian manifolds with bounded mass.

Reformulated Markov's conjecture in combinatorial terms.

Paper confirms Whitehead's conjecture for aspherical 2-complexes.

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and app…

Verifies a conjecture for the figure eight knot.

Fukaya-Yamaguchi conjecture holds in 4D manifolds with nonnegative curvature.

Proved a combinatorial conjecture in machine learning.

In their study of the Yamabe problem in the presence of isometry group, Hebey and Vaugon announced a conjecture. This conjecture generalizes Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In this paper, we generalize Aubin's theorem and we prove the Hebey--Vaugon conjec…

There is a relation between the generalized Property R Conjecture and the Schoenflies Conjecture that suggests a new line of attack on the latter. The approach gives a quick proof of the genus 2 Schoenflies Conjecture and suffices to prove the genus 3 case, even in the absence of new progress on the generalized Propert…

Paper proves contractible fake surfaces up to complexity 6 are deformable.

Paper analyzes Bartnik's quasi-local mass conjectures and their validity.

This memoire consists of two main results. In the first one we describe Ricci flow theory and we give an educative way for proving Elliptization Conjecture and then we prove Poincare conjecture which is the second proof of Perelman for Poincare conjecture. In the second one which is the main propose of our memoire, we …

Proves Stolz conjecture for specific types of manifolds.

In this paper we propose counterexamples to the Geometrization Conjecture and the Elliptization Conjecture.

Established a version of the Atiyah-Floer conjecture for SO(3)-bundles.

The slope conjecture gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this note we propose a generalization of the slope conjecture to links. We prove the conjecture for all alternating and more generally adequ…