Related papers: A formal proof of the Kepler conjecture

We upgrade [1] to a complete proof of the conjecture NP = PSPACE. [1]: L. Gordeev, E. H. Haeusler, Proof Compression and NP Versus PSPACE, Studia Logica (107) (1): 55-83 (2019)

Compact packings are specific packings of spheres which can be seen as tilings and are good candidates to maximize the density. We show that the compact packings of the Euclidean space with two sizes of spheres are exactly those obtained by…

An Isabelle/HOL formalisation of G\"odel's two incompleteness theorems is presented. The work follows \'Swierczkowski's detailed proof of the theorems using hereditarily finite (HF) set theory. Avoiding the usual arithmetical encodings of…

The famous Kepler conjecture has a less spectacular, two-dimensional equivalent: The theorem of Thue states that the densest circle packing in the Euclidean plane has a hexagonal structure. A common proof uses Voronoi cells and analyzes…

We prove multiple generalizations of Fan's combinatorial labeling result for sphere triangulations. This can be seen as a comprehensive extension of the Borsuk--Ulam theorem. In typical applications, the Borsuk--Ulam theorem gives…

In \cite{G3}, Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. Glickenstein's discrete conformal structures include Thurston's circle packings,…

We develop a version of controlled algebra for simplicial rings. This generalizes the methods which lead to successful proofs of the algebraic K- theory isomorphism conjecture (Farrell-Jones Conjecture) for a large class of groups. This is…

We prove that for any discrete curvature satisfying Gauss-Bonnet formula, there exist a unique up to scaling inversive distance circle packing in the discrete conformal equivalent class, whose polyhedral metric meets the target curvature.…

This paper is an exposition, written for the Nieuw Archief voor Wiskunde, about the two recent breakthrough results in the theory of sphere packings. It includes an interview with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Maryna…

The first version of this paper gave another proof of the Kropholler Conjecture, which gives a relative version of Stallings Ends Theorem, following an earlier incorrect proof. It has been pointed out by Sam Shepherd that the the second…

This note is concerned with the disproof of the most general case of Parker's conjecture. The conjecture relates a certain group theoretic objects to the field of moduli of a Dessin d'enfant.

We investigate how many hyperplanes with independent standard Gaussian directions one needs to produce a $\delta$-uniform tessellation of a subset $S$ of the Euclidean sphere, meaning that for any pair of points in $S$ the fraction of…

In this paper, we are concerned with the effective elastic property of a two-phase high-contrast periodic composite with densely packed inclusions. The equations of linear elasticity are assumed. We first give a novel proof of the…

Mechanized theorem proving is becoming the basis of reliable systems programming and rigorous mathematics. Despite decades of progress in proof automation, writing mechanized proofs still requires engineers' expertise and remains labor…

Several results about the union-closed sets conjecture are presented.

In his talk "Integral Apollonian disk Packings" Peter Sarnak asked if there is a "proof from the Book" of the Descartes theorem on circles. A candidate for such a proof is presented in this note

We formulate a "correct" version of the Quillen conjecture on linear group homology for certain arithmetic rings and provide evidence for the new conjecture. In this way we predict that the linear group homology has a direct summand looking…

We give an elementary proof of a version of the implicit function theorem over Henselian valued fields $K$. It yields a density property for such fields (introduced in a joint paper with J. Koll{\'a}r), which is indispensable for ensuring…

In this paper, we give a detailed account of Goldfeld's proof of Siegel's theorem. Particularly, we present complete proofs of the nontrivial assumptions made in his paper.

We show that the Volume Conjecture for polyhedra implies a weak version of the Stoker Conjecture; in turn we prove that this weak version of the Stoker conjecture implies the Stoker conjecture. The main tool used is an extension of a result…