Thomas C. Hales

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This article gives an introduction to a long-term project called Flyspeck, whose purpose is to give a formal verification of the Kepler Conjecture. The Kepler Conjecture asserts that the density of a packing of equal radius balls in three dimensions cannot exceed π/ √ 18. The original proof of the Kepler Conjecture, from 1998, relies extensively on computer(More)
W hen Hilbert introduced his famous list of 23 problems, he said a test of the perfection of a mathematical problem is whether it can be explained to the first person in the street. Even after a full century, Hilbert's problems have never been thoroughly tested. Who has ever chatted with a telemarketer about the Riemann hypothesis or discussed general(More)
The Kepler conjecture asserts that no packing of congruent balls in three-dimensional Euclidean space has density greater than that of the face-centered cubic packing. The original proof, announced in 1998 and published in 2006, is long and complex. The process of revision and review did not end with the publication of the proof. This article summarizes the(More)
1. INTRODUCTION. The Jordan curve theorem states that every simple closed pla-nar curve separates the plane into a bounded interior region and an unbounded exterior. One hundred years ago, Oswald Veblen declared that this theorem is " justly regarded as a most important step in the direction of a perfectly rigorous mathematics " [13, p. 83]. Its position as(More)
We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main(More)
We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay(More)