From hales@math.lsa.umich.edu Wed Aug 19 02:43:02 1998 Date: Sun, 9 Aug 1998 09:54:56 -0400 (EDT) From: Tom Hales To: Subject: Kepler conjecture Dear colleagues, I have started to distribute copies of a series of papers giving a solution to the Kepler conjecture, the oldest problem in discrete geometry. These results are still preliminary in the sense that they have not been refereed and have not even been submitted for publication, but the proofs are to the best of my knowledge correct and complete. Nearly four hundred years ago, Kepler asserted that no packing of congruent spheres can have a density greater than the density of the face-centered cubic packing. This assertion has come to be known as the Kepler conjecture. In 1900, Hilbert included the Kepler conjecture in his famous list of mathematical problems. In a paper published last year in the journal "Discrete and Computational Geometry," (DCG), I published a detailed plan describing how the Kepler conjecture might be proved. This approach differs significantly from earlier approaches to this problem by making extensive use of computers. L. Fejes T'oth was the first to suggest the use of computers.) The proof relies extensively on methods from the theory of global optimization, linear programming, and interval arithmetic. The full proof appears in a series of papers totaling well over 250 pages. The computer files containing the computer code and data files for combinatorics, interval arithmetic, and linear programs require over 3 gigabytes of space for storage. Samuel P. Ferguson, who finished his Ph.D. last year at the University of Michigan last year under my direction, has contributed significantly to this project. The papers containing the proof are: An Overview of the Kepler Conjecture, Thomas C. Hales A Formulation of the Kepler Conjecture, Samuel P. Ferguson and Thomas C. Hales Sphere Packings I, Thomas C. Hales (published in DCG, 1997) Sphere Packings II, Thomas C. Hales (published in DCG, 1997) Sphere Packings III, Thomas C. Hales Sphere Packings IV, Thomas C. Hales Sphere Packings V, Samuel P. Ferguson The Kepler Conjecture (Sphere Packings VI), Thomas C. Hales Postscript versions of the papers and more information about this project can be found at http://www.math.lsa.umich.edu/~hales Tom Hales samf@math.lsa.umich.edu hales@math.lsa.umich.edu