Top-10

Top 10 publications (author’s choice)

  1. A proof of the Kepler conjecture. Ann. of Math. 162 (2005), no. 3, 1065–1185, Annals-link. This solves the 400-year-old Kepler conjecture. It is a major part of Hilbert’s 18th problem and the oldest problem in discrete geometry. T. Hales and S. Ferguson were awarded the first Robbins Prize of the AMS in 2007 because of this work. They also received the 2006 Fulkerson prize of the AMS and Mathematical Optimization Society (MOS) for this work. Hales was also awarded the Moore Prize for Applications of Interval analysis. {D}{T}
  2. (with 21 other coauthors including A. Solovyev and Hoang Le Truong) A Formal proof of the Kepler conjecture, preprint 2014, arxiv:1501.02155. This paper, with nearly two dozen co-authors, was over a decade in the making. It reports on the completion of the formal proof of the Kepler conjecture, which checks every logical inference in the proof of the Kepler conjecture down to the fundamental axioms of mathematical and primitive inference rules of logic. The project created over half a million lines of proof scripts. It is one of the largest formal proof projects ever completed. {D}{F}{R}{T}
  3. Dense Sphere Packings: a blueprint for formal proofs, Cambridge University Press, LMS volume 400, 2012, pdf-link. This book contains a second-generation proof of the Kepler conjecture. It is written in a style that is conducive to formalization. The book was used as the blueprint of the formal proof of the Kepler conjecture. ACM’s Computing Reviews named it a notable book in computing in 2012 in the Theory of Computation category. {F}{D}{B}{R}{T}
  4. The honeycomb conjecture. Discrete Comput. Geom. 25 (2001), no. 1, 1–22, arXiv:math/9906042. This is a problem of ancient origin. The introduction to the paper gives a history of the problem, starting in 36 B.C. Because of the mathematical optimality of the honeycomb, there is a traditional extending through the centuries of the wisdom of the bees. The architecture of the bee’s cell is mentioned in “Arabian nights.” Darwin took interest in the mathematical optimality of the hexagonal cells of the honeycomb from the point of view of natural selection. The mathematical proof of the optimality of the hexagonal arrangement first appeared in this paper. {D}{T}
  5. Cannonballs and honeycombs. Notices Amer. Math. Soc. 47 (2000), no. 4, 440–449, AMS-link. This paper describes the main ideas going into the proof of the Kepler Conjecture and the Honeycomb conjecture, and their relationship to the Kelvin conjecture. Hales was awarded the Chauvenet Prize for this paper. {D}{E}{T}
  6. The Jordan curve theorem, formally and informally. Amer. Math. Monthly 114 (2007), no. 10, 882–894, MAA-link. In 2005, Hales completed a formal proof of the Jordan Curve theorem. At the time, this was one of the most difficult formal proof projects ever completed. This paper describes the project to a general mathematical audience. This article was awarded the Lester Ford award of the MAA in 2008. {D}{T}
  7. What is motivic measure? Bull. Amer. Math. Soc. 42 (2005), 119-135, arxiv-link. This article was solicited by D. Eisenbud and was presented at the annual meetings of the American Mathematical Session at the special session on “Current Events” in Mathematics. It is an expository article on Denef and Loeser’s theory of motivic integration. {M}{E}{T}
  8. Formal proof. Notices Amer. Math. Soc. 55 (2008), no. 11, 1370–1380, AMS-link. This was the lead article of a special issue of the Notices of the AMS on Formal Proof that Hales helped to edit, with other contributions to the issue from J. Harrison, G. Gonthier, and F. Wiedijk. This issue helped to move the subject of Formal Proof from computer science departments into the mathematical mainstream. {F}{E}{T}
  9. On the fundamental lemma for standard endoscopy: reduction to unit elements. Canad. J. Math. 47 (1995), no. 5, 974–994, pdf-link. The fundamental lemma, as originally conjectured by Langlands and Shelstad, was a statement about transfer of functions in the spherical Hecke algebra. This article, based on some ideas of Clozel, used the trace formula to reduce the fundamental lemma to the transfer of the unit element in the Hecke algebra. It was in this form that the fundamental lemma was eventually proved by Ngo Bao Chau. This article also shows that the fundamental lemma holds at all places, if it holds at almost all places. This removes the mild restrictions on the residue field characteristic required by Ngo Bao Chau’s methods. The proofs are based on the trace formula. Recent work (2015) by J.-L. Waldspurger has generalized the results of these papers to twisted endoscopy. {L}{T}
  10. (with R. Cluckers and F. Loeser) Transfer principle for the fundamental lemma. Stabilization of the trace formula, Shimura varieties, and arithmetic applications ed. M. Harris, arXiv:0712.0708. This paper shows how to transfer the Fundamental Lemma from positive characteristic to characteristic zero.
    (This result was obtained independently by J.-L. Waldspurger.)
    The fundamental lemma was conjectured by Langlands and Shelstad and proved by Ngo Bao Chau in work that led to Ngo Bao Chau’s Field’s medal. This article shows that motivic integration and transfer principles of Cluckers and Loeser give a general framework for lifting results to characteristic zero. {M}{L}{T}

Runners-up

  • A proof of the dodecahedral conjecture (with S. McLaughlin) arxiv-link.  This paper solves a conjecture that is over 70 years old by L. Fejes Toth about the smallest possible cells in a packing of congruent spheres. Sean McLaughlin was awarded the Morgan Prize of the AMS-MAA-SIAM for this work.
  • Mathematics in the Age of the Turing Machine. arxiv-link. The book in which this essay appears, “Turing’s legacy: developments from Turing’s ideas in logic,” was named as a notable book in computing in ACM’s 2014 Computing Reviews,  in the general literature category.

Top 10 publications (by Google Scholar)

  • A proof of the Kepler conjecture
  • The honeycomb conjecture
  • Cannonballs and honeycombs
  • Sphere Packings, I
  • Historical overview of the Kepler conjecture
  • The sphere packing problem
  • Formal proof
  • A revision of the proof of the Kepler conjecture (with J. Harrison, S. McLaughlin et al.)
  • Sphere packings, II
  • Introduction to the Flyspeck project

Top 10 publications (by MathSciNet)

  1. A proof of the Kepler conjecture
  2. A simple definition of transfer factors for unramified groups
  3. Sphere Packings, I
  4. Cannonballs and honeycombs
  5. On the fundamental lemma for standard endoscopy: reduction to unit elements
  6. The honeycomb conjecture
  7. Sphere Packings, II
  8. The sphere packing problem
  9. What is motivic measure?
  10. The status of the Kepler conjecture
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