Thomas Hales

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Thomas Callister Hales (born June 4, 1958 in San Antonio , Texas , USA ) is an American mathematician . He is particularly concerned with problems in the field of algebra and geometry . In 1998, Hales became known beyond the limits of the mathematical community through his computer proof of the Keplerian conjecture .

Hales 2012

Life

Hales completed his studies in mathematics and engineering-economic systems at Stanford University with a Bachelor of Sciences or a Master of Sciences . This was followed by a one-year stay at Cambridge University , where he received the Certificate of Advanced Study in Mathematics (Part III of the Mathematical Tripos ). From 1983 he worked at Princeton University with Robert Langlands on his doctorate on The Subregular Germ of Orbital Integrals , which he completed in 1986. After receiving his doctorate, he worked at the Mathematical Sciences Research Institute (MSRI) in Berkeley. After positions as assistant professor or visiting scholar at Harvard University (1987–1989), at the School of Mathematics at the Institute for Advanced Study in Princeton , New Jersey , (1989–1990 and 1994–1995) and the University of Chicago with Paul J Sally (1990–1993) was first assistant professor in 1993 and later professor at the University of Michigan at Ann Arbor . In 2001 he became an Andrew Mellon Professor at the University of Pittsburgh .

In 2009 he received the Fulkerson Prize for proving the Kepler conjecture (like his former doctoral student Ferguson). In 2002 he was invited speaker at the International Congress of Mathematicians in Beijing (A computer verification of the Kepler conjecture). He has been a Fellow of the American Mathematical Society since 2012 . In 2019 Hales held the Tarski Lectures and in 2020 he was awarded the Senior Berwick Prize .

plant

Prior to his work on geometry, Thomas C. Hales examined topics in the Langlands program (automorphic forms and p-adic groups).

Proof of the Kepler Conjecture

Wu-Yi Hsiang tried to prove the Kepler conjecture from around 1990. His solution was heavily criticized by Hales and others. Around 1997 the mathematicians working on it rejected the attempted proof by overwhelming consensus as insufficient, whereas Hsiang remained convinced of the validity.

The evidence subsequently presented by Thomas C. Hales and his PhD student Samuel P. Ferguson comprised an extremely large amount of data. The experts around Gábor Fejes Tóth saw themselves in their own words "99 percent" convinced of the correctness of the evidence, but gave up after years of intensive work exhausted. The proof follows a linear programming approach suggested by László Fejes Tóth . The reviewers also objected in some cases to only sketchy evidence documents in preprint form with a volume of around 200 pages, without the computer printouts. Hales and Ferguson stated that after years of working on the evidence they were too exhausted to bring this to a polished form, which Hales made up for after the appraisers' verdict. The Annals of Mathematics published the proof in spite of the admission of the reviewers' failure in 2005. The editors of the Annals wrote that they will print the human part of computer-aided proofs of particularly important mathematical theorems in the future, even if the computer code (which the Annals on their Website) was not checked completely satisfactorily. The Annals of Mathematics paper was an overview. A more complete publication, which revised the preprints from 1998, took place in 2006 in a special issue of the journal Discrete & Computational Geometry , in which the editors Gabor Fejes Toth and Jeffrey Lagarias thank the peer reviewers and sometimes list them by name, which is unusual because peer Reviewers remain anonymous.

This gave impetus to discussions about the extent to which computer-reliant evidence is acceptable. Similar discussions have been held since the proof of the four-color theorem by Kenneth Appel and Wolfgang Haken in 1977.

Hales is therefore looking for ways to generally produce rigorous mathematical proofs in areas where computers are required to check too many intermediate steps. In the FlysPecK project , he formalized his proof of Kepler's conjecture so that automatic theorem provers such as B. John Harrison's HOL light can test it. Such tests are to be carried out in the future - for example in the SETI @ home project - via distributed computing on the web.

In August 2014, Hales announced that the transfer of the evidence into computerized form had ended and the software had confirmed the veracity of the evidence.

Proof of the honeycomb conjecture

Hales also proved several other famous conjectures in geometry. In 1999, Hales proved the honeycomb conjecture, which goes back to antiquity and assumes that when the plain is divided into areas of the same area, the total circumference of the edges corresponds at least to that of the regular hexagonal honeycomb division.

Proof of the dodecahedron conjecture

In 1998, together with his student Sean McLaughlin, he proved the dodecahedron conjecture by Laszlo Fejes-Toth, which, based on the kiss number 12 of spheres in three dimensions, assumes that the Voronoi polygon derived from the configuration has at least the volume of a regular dodecahedron (the corresponding the problem is scaled). At that time, McLaughlin was only a pre-graduate student (majoring in music, clarinet). In 1999 he received the Frank and Bennie Morgan Prize for outstanding work by mathematics students.

Hales himself wrote a review of the evidence and history of the Kepler conjecture and related conjectures in the Notices of the AMS. The essay won the 2003 Chauvenet Prize .

Disclosure of the back door in the NIST standard for random number generation

Hales also received attention for his analysis of the US standard for the generation of random numbers ( Dual EC DRBG ), in which he mathematically described the weakness of the method previously found by Dan Shumow and Nils Ferguson. The standardization authority NIST had to remove the Dual_EC_DRBG procedure from the SP 800-90A standard due to its publication.

Fonts

  • Jeffrey Lagarias (Editor), Hales, Ferguson: The Kepler conjecture. The Hales-Ferguson proof , Springer Verlag 2011
  • Thomas Hales: A proof of the Kepler Conjecture . In: Annals of Mathematics. Volume 162, 2005, pp. 1063–1183 (Section 5 is written with Ferguson, the article received the Robbins Prize of the AMS in 2007)
  • Thomas Hales, Samuel Ferguson (editors Gábor Fejes Tóth, Jeffrey Lagarias): special issue of Discrete & Computational Geometry, Volume 36, 2006, No. 1 to prove the Kepler conjecture. In this:
    • Hales: Historical Overview of the Kepler Conjecture , pp. 5-20, Hales, Ferguson A Formulation of the Kepler Conjecture , pp. 21-69, Hales Sphere Packing, III. Extremal Cases , pp. 71-110, Hales Sphere Packing, IV. Detailed Bounds , pp. 111-166, Hales Sphere Packings, VI. Tame Graphs and Linear Programs , pp. 205–265
  • Thomas C. Hales, John Harrison, Sean McLaughlin, Tobias Nipkow, Steven Obua, Roland Zumkeller: A Revision of the Proof of the Kepler Conjecture . Discrete & Computational Geometry, Volume 44, 2010, pp. 1-34

Broadcast contributions

Web links

Individual evidence

  1. ↑ Thomas Hales in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used
  2. ^ Hales: The status of the Kepler Conjecture. In: Mathematical Intelligencer. Volume 16, No. 3, 1994, p. 47
  3. ↑ 162-3 | Annals of Mathematics. Retrieved March 18, 2019 (American English).
  4. ^ A b Thomas C. Hales, Samuel P. Ferguson: A Formulation of the Kepler Conjecture . In: Discrete & Computational Geometry . tape 36 , no. 1 , July 2006, ISSN  0179-5376 , p. 21–69 , doi : 10.1007 / s00454-005-1211-1 ( springer.com [accessed March 18, 2019]).
  5. ^ Thomas C. Hales: An overview of the Kepler conjecture . In: arXiv: math / 9811071 . November 11, 1998 (arXiv = math / 9811071 [accessed March 18, 2019]).
  6. ↑ Eric W. Weisstein: Kepler Conjecture. Retrieved March 18, 2019 .
  7. ^ The literal description of the editors of the Annals, human part of the proof
  8. ↑ JSTOR. Retrieved March 18, 2019 .
  9. ↑ Fejes-Toth and Lagarias perform: Andras Bezdek, Michael Bleicher, Karoly Böröczky, Karoly Böröczky Junior, Aladar Heppes, Wlodek Kuperberg, Endre Makai, Attila Por, Günter Rote, Istvan Talata, Bela Uhrin, Zoltan Ujvary-Menyhard.
  10. ^ A. Bundy (editor) The nature of mathematical proof , Philosophical Transactions Royal Society A, Volume 363, October 2005, pp. 2331-2461
  11. ↑ For example William Thurston On proof and progress in mathematics , Bulletin AMS, Volume 30, 1994, No. 2
  12. ↑ derStandard.at: Proof of a 400 year old stacking problem confirmed . Article dated August 15, 2014, accessed August 16, 2014.
  13. ^ Hales: The Honeycomb Conjecture. In: Discrete and computational geometry. Volume 25, 2001, pp. 1-22; Honeycomb Conjecture at Mathworld
  14. ↑ Hales, McLaughlin: Proof of the Dodecaedral Conjecture , Preprint 1998; Dodecahedron conjecture at Mathworld .
  15. ↑ Hales: Cannonballs and Honeycombs. In: Notices of the AMS. Volume 47, April 2000, pp. 440-449 ( online , PDF file; 145 kB).
  16. ↑ Monika Ermert: US standard authority NIST and encryption: corrections, improvements, justifications
  17. ↑ Christoph Pöppe: friend reads along
  18. ^ Hales, TC: The NSA Back Door to NIST. In: Notices of the American Mathematical Society 61, pp. 190–192, 2014
  19. ↑ Notices AMS, 2007, No. 4, Robbins Prize, pdf
personal data
SURNAME Hales, Thomas
ALTERNATIVE NAMES Hales, Thomas Callister
BRIEF DESCRIPTION American mathematician and Mellon Professor of Mathematics at the University of Pittsburgh
DATE OF BIRTH June 4th 1958
PLACE OF BIRTH San Antonio , Texas