Wu-Yi Hsiang

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Wu-Yi Hsiang

Wu-Yi Hsiang (* 1937 in China ) is a Chinese-American mathematician who mainly deals with geometry.


Hsiang studied at the National University of Taiwan (Bachelor 1959) and went to the military service in the US, where he attended the 1964 Princeton University with John Coleman Moore with the dissertation On the Classification of differentiable actions of the Classical Groups on Pi manifolds Ph.D. has been. In 1968 he became a professor at the University of California, Berkeley . In 1997 he retired there and went to the Hong Kong University of Science and Technology. From 1965 to 1967 and 1968/69 he was at the Institute for Advanced Study . 1968 received a research grant from the Alfred P. Sloan Foundation ( Sloan Research Fellowship ).

In 1974 he gave the lecture Local and global characteristic class theory in topological transformation groups as invited speaker at the International Congress of Mathematicians in Vancouver .

Bruce Kleiner is one of his PhD students .

His brother Wu-Chung Hsiang is also a well-known mathematician.


Hsiang dealt with classical geometry, differential topology, differential geometry, transformation groups and in the 2000s with applications of differential geometry in celestial mechanics, for example in the three-body problem.

In 2001, Hsiang announced a new proof of the kissing number problem in three dimensions (that a maximum of twelve spheres of the same size touch a central sphere of the same size). He also claimed in 1992 to have solved the kiss number problem in four dimensions (with 24 balls).

Kepler assumption

Hsiang became known after he sent preprints around 1990 in which he claimed to have proven Kepler's conjecture with "elementary" geometric and analytical methods (vector algebra and spherical trigonometry) (and also another, related conjecture, the dodecahedron conjecture ). His proof of around 100 pages, which he published in the International Journal of Mathematics in 1993 , initially met with a positive response. Hsiang even gave a plenary lecture at the joint meeting of the American Mathematical Society and the Mathematical Association of America in January 1993 . On closer inspection, however, doubts arose about his evidence, which were furthered by the fact that the review process only lasted four months, unusual for such extensive work. Among other things, the respected specialist Gabor Fejes-Toth criticized him in Mathematical Reviews . Hsiang's evidence was examined in Budapest for over a year, or attempts were made to complete many of the details only hinted at and to correct errors. Such a procedure is not unusual in itself, for example the proof of the Fermat conjecture by Andrew Wiles required several attempts as part of a critical review. The mathematician Karoly Bezdek worked closely with Hsiang. Bezdek gave up in 1997 and even published a counterexample to one of Hsiang's central sentences. Finally, the majority of the mathematicians involved became convinced that Hsiang's proof was false. A corresponding open letter was published in the Mathematical Intelligencer in 1994 by John Horton Conway , Thomas Hales , Doug Muder, and Neil Sloane , followed by sharp criticism from Hales. Hales asked Hsiang in detail to clarify gaps in his evidence, whereupon Hsiang reacted irritably and published a sharp reply in the Mathematical Intelligencer , in which he was indignant about "fabricated" counterexamples. In addition, Hales, Barry Cipra , who wrote for Science , and Ian Stewart , who had recently found Hsiang's proof to be promising, publicly pointed out apparently false elementary geometrical claims in Hsiang's preprint, which were then eliminated in later versions. Hsiang himself remained convinced that his proof was fundamentally valid. In late 2001, he published a more detailed version of his evidence in book form.

The background to it all is that Hales himself was working on his own proof of the Kepler conjecture at the time Hsiang made his first announcements, for which he then published preparatory publications between 1992 and 2002. The proof was based essentially on the use of extensive computer calculations and was initially over 200 pages long even in overview form. It was later also controversial - an independent team of mathematicians who were appointed as speakers by the editor of the Annals of Mathematics Robert MacPherson had to admit after four years of intensive scrutiny in 2003 that they were only "99%" certain that the proof was correct , and that they lack the energy for further testing - an incidentally ungrateful task. The article was published in the Annals in 2005 . Originally, they wanted to make a note of the failed full review, which was then dropped. Hales then announced its own lengthy computerized review project.

Computer-aided evidence was not new at the time; it had been used, for example, with the four-color problem and the fig tree conjecture. The discussion about their assessment continues, however, as does the question of the acceptance of mathematical proofs raised by the discussion about Hsiang's and Hale's proofs.


  • Cohomology theory of topological transformation groups , Springer, Berlin / Heidelberg 1975 (results of mathematics and its border areas), ISBN 978-3-540-07100-6
  • Least Action Principle of Crystal Formation of Dense Packing Type and Kepler's Conjecture , World Scientific, Singapore / New Jersey 2001 (Nankai Tracts in Mathematics 3), ISBN 981-02-4670-6
  • Lectures on Lie groups , World Scientific, Singapore / New Jersey 2000 (Series on University Mathematics 2), ISBN 981-02-3522-4


Web links


  1. The original proof by Bartel Leendert van der Waerden and Schütte from 1953 appears to many mathematicians to be too complicated.
  2. ^ Proved by Oleg Musin 2003.
  3. Proven in 1998 by Thomas Hales and his then pre-graduate student Sean McLaughlin, who received the Frank and Bennie Morgan Prize for it in 1999 . McLaughlin was actually a music student (clarinet).
  4. ^ Hsiang: On the sphere packing problem and proof of the Kepler Conjecture , International Journal of Mathematics, Vol. 4, pp. 739-831.
  5. The journal was also published by colleagues from Berkeley. The Bulletin of the American Mathematical Society , to which he had first sent his manuscript, refused to print it because too many details were missing.
  6. Bezdek: Isoperimetric Inequalities and the Dodecaedral Conjecture , International Journal of Mathematics, Vol. 8, 1997, p. 759
  7. Muder found in 1988 and 1993 a series of the best upper bounds for the density of spherical packings in three dimensions
  8. Conway et al. a .: The Kepler Conjecture , Mathematical Intelligencer, Vol. 16, Spring 1994, p. 5
  9. Hales: The status of the Kepler Conjecture , Mathematical Intelligencer Vol. 16, 1994, No. 3, p. 47
  10. Hsiang: A Rejoinder of Hales's Article , Mathematical Intelligencer, Vol. 17, 1995, No. 1, pp. 35-42.
  11. ^ Cipra: "Gaps in a Sphere Packing Proof?", Science, Vol. 259, 1993, p. 895
  12. Stewart: Has the sphere packing problem been solved? , New Scientist, May 2, 1992
  13. For example in the article "Mathematics" in the Encyclopedia Britannica Yearbook 1992
  14. After Frank Morgan's review of Szpiro's book (Notices AMS 2005), the question of whether Hsiang's proof can still be completed is still open, since errors found have so far been "repaired". Other mathematicians, such as Bezdek and Conway, also expressed their opinion after Szpiro, Kepler's Conjecture , 2003, p. 155
  15. Hales: A proof of the Kepler Conjecture , Annals of Mathematics, Vol. 162, 2005, pp. 1063-1183; Hales, Samuel Ferguson: A formulation of the Kepler conjecture , Discrete and computational geometry, Vol. 36, 2006, pp. 21-69.
  16. On the struggle for its formulation see Steven Krantz: Mathematical Apocrypha Redux , Mathematical Association of America, 2005, p. 78f
  17. He had also added an improved and shorter manuscript after the examiners complained about the "work in progress" character of the documents. According to his own statements, he and his then student Ferguson lacked the energy and will to deal with the problem further.
  18. e.g. Bonnie Gold, Roger Simons (editor): Proofs and other dilemmas: Mathematics and Philosophy , Mathematical Association of America 2008