Heisuke Hironaka

Heisuke Hironaka (Japanese広中平佑, Hironaka Heisuke, born April 9, 1931 in Yuu, Kuga -gun (now Iwakuni ), Yamaguchi Prefecture, Japan) is a Japanese mathematician and winner of the Fields Medal.

Life and work

Hironaka was born as one of 15 children of a clothes dealer ( and temporary textile manufacturer ) in a 3000 - inhabitant village in Hiroshima. 1949-1954 he studied at the Kyoto University physics first, but then switched to mathematics, which focused there under Yasuo Akizuki in abstract algebra. In 1957 he was invited by Oscar Zariski, who was in Kyoto the year before, according to Harvard, where simultaneously studied other leading algebraic geometers later as David Mumford, Steven Kleiman and Michael Artin. In 1959 he was invited by Alexander Grothendieck, with whom he at Harvard 1958/9 befriended, at IHES in Paris. His acquaintance with Grothendieck gave him according to their own word significant stimulus - a "global " view - for his later proof of the solvability of singularities. After receiving his doctorate in 1960 by Zariski he went to Brandeis University, from 1964 to Columbia University in New York and from 1968 as a professor at Harvard. Already 1975-1988 he was also a professor in Kyoto, where he was also director of the 1983-1985 " Research Institute for Mathematical Sciences" (RIMS ). In Japan, he is so respected and influential that his name is a term with many non- mathematicians. 1996-2002 he was Director of Yamaguchi University in his home prefecture.

He worked in the field of algebraic geometry, as well as the other two Fields medalists from Japan Kunihiko Kodaira and Shigefumi Mori.

Hironaka proved in 1964 that one can resolve the singularities of an algebraic variety of any dimension in characteristic zero.

Before Hironaka Robert Walker already had on work done by Giacomo Albanese and others, dating back to the 19th century, in 1935 demonstrated the solvability of algebraic surfaces over the complex numbers, and even Zariski proved this in 1939 with purely algebraic methods for field of characteristic 0 (for surfaces and curves ). He also proved 1944, the solvability in characteristic 0 and dimension 3

For its proof, which is nearly 200 pages long and very difficult to understand, Hironaka 1970 received the Fields Medal (Presentation: Desingularization of complex analytic varieties ). The proof that even Hironaka not look as complicated, is now by Orlando Villamayor, Santiago Encinas, beer Edward Stone, Pierre Milman, Steven Dale Cutkosky, Herwig Hauser, Janos Kollar, inter alia, been simplified - it now fits on about 20 pages. An alternative proof of his method of alteration was de Jong 1997. ( Ie for varieties over finite fields ) Whether one can resolve the singularities in positive characteristic, is only in dimension 2, ie for algebraic surfaces, known ( proof of SSAbhyankar 1956), in general, but to date open.

In 1962 he was invited speaker at the International Congress of Mathematicians in Stockholm (On resolution of singularities ( characteristic zero) ).

Hironaka is married to the politician Wakako Hironaka and has two children.

It was in 1967 awarded the Asahi Prize and he holds honorary doctorates from the University of Nice.

Quotes

The world is due to their singularities interesting ... Smooth objects can be seen from a distance and recognize their shape in singularities have to come closer and closer .. Hawking said that in a black hole is another universe. A singularity is something like this: If you look at it closer, you notice a large universe. The problem in the treatment of singularities is that these are indeed only points, but contain very many things. To see what is in it, you have to blow them up, enlarge, make it smooth, then you can see the whole picture. ( Hironaka, Interview, Notices AMS 2005)

Works

Resolution of singularities of an algebraic variety over a field of characteristic zero, Part I, II, Annals of Math ( 2) Bd.79, 1964, p.109 - 203; Pp. 205-326.
With Hideyuki Matsumura Formal functions and formal imbeddings, 1967
On the characters and of singularities
Lectures on introduction to the theory of infinitely near singular points, Madrid, 1971
José M. Aroca, José Vicente L. The theory of the maximum contact, 1975
Jose M. Aroca, Jose L. Vicente Desingularization theorems, 1977
Edited by Stanislaw Janeczko Geometric singularity theory, Warsaw 2004 ( memory band Lojasiewicz )
Fame, Sweet and Bitter in Michael Atiyah et al Miscellanea Mathematica, Springer Verlag 1991

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