Pólya enumeration theorem

2013-09-19T15:54:56Z

85.127.75.157: removed a comma from the end of the formula, which effected a parse error

[[File:Mitsuhiro Shishikura.jpg|thumb|220px|{{PAGENAME}}]]
{{nihongo|'''Mitsuhiro Shishikura'''|宍倉光広|Shishikura Mitsuhiro|born November 27, 1960}} is a [[Japan]]ese [[mathematician]] working in the field of [[complex dynamics]]. He is professor at [[Kyoto University]] in Japan.

Shishikura became internationally recognized<ref>This recognition is evidenced e.g. by the prizes he received (see below) as well as his invitation as an invited speaker in the Real & Complex Analysis Section of the 1994 [[International Congress of Mathematicians]]; see http://www.mathunion.org/o/ICM/Speakers/SortedByCongress.php.</ref> for two of his earliest contributions, both of which solved long-standing [[open problems]].
* In his Master's thesis, he proved a conjectured of [[Pierre Fatou|Fatou]] from 1920<ref>P. Fatou, ''Sur les équations fonctionelles'', Bull. Soc. Math. Fr., 1920</ref> by showing that a [[rational function]] of degree <math>d\,</math> has at most <math>2d-2\,</math> nonrepelling [[periodic cycle]]s.<ref>M. Shishikura, ''On the quasiconformal surgery of rational functions,'' Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 1–29.</ref>
* He proved<ref>M. Shishikura, ''The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets'', Ann. of Math. (2) 147 (1998), no. 2, 225–267 ([http://arxiv.org/pdf/math/9201282v1.pdf preprint])</ref> that the boundary of the [[Mandelbrot set]] has [[Hausdorff dimension]] two, confirming a conjecture stated by [[Benoit Mandelbrot|Mandelbrot]]<ref>B. Mandelbrot, ''On the dynamics of iterated maps V: Conjecture that the boundary of the M-set has a fractal dimension equal to 2'', in: Chaos, Fractals and Dynamics, Eds. Fischer and Smith, Marcel Dekker, 1985, 235-238</ref> and [[John Milnor|Milnor]].<ref>J. Milnor, ''Self-similarity and hairiness in the Mandelbrot set'', in: Computers in Geometry and Topology, ed. M. C. Tangora, Lect. Notes in Pure and Appl. Math., Marcel
Dekker, Vol. 114 (1989), 211-257</ref>

For his results, he was awarded the [[Salem Prize]] in 1992, and the Iyanaga Spring Prize of the [[Mathematical Society of Japan]] in 1995.

More recent results of Shishikura include
* ''(in joint work with Kisaka<ref>M. Kisaka and M. Shishikura, ''On multiply connected wandering domains of entire functions'', in: Transcendental dynamics and complex analysis, London Math. Soc. Lecture Note Ser., 348, Cambridge Univ. Press, Cambridge, 2008, 217–250</ref>)'' the existence of a [[entire function|transcendental entire function]] with a [[doubly connected]] [[wandering domain]], answering a question of Baker from 1985;<ref>I. N. Baker, ''Some entire functions with multiply-connected wandering domains'', Ergodic Theory Dynam. Systems 5 (1985), 163-169</ref>
* ''(in joint work with Inou<ref>H. Inou and M. Shishikura, ''The renormalization of parabolic fixed points and their perturbation'', Preprint, 2008, http://www.math.kyoto-u.ac.jp/~mitsu/pararenorm/</ref>)'' a study of ''near-parabolic renormalization'' which is essential in Buff and Chéritat's recent proof of the existence of polynomial [[Julia set]]s of positive planar [[Lebesgue measure]].

One of the main tools pioneered by Shishikura and used throughout his work is that of [[quasiconformal mapping|quasiconformal]] surgery.

==References==
{{Reflist}}

==External links==
*[http://www.math.kyoto-u.ac.jp/~mitsu/ Faculty home page] at Kyōto University

{{Persondata 
|name= Shishikura, Mitsuhiro
|alternative names=
|short description= Japanese mathematician
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{{DEFAULTSORT:Shishikura, Mitsuhiro}}
[[Category:1960 births]]
[[Category:Living people]]
[[Category:Japanese mathematicians]]
[[Category:Tokyo Institute of Technology faculty]]
[[Category:University of Tokyo faculty]]
[[Category:Kyoto University faculty]]
[[Category:Kyoto University alumni]]
[[Category:20th-century mathematicians]]
[[Category:21st-century mathematicians]]

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Pólya enumeration theorem