フィールズ賞2006
ガウス賞が新設され，日本人が受賞したことで予想以上に報道された国際数学者会議ですが，アーベル賞に対抗するかのように名誉賞として創設されたガウス賞に，関心を持つ若い数学者は多くないです．
（受賞された伊藤先生自身については，業績と人柄から私も含めて世界中から尊敬を集めているので，ガウス賞を取ったから，というわけではなく，何の賞を取ったとしても，素直に喜んでいる数学者は多いとは思います）
なんと言っても，話題はPerelman博士です．
ポアンカレ予想を解決したと完全に認められてはいないにも関わらず，比較的新しい学問であるトポロジーと，古典的な微分幾何学を結びつけた衝撃は忘れられません．
当時，数理科学メーリングリストで，「食堂で見かけたけど，とてもヘンな人でした」と言っておられた方がおりましたが，40歳未満が対象のフィールズ賞で，40歳で辞退するとは．．．
（あれ？40歳なら，40歳未満なら対象外？その辺，調べてません．後日調べます）
さらっと日本のニュースサイトを見ても，ずいぶんありますね．
http://www.asahi.com/science/news/TKY200608220453.html
http://www.asahi.com/international/update/0822/020.html
http://www.asahi.com/international/update/0822/019.html
http://www.asahi.com/national/update/0822/TKY200608220453.html
http://www.asahi.com/international/update/0822/017.html
第１回「ガウス賞」に伊藤清氏
京大名誉教授 確率微分方程式を創始
氏のページによると表現論の人のようですが，下記の受賞講演アブストラクトには，グロモフ-ウィッテンの名前なんかがあって，純粋数学の中にも数理科学の香りが．
Enumerative geometry of curves in threefolds
Andrei Okounkov
Department of Mathematics, Princeton University, Princeton, New Jersey, 08544,
U.S.A.
okounkov@math.princeton.edu
2000 Mathematics Subject Classification. 14N35
Let X be an algebraic variety. Enumerative questions of the kind How many algebraic curves of given degree and genus does X contain ? have a long history in algebraic geometry. When this number is infinite, one may further constrain
the curves, e.g. to meet some number of general points of X. In recent years, this classical field has been enriched by new ideas and techniques coming from theoretical physics. The new structures thus discovered link enumerative geometry
to a variety of different areas of mathematics.
The case dimX = 3 is special. For threefolds one can develop two natural enumerative theories, know as the Gromov-Witten and the Donaldson-Thomas theories.
In oversimplified terms, the former views a curve as given by a parametrization while the latter views curves as cut out by their equations. While this leads to different enumerative answers, a precise correspondence between the two theories has been conjectured in [1].
In my talk I will review these conjectures, discuss some of their implications, and survey recent progress toward their proof.
1. Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R., Gromov-Witten theory
and Donaldson-Thomas theory, I. & II., math.AG/0312059, math.AG/0406092.
1
- パリ-Sud大学Werner Wendelin氏
フラクタルな図形がそそります．下記のアブストラクトによると，やはり数理科学的な．
Random planar loops and conformal restriction
Wendelin Werner
Laboratoire de Math´ematiques, Universit´e Paris-Sud, Bˆat. 425, 91405 Orsay cedex,
France
2000 Mathematics Subject Classification. 60D05, 82B41, 82B43, 30C99, 60J65
The goal of this lecture will be to describe some of the basic ideas that have improved
our mathematical understanding of two-dimensional conformally invariant random structures. Motivations for looking at such objects come from physics, as it is believed (and in some cases proved) that planar models from statistical
mechanics behave on large scale in a conformally invariant way at their critical point.
We will in particular focus on the “conformal restriction property”. We shall see how it enables to relate the geometry of two-dimensional Brownian paths to that of the scaling limit of two-dimensional percolation clusters, and to derive some of their properties. Much of the material of this lecture will be based on joint work
with Greg Lawler and Oded Schramm. We will also mention recent and ongoing work with Scott Sheffield on random systems of interacting loops.
This lecture will be related and complementary to (but independent of) the plenary lecture of Oded Schramm on the same day. It will be of very introductory nature and requires no prerequisite.
- カリフォルニア大学Terence Tao氏
この方も幾何学全般，代数幾何やら幾何的数論やら多岐に渡っています．
Long arithmetic progressions in the primes
Terence Tao
Department of Mathematics, UCLA, Los Angeles, CA 90095, U.S.A.
2000 Mathematics Subject Classification. Primary 11P32, 37A45, 05C65, 05C75,
42A99.
Key words. Szemer´edi’s theorem, ergodic theory, graph theory, hypergraph theory,
arithmetic combinatorics, arithmetic progressions, prime numbers.
A famous theorem of Szemer´edi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly
speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemer
´edi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power
of this dichotomy is evidenced by the fact that the Green-Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemer´edi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics,
harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different.
しかも，ネヴァンリンナ賞は，ほとんどサイエンス社の数理科学誌が予言したみたいな感じになっちゃってます．
こうして数理科学者は，「フィールズ賞も決まったし，ワインでも飲みますか」とか，「ノーベル物理学賞も決まったし，呑みますか」とやってるわけです．
