現代数学の系譜11 ガロア理論を読む27 [無断転載禁止]©2ch.net

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262: 現代数学の系譜11 ガロア理論を読む [sage] 2017/01/02(月) 13:15:32.06 ID:MUXssChK

>>242 関連

https://en.wikipedia.org/wiki/Landau_damping
(抜粋)
In physics, Landau damping, named after its discoverer,[1] the eminent Soviet physicist Lev Landau (1908?68), is the effect of damping (exponential decrease as a function of time) of longitudinal space charge waves in plasma or a similar environment.[2]
This phenomenon prevents an instability from developing, and creates a region of stability in the parameter space.

Landau damping can be manipulated exactly in numerical simulations such as particle-in-cell simulation.[5] It was proved to exist experimentally by Malmberg and Wharton in 1964,[6] almost two decades after its prediction by Landau in 1946.[7]

Mathematical theory: the Cauchy problem for perturbative solutions

The rigorous mathematical theory is based on solving the Cauchy problem for the evolution equation (here the partial differential Vlasov?Poisson equation) and proving estimates on the solution.

First a rather complete linearized mathematical theory has been developed since Landau.[14]

In a recent paper[17] the initial data issue is solved and Landau damping is mathematically established for the first time for the non-linear Vlasov equation.
It is proved that solutions starting in some neighborhood (for the analytic or Gevrey topology) of a linearly stable homogeneous stationary solution are (orbitally) stable for all times and are damped globally in time.
The damping phenomenon is reinterpreted in terms of transfer of regularity of f {\displaystyle f} f as a function of x {\displaystyle x} x and v {\displaystyle v} v, respectively, rather than exchanges of energy.

17 Mouhot, C., and Villani, C. "On Landau damping", Acta Math. 207, 1 (2011), 29?201 (quoted for the Fields Medal awarded to Cedric Villani in 2010)

http://rio2016.5ch.net/test/read.cgi/math/1483075581/262

267: 現代数学の系譜11 ガロア理論を読む [sage] 2017/01/02(月) 13:32:40.13 ID:MUXssChK

>>69 関連

2014年フィールズ賞マルティン・ハイラー　特に確率偏微分方程式ってあるね。応用面も強そうだ
https://ja.wikipedia.org/wiki/%E3%83%9E%E3%83%AB%E3%83%86%E3%82%A3%E3%83%B3%E3%83%BB%E3%83%8F%E3%82%A4%E3%83%A9%E3%83%BC
マルティン・ハイラー（Hairer Martin、1975年11月14日 - ）はオーストリア国籍の数学者。王立協会会員。英国のウォーリック大学、米国のニューヨーク大学を経て、2010年よりウォーリック大学教授。専門は確率解析、特に確率偏微分方程式。父はジュネーブ大学の数学者、アーネスト・ハイラー、配偶者は同じくウォーリック大学の数学者である、Xuemei Li。

https://en.wikipedia.org/wiki/Martin_Hairer
(抜粋)
Research

Hairer's nomination for the Royal Society reads:
“ Professor Martin Hairer is one of the world's foremost leaders in the field of stochastic partial differential equations in particular, and in stochastic analysis and stochastic dynamics in general.
By bringing new ideas to the subject he made fundamental advances in many important directions such as the study of variants of Hormander's theorem, systematisation of the construction of Lyapunov functions for stochastic systems,
development of a general theory of ergodicity for non-Markovian systems, multiscale analysis techniques, theory of homogenisation, theory of path sampling and, most recently, theory of rough paths and the newly introduced theory of regularity structures.[8] ”
Under the name HairerSoft, he develops Macintosh software.[9]

http://rio2016.5ch.net/test/read.cgi/math/1483075581/267

270: 現代数学の系譜11 ガロア理論を読む [sage] 2017/01/02(月) 13:43:43.67 ID:MUXssChK

>>69 関連

2014年フィールズ賞Artur Avila Cordeiro de Melo　物理数理系か
https://en.wikipedia.org/wiki/Artur_Avila
(抜粋)
Artur Avila Cordeiro de Melo (born 29 June 1979) is a Brazilian and French mathematician working primarily on dynamical systems and spectral theory. He is one of the winners of the 2014 Fields Medal,[2] being the first Latin American to win such award. He is a researcher at both the IMPA and the CNRS (working a half-year in each one).

At 19 he began his doctoral thesis on the theory of dynamical systems. Completed in 2001, when he traveled to France to do post-doctoral.[6] It works in the areas of dimensional dynamic and holomorfa.[7]
Since 2003 works at the Centre National de la Recherche Scientifique (CNRS) in France, and since 2008 is Directeur de recherche at this property. His post-doctoral supervisor was Jean-Christophe Yoccoz.[8]

Considered a prodigy as a teenager in 2005, at age 26, Arthur became known among mathematicians can prove by the "Conjecture of the ten martinis" problem proposed in 1980 by the American Barry Simon.
Simon promised to pay ten martini doses who explained his theory about the behavior of "Schrodinger operators", mathematical tools related to quantum physics. Artur solved the problem with mathematician Svetlana Jitomirskaya [9] and was presented with a few rounds of martini.

Prizes

In 2011, he was awarded the Michael Brin Prize in Dynamical Systems. He received the Early Career Award from the International Association of Mathematical Physics in 2012,[11] TWAS Prize in 2013[12] and the Fields Medal in 2014.[13]

http://rio2016.5ch.net/test/read.cgi/math/1483075581/270

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