en.wikipedia.org/wiki/Emmy_Noether Ascending and descending chain conditions In this epoch, Noether became famous for her deft use of ascending (Teilerkettensatz) or descending (Vielfachenkettensatz) chain conditions. A sequence of non-empty subsets A1, A2, A3, etc. of a set S is usually said to be ascending, if each is a subset of the next A1⊂A2⊂A3⊂・・・ Conversely, a sequence of subsets of S is called descending if each contains the next subset: A1⊃A2⊃A3⊃・・・ A chain becomes constant after a finite number of steps if there is an n such that for all m ? n. A collection of subsets of a given set satisfies the ascending chain condition if any ascending sequence becomes constant after a finite number of steps. It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.
Ascending and descending chain conditions are general, meaning that they can be applied to many types of mathematical objects?and, on the surface, they might not seem very powerful. Noether showed how to exploit such conditions, however, to maximum advantage: for example, how to use them to show that every set of sub-objects has a maximal/minimal element or that a complex object can be generated by a smaller number of elements. These conclusions often are crucial steps in a proof. （つづく）
>>253 en.wikipedia.org/wiki/Emmy_Noether つづき Many types of objects in abstract algebra can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called Noetherian in her honor. By definition, a Noetherian ring satisfies an ascending chain condition on its left and right ideals, whereas a Noetherian group is defined as a group in which every strictly ascending chain of subgroups is finite. A Noetherian module is a module in which every strictly ascending chain of submodules breaks off after a finite number. A Noetherian space is a topological space in which every strictly increasing chain of open subspaces breaks off after a finite number of terms; this definition is made so that the spectrum of a Noetherian ring is a Noetherian topological space.
The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space, are Noetherian themselves; all subgroups and quotient groups of a Noetherian group are likewise, Noetherian; and, mutatis mutandis, the same holds for submodules and quotient modules of a Noetherian module. All quotient rings of a Noetherian ring are Noetherian, but that does not necessarily hold for its subrings. The chain condition also may be inherited by combinations or extensions of a Noetherian object. For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring. つづく
>>254 en.wikipedia.org/wiki/Emmy_Noether つづき Another application of such chain conditions is in Noetherian induction?also known as well-founded induction?which is a generalization of mathematical induction. It frequently is used to reduce general statements about collections of objects to statements about specific objects in that collection. Suppose that S is a partially ordered set. One way of proving a statement about the objects of S is to assume the existence of a counterexample and deduce a contradiction, thereby proving the contrapositive of the original statement. The basic premise of Noetherian induction is that the every non-empty subset of S contains a minimal element. In particular, the set of all counterexamples contains a minimal element, the minimal counterexample. In order to prove the original statement, therefore, it suffices to prove something seemingly much weaker: For any counterexample, there is a smaller counterexample. (引用おわり)
英語版（Henri Poincare 1895が入っている） en.wikipedia.org/wiki/Homology_(mathematics) History Homology classes were first defined rigorously by Henri Poincare in his seminal paper "Analysis situs", J. Ecole polytech. (2) 1. 1?121 (1895). The homology group was further developed by Emmy Noether[1][2] and, independently, by Leopold Vietoris and Walther Mayer, in the period 1925?28.[3] Prior to this, topological classes in combinatorial topology were not formally considered as abelian groups. The spread of homology groups marked the change of terminology and viewpoint from "combinatorial topology" to "algebraic topology".[4]
ja.wikipedia.org/wiki/%E7%82%B9%E7%BE%A4 数学における点群（てんぐん、英: point group）とはある図形の形を保ったまま行う移動操作のうち、少なくとも1つの不動点を持つものを元とする群のこと。 このような群によって物理学や化学における分子や結晶の対称性を数学的に記述することができる。そのような応用との関係からふつう3次元ユークリッド空間における変換の範疇で考えることが多い。
en.wikipedia.org/wiki/Auguste_Bravais Auguste Bravais (French pronunciation: [o?yst b?av?]; 23 August 1811, Annonay, Ardeche ? 30 March 1863, Le Chesnay, France) was a French physicist, well known for his work in crystallography (the Bravais lattices, and the Bravais laws).
He is best remembered for pointing out in 1845 that there are 14 unique Bravais lattices in three dimensional crystalline systems, adjusting the previously existent result (15 lattices) by Frankenheim, obtained three years before. Bravais published a memoire about crystallography in 1847.
Bravias worked continuously for 4 years for crystallography and declared his result at just an age of 26. But he was not at all appreciated for his work at that time and all the great scientists at that time said that his work is of no use. His sentiments were hurt and he started hating his life. At last he died at an age of 51. Later on, in the middle of 20th century, Germany and England raced for the discovery of crystallography and Germany gave heavy sum to 15 scientists to research for it. After 15 years, they came up with the result and at the press conference, when the presented, one of the reporters reported that the work which 15 scientists had done in 15 years, has already been done by a young man, in only 4 years, that too in 1850s, when science was not developed that much. When the records were checked, it was found that the reporter was saying true. [1]
1901年 ヴィルヘルム・レントゲン Wilhelm Conrad Rontgen 1914年 マックス・フォン・ラウエ Max von Laue 結晶によるX線回折現象の発見 1915年 ヘンリー・ブラッグ Sir William Henry Bragg ローレンス・ブラッグ William Lawrence Bragg X線による結晶構造解析に関する研究[11]
ガロアの原論文（「現代数学の系譜11、アーベル、ガロア、群と代数方程式、守屋美賀雄訳」）を読むための３つのポイントは １．ガロア分解式（リゾルベント） V=Aa+Bb+Cc+・・・ a,b,c・・・は、（重根を持たない）で問題の方程式の根、係数A,B,C・・・は根の置換で異なる値をとるように定める ２．置換群のガロア記法 a b c d・・・・k b c d・・・・k a c d・・・・k a b ・・・・・・・・・・・ k a b・・・・・i
注）今日、置換は普通はコーシーの記法 （a b c d・・・・k） （a b c d・・・・k） （直上の２行は大きな括弧で括られていると思ってください）
Wittenさんこれですね en.wikipedia.org/wiki/Edward_Witten Witten is a researcher in superstring theory, a theory of quantum gravity, supersymmetric quantum field theories and other areas of mathematical physics.[1] He has made contributions in mathematics and helped bridge gaps between fundamental physics and other areas of mathematics. In 1990 he became the first physicist to be awarded a Fields Medal by the International Union of Mathematics. In 2004, Time magazine stated that Witten was widely thought to be the world's greatest living theoretical physicist.
Birth and education Witten was born in Baltimore, Maryland. He is the son of Lorraine W. Witten and Louis Witten, a theoretical physicist specializing in gravitation and general relativity. Witten attended the Park School of Baltimore (class of '68), and received his Bachelor of Arts with a major in history and minor in linguistics from Brandeis University in 1971. He published articles in The New Republic and The Nation. In 1968 Witten published an article in The Nation arguing that the New Left had no strategy. He worked briefly for George McGovern, a Democratic presidential nominee in 1972. McGovern lost the election in a landslide to Richard Nixon.
Witten attended the University of Wisconsin?Madison for one semester as an economics graduate student before dropping out.[citation needed] He returned to academia, enrolling in applied mathematics at Princeton University[citation needed] then shifting departments and receiving a Ph.D. in physics in 1976 under David Gross, the 2004 Nobel laureate in Physics. He held a fellowship at Harvard University (1976?77), was a junior fellow in the Harvard Society of Fellows (1977?80), and held a MacArthur Foundation fellowship (1982).
en.wikipedia.org/wiki/Edward_Witten Awards and honors Witten has been honored with numerous awards including a MacArthur Grant (1982), the Fields Medal (1990), the Nemmers Prize in Mathematics (2000), the National Medal of Science[6] (2002), Pythagoras Award[7] (2005), the Henri Poincare Prize (2006), the Crafoord Prize (2008), the Lorentz Medal (2010) the Isaac Newton Medal (2010) and the Fundamental Physics Prize (2012).
Pope Benedict XVI appointed Witten as a member of the Pontifical Academy of Sciences (2006). He also appeared in the list of TIME magazine's 100 most influential people of 2004.
In July 2012, he was an inaugural awardee of the Fundamental Physics Prize, the creation of physicist and internet entrepreneur, Yuri Milner.[8]
>In July 2012, he was an inaugural awardee of the Fundamental Physics Prize, the creation of physicist and internet entrepreneur, Yuri Milner.[8]
ノーベル賞の３倍の賞金、$3 millionだそうです
en.wikipedia.org/wiki/Fundamental_Physics_Prize The Fundamental Physics Prize is awarded by the The Fundamental Physics Prize Foundation which is a not-for-profit organization dedicated to awarding physicists involved in fundamental research which was founded in July 2012 by physicist and internet entrepreneur Yuri Milner.[1] This prize is the most lucrative academic prize in the world as of July 2012 and it is three times bigger than the amount given to the Nobel awardees.[2][3] This prize is also dubbed by the media as 'Russian Nobel'.[4] This is world's biggest prize in physics as of July 2012.[5]
Award money As of July 2012, each award is worth $3 million, the monetary value exceeds that of the prestigious Nobel Prize, which last year stood at slightly more than $1 million.[2][4]
>>295 つづき 2012年が最初で、受賞者下記9名。 'Russian Nobel'ということであまり報道されていませんね。”"9 Scientists Receive a New Physics Prize". The New York Times. 31 July 2012. Retrieved 2012-08-04.”最近の情報ですね en.wikipedia.org/wiki/Fundamental_Physics_Prize Nima Arkani-Hamed Original approaches to outstanding problems in particle physics
Alan Guth Invention of inflationary cosmology, and for contributions to the theory for the generation of cosmological density fluctuations arising from quantum fluctuations
Alexei Kitaev For robust quantum memories and fault-tolerant quantum computation using topological quantum phases with anyons and unpaired Majorana modes.
Maxim Kontsevich Numerous contributions including development of homological mirror symmetry, and the study of wall-crossing phenomena.
Andrei Linde For development of inflationary cosmology, including the theory of new inflation, eternal chaotic inflation and the theory of inflationary multiverse, and for contributing to the development of vacuum stabilization mechanisms in string theory.
Juan Maldacena Contributions to gauge/gravity duality, relating gravitational physics in a spacetime and quantum field theory on the boundary of the spacetime
Nathan Seiberg Contributions to our understanding of quantum field theory and string theory. Ashoke Sen Opening the path to the realization that all string theories are different limits of the same underlying theory.
Edward Witten For applications of topology to physics, non-perturbative duality symmetries, models of particle physics derived from string theory, dark matter detection, and the twistor-string approach to particle scattering amplitudes, as well as numerous applications of quantum field theory to mathematics.
Ｃｏｌｅｍａｎさんは１９６６年から１９７９年の間に８回エリチェに行かれて講義をされました。 どれも名講義ですが、特に、"Classical lumps and their quantum descendants," "The uses of instantons," "1/N" は、今読んでも勉強になります。
これらの講義録は１９８５年に"Aspects of Symmetry"と名付けられた本にまとめられています。
