en.wikipedia.org/wiki/Edward_Witten Edward Witten (born August 26, 1951) is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study. 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 various areas of mathematics. In 1990 he was the world's first physicist to be awarded a Fields Medal by the International Union of Mathematics. In 2004, Time magazine wrote that Witten was "generally considered the greatest theoretical physicist in the world."[2] （引用おわり）
”東京大学 博士( 理学系研究科 数学) 1990” ＆ ”In 1990 he was the world's first physicist to be awarded a Fields Medal by the International Union of Mathematics. ” なので、1990より前ですな 橋本 義武さん、がんばって！
It is easy to find a homeomorphism that is not a diffeomorphism, but it is more difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2, 3, any pair of homeomorphic smooth manifolds are diffeomorphic. In dimension 4 or greater, examples of homeomorphic but not diffeomorphic pairs have been found. The first such example was constructed by John Milnor in dimension 7. He constructed a smooth 7-dimensional manifold (called now Milnor's sphere) that is homeomorphic to the standard 7-sphere but not diffeomorphic to it. There are in fact 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is a total space of the fiber bundle over the 4-sphere with the 3-sphere as the fiber).
Much more extreme phenomena occur for 4-manifolds: in the early 1980s, a combination of results due to Simon Donaldson and Michael Freedman led to the discovery of exotic R4s: there are uncountably many pairwise non-diffeomorphic open subsets of R4 each of which is homeomorphic to R4, and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to R4 that do not embed smoothly in R4.
>>62 なるほど・・ en.wikipedia.org/wiki/Exotic_R4 In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic to the Euclidean space R4, but not diffeomorphic. The first examples were found by Robion Kirby and Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds. There is a continuum of non-diffeomorphic differentiable structures of R4, as was shown first by Clifford Taubes.
Prior to this construction, non-diffeomorphic smooth structures on spheres ? exotic spheres ? were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open. For any positive integer n other than 4, there are no exotic smooth structures on Rn; in other words, if n ≠ 4 then any smooth manifold homeomorphic to Rn is diffeomorphic to Rn.
Small exotic R4s An exotic R4 is called small if it can be smoothly embedded as an open subset of the standard R4. Small exotic R4s can be constructed by starting with a non-trivial smooth 5-dimensional h-cobordism (which exists by Donaldson's proof that the h-cobordism theorem fails in this dimension) and using Freedman's theorem that the topological h-cobordism theorem holds in this dimension.
Large exotic R4s An exotic R4 is called large if it cannot be smoothly embedded as an open subset of the standard R4. Examples of large exotic R4s can be constructed using the fact that compact 4 manifolds can often be split as a topological sum (by Freedman's work), but cannot be split as a smooth sum (by Donaldson's work). Michael Hartley Freedman and Laurence R. Taylor (1986) showed that there is a unique maximal exotic R4, into which all other R4s can be smoothly embedded as open subsets. （つづく）
en.wikipedia.org/wiki/Exotic_R4 Related exotic structures Casson handles are homeomorphic to D2×R2 by Freedman's theorem (where D2 is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to D2×R2. In other words, some Casson handles are exotic D2×R2s. It is not known (as of 2009) whether or not there are any exotic 4-spheres; such an exotic 4-sphere would be a counterexample to the smooth generalized Poincare conjecture in dimension 4. Some plausible candidates are given by Gluck twists.
en.wikipedia.org/wiki/Gluck_twist#4-dimensional_exotic_spheres_and_Gluck_twists In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic").
The first exotic spheres were constructed by John Milnor (1956) in dimension n = 7 as S3-bundles over S4. He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension Milnor (1959) showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finite abelian group if the dimension is not 4. The classification of exotic spheres by Michel Kervaire and John Milnor (1963) showed that the oriented exotic 7-spheres are the non-trivial elements of a cyclic group of order 28 under the operation of connected sum.
>>65 つづき ここらも面白いね en.wikipedia.org/wiki/Gluck_twist#4-dimensional_exotic_spheres_and_Gluck_twists Explicit examples of exotic spheres One of the first examples of an exotic sphere found by Milnor (1956, section 3) was the following: Take two copies of B4×S3, each with boundary S3×S3, and glue them together by identifying (a,b) in the boundary with (a, a2ba?1), (where we identify each S3 with the group of unit quaternions). The resulting manifold has a natural smooth structure and is homeomorphic to S7, but is not diffeomorphic to S7. Milnor showed that it is not the boundary of any smooth 8-manifold with vanishing 4th Betti number, and has no orientation-reversing diffeomorphism to itself; either of these properties implies that it is not a standard 7-sphere. Milnor showed that this manifold has a Morse function with just two critical points, both non-degenerate, which implies that it is topologically a sphere.
As shown by Egbert Brieskorn (1966, 1966b) (see also (Hirzebruch & Mayer 1968)) the intersection of the complex manifold of points in C5 satisfying （式省略） with a small sphere around the origin for k = 1, 2, ..., 28 gives all 28 possible smooth structures on the oriented 7-sphere. Similar manifolds are called Brieskorn spheres.
Twisted spheres Given an (orientation-preserving) diffeomorphism f: Sn?1→Sn?1, gluing the boundaries of two copies of the standard disk Dn together by yields a manifold called a twisted sphere (with twist f). （面白いが省略）
ああ、そうそう、これは落とせないね。直接関係するから ”The statement that they do not exist is known as the "smooth Poincare conjecture", and is discussed by Michael Freedman, Robert Gompf, and Scott Morrison et al. (2010) who say that it is believed to be false.”だと en.wikipedia.org/wiki/Gluck_twist#4-dimensional_exotic_spheres_and_Gluck_twists 4-dimensional exotic spheres and Gluck twists In 4 dimensions it is not known whether there are any exotic smooth structures on the 4-sphere. The statement that they do not exist is known as the "smooth Poincare conjecture", and is discussed by Michael Freedman, Robert Gompf, and Scott Morrison et al. (2010) who say that it is believed to be false.
Some candidates for exotic 4-spheres are given by Gluck twists (Gluck 1962). These are constructed by cutting out a tubular neighborhood of a 2-sphere S in S4 and gluing it back in using a diffeomorphism of its boundary S2×S1. The result is always homeomorphic to S4. But in most cases it is unknown whether or not the result is diffeomorphic to S4. (If the 2-sphere is unknotted, or given by spinning a knot in the 3-sphere, then the Gluck twist is known to be diffeomorphic to S4, but there are plenty of other ways to knot a 2-sphere in S4.)
Akbulut (2009) showed that a certain family of candidates for 4-dimensional exotic spheres constructed by Cappell and Shaneson are in fact standard.
なるほど下記ですな en.wikipedia.org/wiki/Connes 抜粋 Work Alain Connes is one of the leading specialists on operator algebras. In his early work on von Neumann algebras in the 1970s, he succeeded in obtaining the almost complete classification of injective factors. Following this he made contributions in operator K-theory and index theory, which culminated in the Baum-Connes conjecture. He also introduced cyclic cohomology in the early 1980s as a first step in the study of noncommutative differential geometry. Connes has applied his work in areas of mathematics and theoretical physics, including number theory, differential geometry and particle physics.[1]
Awards and honours Connes was awarded the Fields Medal in 1982, the Crafoord Prize in 2001 and the gold medal of the CNRS in 2004.
See also Cyclic homology C*-algebra M Theory Groupoid External links 1.^ Scientific Americain, The Geometry of Particle Physics, July 24, 2006 Alain Connes Official Web Site containing downloadable papers, and his book Non-commutative geometry, ISBN 0-12-185860-X. www.alainconnes.org/ nlab about Alain Connes Alain Connes' Standard Model
In "Brisure de symetrie spontanee et geometrie du point de vue spectral", Journal of Geometry and Physics 23 ('97), 206?234, Alain Connes wrote: "The answer given by non-standard analysis, namely a nonstandard real, is equally disappointing: every non-standard real canonically determines a (Lebesgue) non-measurable subset of the interval [0, 1], so that it is impossible (Stern, 1985) to exhibit a single [nonstandard real number]. The formalism that we propose will give a substantial and computable answer to this question." In his '95 article "Noncommutative geometry and reality" Connes develops a calculus of infinitesimals based on operators in Hilbert space. He proceeds to "explain why the formalism of nonstandard analysis is inadequate" for his purposes. Connes points out the following three aspects of Robinson's hyperreals:
(1) a nonstandard hyperreal "cannot be exhibited" (the reason given being its relation to non-measurable sets); (2) "the practical use of such a notion is limited to computations in which the final result is independent of the exact value of the above infinitesimal. This is the way nonstandard analysis and ultraproducts are used [...]". (3) the hyperreals are commutative.
In the view of M. Katz and K. Katz Connes' comments are critical of non-standard analysis, and they challenge these specific claims.[6] With regard to (1), Connes' own infinitesimals similarly rely on non-constructive foundational material, such as the existence of a Dixmier trace. With regard to (2), Connes presents the independence of the choice of infinitesimal as a feature of his own theory.
ソリトンも落とせないかな フェルミ・パスタ・ウラムの問題→ソリトン→可積分系 ja.wikipedia.org/wiki/%E3%83%95%E3%82%A7%E3%83%AB%E3%83%9F%E3%83%BB%E3%83%91%E3%82%B9%E3%82%BF%E3%83%BB%E3%82%A6%E3%83%A9%E3%83%A0%E3%81%AE%E5%95%8F%E9%A1%8C 950年代にロスアラモス研究所で電子計算機を用いて、この問題に取り組んだ3人の物理学者エンリコ・フェルミ、ジョン・パスタ、スタニスワフ・ウラムに名に因む。 当初の予想では相互作用が非線形な系ではエルゴード性によって、長時間経過後に各モードにエネルギーが等分配された平衡状態に達するはずであったが、 計算機実験の結果はそれに反し、初期状態のモードに戻る再帰現象が観測された。後に、この再帰現象はKdV方程式の研究から可積分系におけるソリトンと関連した現象であることが明らかにされた。 en.wikipedia.org/wiki/Soliton In 1965 Norman Zabusky of Bell Labs and Martin Kruskal of Princeton University first demonstrated soliton behaviour in media subject to the Korteweg?de Vries equation (KdV equation) in a computational investigation using a finite difference approach. They also showed how this behavior explained the puzzling earlier work of Fermi, Pasta and Ulam.[3] （つづく）
en.wikipedia.org/wiki/Simon_Donaldson Biography Donaldson's father was an electrical engineer in the physiology department at the University of Cambridge[citation needed]. Donaldson gained a BA degree in mathematics from Pembroke College, Cambridge in 1979, and in 1980 began postgraduate work at Worcester College, Oxford, at first under Nigel Hitchin and later under Michael Atiyah's supervision. Still a graduate student, Donaldson proved in 1982 a result that would establish his fame. He published the result in a paper Self-dual connections and the topology of smooth 4-manifolds which appeared in 1983. In the words of Atiyah, the paper "stunned the mathematical world" (Atiyah 1986).
Whereas Michael Freedman classified topological four-manifolds, Donaldson's work focused on four-manifolds admitting a differentiable structure, using instantons, a particular solution to the equations of Yang-Mills gauge theory which has its origin in quantum field theory. One of Donaldson's first results gave severe restrictions on the intersection form of a smooth four-manifold. As a consequence, a large class of the topological four-manifolds do not admit any smooth structure at all. Donaldson also derived polynomial invariants from gauge theory. These were new topological invariants sensitive to the underlying smooth structure of the four-manifold. They made it possible to deduce the existence of "exotic" smooth structures?certain topological four-manifolds could carry an infinite family of different smooth structures. （つづく）
en.wikipedia.org/wiki/Simon_Donaldson Donaldson's work（抜粋） A thread running through Donaldson's work is the application of mathematical analysis (especially the analysis of elliptic partial differential equations) to problems in geometry. The problems mainly concern 4-manifolds, complex differential geometry and symplectic geometry. The following theorems rank among his most striking achievements: The diagonalizability theorem (Donaldson 1983a, 1983b): if the intersection form of a smooth, closed, simply connected 4-manifold is positive- or negative-definite then it is diagonalizable over the integers. (The simple connectivity hypothesis has since been shown to be unnecessary using Seiberg-Witten theory.) This result is sometimes called Donaldson's theorem. A smooth h-cobordism between 4-manifolds need not be trivial (Donaldson 1987a). This contrasts with the situation in higher dimensions. A stable holomorphic vector bundle over a non-singular projective algebraic variety admits a Hermitian-Einstein metric (Donaldson 1987b). This was proved independently by Karen Uhlenbeck and Shing-Tung Yau (Uhlenbeck & Yau 1986).
Donaldson's recent work centers on a difficult problem in complex differential geometry concerning a conjectural relationship between algebro-geometric "stability" conditions for smooth projective varieties and the existence of "optimal" Kahler metrics, typically those with constant scalar curvature. Definitive results have not yet been obtained, but substantial progress has been made (see for example Donaldson 2001).
See also Donaldson theory.
External links O'Connor, John J.; Robertson, Edmund F., "Simon Donaldson", MacTutor History of Mathematics archive, University of St Andrews. Simon Donaldson at the Mathematics Genealogy Project. Home page at Imperial College
概略 群の概念は、数学的対象 X から X への自己同型の集まりの満たす性質を代数的に抽象化することによって得られる。 この集まりは X の対称性を表現していると考えられ、結合法則・恒等変換の存在・逆変換の存在などがなりたっている。 集合論にもとづき X が集合として実現されている場合には、自己同型として X からそれ自身への全単射写像を考えることになるが、空間や対象の持つ構造に応じてさらに付加条件を課すことが多い。 例えば、ベクトル空間 X に対してその自己同型写像の集まりを考えると群が得られる。 また、平面上に正三角形など何らかの対称性を持った図形が与えられているとき、平面全体の変換のうちでその図形を保つようなものだけを考えることによって、図形の対称性を表す群を取り出すことができる。 （つづく）
>>87 英語版 en.wikipedia.org/wiki/Group_(mathematics) In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group. However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.[1][2]
Groups share a fundamental kinship with the notion of symmetry. A symmetry group encodes symmetry features of a geometrical object: it consists of the set of transformations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in many academic disciplines. Matrix groups, for example, can be used to understand fundamental physical laws underlying special relativity and symmetry phenomena in molecular chemistry.
The concept of a group arose from the study of polynomial equations, starting with Evariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory?a very active mathematical discipline?studies groups in their own right.a[?] （略）
>>88 つづき なお”Main article: History of group theory en.wikipedia.org/wiki/History_of_group_theory”がまた面白いんだ en.wikipedia.org/wiki/Group_ History The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Evariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 (1854) gives the first abstract definition of a finite group.
Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.
The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker. In 1847, Ernst Kummer led early attempts to prove Fermat's Last Theorem to a climax by developing groups describing factorization into prime numbers.
The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traite des substitutions et des equations algebriques (1870).
en.wikipedia.org/wiki/Galois_theory In mathematics, more specifically in abstract algebra, Galois theory, named after Evariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood.
Originally Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions.
Application to classical problems Galois theory not only provides a beautiful answer to this question, it also explains in detail why it is possible to solve equations of degree four or lower in the above manner, and why their solutions take the form that they do. Further, it gives a conceptually clear, and often practical, means of telling when some particular equation of higher degree can be solved in that manner.
Galois theory originated in the study of symmetric functions ? the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots. For instance, (x ? a)(x ? b) = x2 ? (a + b)x + ab, where 1, a + b and ab are the elementary polynomials of degree 0, 1 and 2 in two variables.
This was first formalized by the 16th century French mathematician Francois Viete, in Viete's formulas, for the case of positive real roots. In the opinion of the 18th century British mathematician Charles Hutton,[1] the expression of coefficients of a polynomial in terms of the roots (not only for positive roots) was first understood by the 17th century French mathematician Albert Girard; Hutton writes: ...[Girard was] the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products.
He was the first who discovered the rules for summing the powers of the roots of any equation.
In this vein, the discriminant is a symmetric function in the roots which reflects properties of the roots ? it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct complex conjugate roots.
See Discriminant: nature of the roots for details. （以下略）
>>110 シローの定理というのが出てくる 四郎さん、アーベルと同じノルウエー人で高校の教師だった、1862年30歳のときに大学で講義”explained Abel's and Galois's work on algebraic equations”をしている。 そのときに、シローの定理に関する問題意識をもったようだ 1872年40歳のときにシローの定理の論文を発表したが、これが大ヒットだったんだ
en.wikipedia.org/wiki/Peter_Ludwig_Mejdell_Sylow Peter Ludwig Mejdell Sylow (12 December 1832 ? 7 September 1918) was a Norwegian Sylow was a high school teacher in Halden, Norway, from 1858 to 1898, It was then that he posed the question that led to his theorems regarding Sylow subgroups. Sylow published the Sylow theorems in 1872, and subsequently devoted eight years of his life, with Sophus Lie, to the project of editing the mathematical works of his countryman, Niels Henrik Abel.
www-history.mcs.st-andrews.ac.uk/Biographies/Sylow.html In 1862 Sylow lectured at the University of Christiania, substituting for Broch. In his lectures Sylow explained Abel's and Galois's work on algebraic equations. A summary of these lectures is presented in [2]. It is worth noting that although he had not proved 'Sylow's theorems' at this time (he published them 10 years later) he did pose a question about them.
Sylow's fame rests on one 10 page paper published in 1872. In this paper Theoremes sur les groupes de substitutions which Sylow published in Mathematische Annalen Volume 5 (pages 584 to 594) appear the three Sylow theorems. Cauchy had already proved that a group whose order is divisible by a prime p has an element of order p. Sylow proved what is perhaps the most profound result in the theory of finite groups.
If pn is the largest power of the prime p to divide the order of a group G then G has subgroups of order pn, G has 1 + kp such subgroups, any two such subgroups are conjugate.
Almost all work on finite groups uses Sylow's theorems.
とすれば、 「この（正規部分群列（方程式のガロア群=G0>G1>G2>・・・>Gs-1>Gs=1 by アルティン>>511 ））について、前の群（Gs-1 by アルティン＝群（G）の直前の群 by ガロア）についてと同様に論ぜられるであろう。 その結果、[群の]分解の順序で第一の群、すなわち方程式の実際の群は xk, xak+b という形の置換だけを含むことができることになる。」とされるべきだったろう。ガロアの原文が悪いのか、訳がどうなのか不明だが
