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
（アブストラクト） We propose that the stability of dark matter is ensured by a discrete subgroup of the U(1)B-L gauge symmetry, Z_2(B-L). We introduce a set of chiral fermions charged under the U(1)B-L in addition to the right-handed neutrinos, and require the anomaly-cancellation conditions associated with the U(1)B-L gauge symmetry. We find that the possible number of fermions and their charges are tightly constrained, and that non-trivial solutions appear when at least five additional chiral fermions are introduced. The Fermat theorem in the number theory plays an important role in this argument. Focusing on one of the solutions, we show that there is indeed a good candidate for dark matter, whose stability is guaranteed by Z_2(B-L). Comments: 12 pages, no figure. v2: references added Subjects: High Energy Physics - Phenomenology (hep-ph); Cosmology and Extragalactic Astrophysics (astro-ph.CO); High Energy Physics - Theory (hep-th) Journal reference: Physics Letters B 699 (2011) 360-363 DOI: 10.1016/j.physletb.2011.04.035 Report number: IPMU 11-0020, TU-878, KEK-TH 1440 Cite as: arXiv:1102.4688v2 [hep-ph] （引用おわり）
フェルマー理論との関係： ”We find that the possible number of fermions and their charges are tightly constrained, and that non-trivial solutions appear when at least five additional chiral fermions are introduced. The Fermat theorem in the number theory plays an important role in this argument. Focusing on one of the solutions, we show that there is indeed a good candidate for dark matter, whose stability is guaranteed by Z_2(B-L). ”
論文のPDFが見つかった（下記） www.citebase.org/abstract?id=oai%3AarXiv.org%3A1102.4688 で、本文を読むと、P9で ”Importantly, the Fermat theorem excludes the case of n = 3, which have forced us to consider n > 3.”と出てくる P7で ”In the case of three new fermions (n = 3), we can easily see that there is no solution to Eq. (4), by noting the famous Fermat theorem in the number theory [7].” ”[7] A. Wiles, Annals of Mathematics 141, 3, 443 (1995); R. Taylor and A. Wiles, Annals of Mathematics 141, 3, 553 (1995).”
下記ご参照 en.wikipedia.org/wiki/B%E2%88%92L In high energy physics, B ? L (pronounced "bee minus ell") is the difference between the baryon number (B) and the lepton number (L).
DetailsThis quantum number is the charge of a global/gauge U(1) symmetry in some Grand Unified Theory models, called U(1)B ? L. Unlike baryon number alone or lepton number alone, this hypothetical symmetry would not be broken by chiral anomalies or gravitational anomalies, as long as this symmetry is global, which is why this symmetry is often invoked. If B ? L exists as a symmetry, it has to be spontaneously broken to give the neutrinos a nonzero mass if we assume the seesaw mechanism. The gauge bosons associated to this symmetry are commonly called X and Y bosons.
The anomalies that would break baryon number conservation and lepton number conservation individually cancel in such a way that B ? L is always conserved. One hypothetical example is proton decay where a proton (B = 1; L = 0) would decay into a pion (B = 0, L = 0) and positron (B = 0; L = ?1).
Weak hypercharge Y W is related to B ? L via:
X + 2Y W = 5(B ? L) where X is the U(1) symmetry Grand Unified Theory-associated conserved quantum number.
en.wikipedia.org/wiki/Left-right_model Left?right symmetry is a general principle in physics which holds that valid physical laws must not produce a different result for a motion that is left-handed than motion that is right-handed. The most common application is expressed as equal treatment of clockwise and counter-clockwise rotations from a fixed frame of reference.
The general principle is often referred to by the name chiral symmetry. The rule is absolutely valid in the classical mechanics of Newton and Einstein, but results from quantum mechanical experiments show a difference in the behavior of left-chiral versus right-chiral subatomic particles.
Particle PhysicsIn theoretical physics, the electroweak model breaks parity maximally. All its fermions are chiral Weyl fermions, which means that the weak gauge bosons only couple to left-handed quarks and leptons. Some theorists found this objectionable, and so proposed a GUT extension of the weak force which has new, high energy W' and Z' bosons which couple with right handed quarks and leptons. （この後が面白いが、数学記号が複雑なので省略する）
en.wikipedia.org/wiki/Grand_Unified_Theory A Grand Unified Theory, (GUT), is a candidate model in particle physics in which at high energy, the three gauge interactions of the Standard Model which define the electromagnetic, weak, and strong interactions, are merged into one single interaction characterized by one larger gauge symmetry and thus one unified coupling constant. In contrast, the experimentally verified Standard Model of particle physics is based on three independent interactions, symmetries and coupling constants.
Models that do not unify all interactions using one simple Lie group as the gauge symmetry, but do so using semisimple groups, can exhibit similar properties and are sometimes referred to as Grand Unified Theories as well.
Unifying gravity with the other three interactions would provide a theory of everything (TOE), rather than a GUT. Nevertheless, GUTs are often seen as an intermediate step towards a TOE.
As of 2012[update], all GUT models which aim to be completely realistic are quite complicated, even compared to the Standard Model, because they need to introduce additional fields and interactions, or even additional dimensions of space. The main reason for this complexity lies in the difficulty of reproducing the observed fermion masses and mixing angles. Due to this difficulty, and due to the lack of any observed effect of grand unification so far, there is no generally accepted GUT model. （この後が面白いが、数学記号が複雑なので省略する）
