https://encyclopediaofmath.org/wiki/Vertex_operator encyclopediaofmath Vertex operator The term "vertex operator" in mathematics refers mainly to certain operators (in a generalized sense of the term) used in physics to describe interactions of physical states at a "vertex" in string theory [a9] and its precursor, dual resonance theory; the term refers more specifically to the closely related operators used in mathematics as a powerful tool in many applications, notably, constructing certain representations of affine Kac?Moody algebras (cf. also Kac?Moody algebra) and other infinite-dimensional Lie algebras, addressing the problems of the "Monstrous Moonshine" phenomena for the Monster finite simple group, and studying soliton equations (cf. also Moonshine conjectures). The term "vertex operator" also refers, more abstractly, to any operator corresponding to an element of a vertex operator algebra or a related operator.
https://ncatlab.org/nlab/show/vertex+operator+algebra ncatlab vertex operator algebra Contents 1. Idea 2. Standard definition 3. Properties Category of vertex operator algebras Modular category of modules over a VOA Goddard-Thorn theorem Relation to conformal nets
https://ncatlab.org/nlab/show/functorial+field+theory ncatlab functorial field theory Redirected from "FQFT".
https://en.wikipedia.org/wiki/Monstrous_moonshine Monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The term was coined by John Conway and Simon P. Norton in 1979.[1][2][3] The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a two-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas.
これに関連して "vertex" dual resonance theory Kac Moody algebra で検索すると、Frenkel 1985 があり、上記1988より早い ”Representations of Kac-Moody Algebras and Dual Resonance Models”がヒット ”j(q) = θL(q)/η(q)^24 =q^-1 + 24 + 196884q +・・ (4.21)”（下記）に言及しているね ここらが発端だろう
https://cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/2/3739/files/2021/06/frenkel_representations_kac_moody.pdf Volume 21, 1985 American Mathematical Society Representations of Kac-Moody Algebras and Dual Resonance Models I. B. Frenkel
Introduction. The theories of Kac-Moody algebras and dual resonance models were born at approximately the same time (1968). The second theory underwent enormous development until 1974 (see reviews [25, 26]) followed by years of decliae, while the first theory moved slowly until the work of Kac [14] in 1974 followed by accelerated progress.
Now both theories have gamed considerable interest in their respective fields, mathematics and physics. Despite the fact that these theories have no common motivations, goals or problems, their formal similarity goes remarkably far. In this paper we discuss primarily the mathematical theory. For a review of the physical theory see the paper of J. Schwarz in this volume [27]
Then in [9, 28] the "vertex construction" was found for the whole class of affine Lie algebras and the similarity became a precise correspondence.
Let us fix a light-cone element c ∈ Δ such that there are no real roots orthogonal to it. Such a vector exists and the set L = [a ∈ ΔR: (a, c) =1} is isomorphic to the unique even unimodular lattice of rank 24, which does not contain elements of length √2 [2]. We denote by V1,c, the space Σα∈L V1,c,α. Then the character of V1,c, is j(q) = θL(q)/η(q)^24 =q^-1 + 24 + 196884q +・・ (4.21) It was noticed by McKay that the number 196884 exceeds by only one the dimension of the minunal representation of F1. Conway and Norton [3] conjectured that there is a natural graded representation of F] with the character (4.21) minus 24. First Garland [12] and Kac [17] independently tried to construct i7i in a space isomorphic to ^ . The first problem was to obtain a representation of one important subgroup C=2^+l ' ・(・0)/±1, where -0 is the automorphism group of the Leech lattice. It is easy to construct another group C' = 224 ・ (-0) (= (2M+1/±1) ・ (-0)). Using one observation of Griess, Kac [18] succeeded in passing from C' to C. The last question is: Where is the whole group F\1 Recently, important progress has been made in answer to this question [10]. Turning again to the dual resonance models gives a hint as to the answer. Physicists know that m the contmuous version of F; ^ the obvious action of the group 0(24) can be extended to the bigger group 0(25). This extension becomes apparent only if we return to the bigger space V1+. Whether this unusual phenomenon corresponds to the extension of C to F1 will become clear in the future. (引用終り) 以上
>>349 関連 >"vertex" dual resonance theory Kac Moody algebra
Kac?Moody Lie algebra（下記） ”E. Date, M. Jimbo, M. Kashiwara, T. Miwa,(1983) The vertex operator constructions were, quite unexpectedly, applied to the theory of soliton equations. This was based on the observation (see [DaJiKaMi]) The vertex operators were introduced in string theory around 1969, but the vertex operator construction entered string theory only at its revival in the mid 1980s. Thus, the representation theory of affine algebras became an important ingredient of string theory (see [GrScWi]).(1987) The vertex operators turned out to be useful even in the theory of finite simple groups. Namely, a twist of the homogeneous vertex operator construction based on the Leech lattice pro
402 名前：duced the 196883-dimensional Griess algebra and its automorphism group, the famous finite simple Monster group (see Sporadic simple group) [FrLeMe].(1989)” そうなんだ。Kac?Moody Lie algebraだったね
(参考) https://encyclopediaofmath.org/wiki/Kac-Moody_algebra 15 November 2017 Kac-Moody algebra A Kac-Moody algebra (also Kac?Moody Lie algebra) is defined as follows: Let A=(aij)ni,j=1 be an (n×n) -matrix satisfying conditions (see Cartan matrix) aii=2;aij?0 aij=0 and aij∈Z for i≠j,⇒ aji=0.}(a1) The associated Kac?Moody algebra g(A) is a Lie algebra over C on 3ngenerators ei, fi, hi (called the Chevalley generators) and the following defining relations: 略
A systematic study of Kac-Moody algebras was started independently by V.G. Kac [Ka] and R.V. Moody [Mo], and subsequently many results of the theory of finite-dimensional semi-simple Lie algebras have been carried over to Kac-Moody algebras. The main technical tool of the theory is the generalized Casimir operator (cf. Casimir element), which can be constructed provided that the matrix A is symmetrizable, i.e. A=DB for some invertible diagonal matrix D and symmetric matrix B [Ka2]. In the non-symmetrizable case more sophisticated geometric methods are required [Ku], [Ma].
One of the most important ingredients of the theory of Kac-Moody algebras are integrable highest-weight representations (cf. also Representation with a highest weight vector).
The numerous applications of Kac-Moody algebras are mainly related to the fact that the Kac-Moody algebras associated to positive semi-definite indecomposable Cartan matrices (called affine matrices) admit a very explicit construction. (A matrix is called indecomposable if it does not become block-diagonal after arbitrary permutation of the index set.) These Kac-Moody algebras are called affine algebras.
This observation leads to geometric applications of affine algebras and the corresponding groups, called the loop groups (see [PrSe]).
The basic representation of g(A(1)) is then defined on V by the following formulas [FrKa]:
π(u(n))=u(n),u∈h π(E(n)α)=Xn(α)cα,π(k)=1; This is called the homogeneous vertex operator construction of the basic representation.
The vertex operators were introduced in string theory around 1969, but the vertex operator construction entered string theory only at its revival in the mid 1980s. Thus, the representation theory of affine algebras became an important ingredient of string theory (see [GrScWi]).
The vertex operators turned out to be useful even in the theory of finite simple groups. Namely, a twist of the homogeneous vertex operator construction based on the Leech lattice produced the 196883-dimensional Griess algebra and its automorphism group, the famous finite simple Monster group (see Sporadic simple group) [FrLeMe].
The vertex operator constructions were, quite unexpectedly, applied to the theory of soliton equations. This was based on the observation (see [DaJiKaMi]) that the orbit of the vector vΛ0 of the basic representation under the loop group satisfies an infinite hierarchy of partial differential equations, the simplest of them being classical soliton equations, like the Korteweg-de Vries equation.
Moreover, the linear span of the functions χΛ(τ,0) for Λ of fixed level k is invariant under the modular transformations
This turned out to be a key fact in the representation theory of affine algebras, as well as its applications to conformal field theory (see [Ve]), to 2 -dimensional lattice models [DaJiKuMiOk], and even to knot theory[YaGe].
References [DaJiKaMi] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, "Transformation groups for soliton equations" M. Jimbo (ed.) T. Miwa (ed.), Proc. RIMS Symp., World Sci. (1983) pp. 39-120 [DaJiKuMiOk] E. Date, M. Jimbo, A. Kuniba, T. Miwa, M. Okado, "Exactly solvable SOS models" Nucl. Phys., B290 (1987) pp. 231-273 MR0910849 Zbl 0679.17010 [FrKa] I.B. Frenkel, V.G. Kac, "Basic representations of affine Lie algebras and dual resonance models" Invent. Math., 62 (1980) pp. 23-66 MR0595581 Zbl 0493.17010 [FrLeMe] I.
406 名前：Frenkel, J. Lepowsky, A. Meurman, "Vertex operator algebras and the Monster", Acad. Press (1989) MR1167718 MR0996026 Zbl 0674.17001 [GrScWi] M.B. Green, J.H. Schwarz, E. Witten, "Superstring theory", Cambridge Univ. Press (1987) MR0922731 MR0915347 MR0878144 MR0878143 Zbl 0637.53111 Zbl 0619.53002 [Ka] V.G. Kac, "Simple irreducible graded Lie algebras of finite growth" Math. USSR Izv., 2 (1968) pp. 1271-1311 Izv. Akad. Nauk USSR Ser. Mat., 32 (1968) pp. 1923-1967 MR0259961 Zbl 0222.17007 [Mo] R.V. Moody, "A new class of Lie algebras" J. of Algebra, 10 (1968) pp. 211-230 MR0229687 Zbl 0191.03005 [YaGe] C.N. Yang (ed.) M.L. Ge (ed.), Braid group, knot theory and statistical mechanics, World Sci. (1989) MR1062420 Zbl 0716.00010 (引用終り) 以上 []
(参考) https://handwiki.org/wiki/Physics:History_of_string_theory Physics:History of string theory Contents 1 1943?1959: S-matrix theory 2 1959?1968: Regge theory and bootstrap models 3 1968?1974: Dual resonance model 4 1974?1984: Bosonic string theory and superstring theory 5 1984?1994: First superstring revolution 6 1994?2003: Second superstring revolution 7 2003?present
https://en.wikipedia.org/wiki/Regge_theory Regge theory In quantum physics, Regge theory (/?r?d?e?/) is the study of the analytic properties of scattering as a function of angular momentum, where the angular momentum is not restricted to be an integer multiple of ? but is allowed to take any complex value. The nonrelativistic theory was developed by Tullio Regge in 1959.[1]
History and implications This observation turned Regge theory from a mathematical curiosity into a physical theory: it demands that the function that determines the falloff rate of the scattering amplitude for particle-particle scattering at large energies is the same as the function that determines the bound state energies for a particle-antiparticle system as a function of angular momentum.[5]
After many false starts, Richard Dolen, David Horn, and Christoph Schmid understood a crucial property that led Gabriele Veneziano to formulate a self-consistent scattering amplitude, the first string theory.
https://en.wikipedia.org/wiki/Ramamurti_Rajaraman Ramamurti Rajaraman (born 11 March 1939)
Regge poles and particle phenomenology At that time, high energy hadron scattering was being analysed using S-matrix and Regge pole techniques. Rajaraman gave the first determination from experimental data of the value of the "Triple Pomeron Vertex" as a function of momentum transfer[12] and also derived the consequences of the vanishing of this vertex on high energy hadron scattering.[13] With Finkelstein, he analysed Exchange Degeneracy in inclusive reactions involving the triple-Reggeon vertex[14][15]
https://en.wikipedia.org/wiki/Vertex_function Vertex function In quantum electrodynamics, the vertex function describes the coupling between a photon and an electron beyond the leading order of perturbation theory. In particular, it is the one particle irreducible correlation function involving the fermion ψ,the antifermion ψ^-, and the vector potential A.
下記のPhysicsの3例を見ると、”交点”が適当かもしれない 特に、”PV (physics) Primary Vertex (i.e. the interaction point)”とあるし
https://en.wikipedia.org/wiki/Vertex Vertex Science and technology Physics ・Vertex (physics), the reconstructed location of an individual particle collision ・Vertex (optics), a point where the optical axis crosses an optical surface ・Vertex function, describing the interaction between a photon and an electron
https://en.wikipedia.org/wiki/Interaction_point Interaction point (Redirected from Vertex (physics)) In particle physics, an interaction point (IP) is the place where particles collide in an accelerator experiment. The nominal interaction point is the design position, which may differ from the real or physics interaction point, where the particles actually collide. A related, but distinct, concept is the primary vertex: the reconstructed location of an individual particle collision.
https://twiki.cern.ch/twiki/bin/view/CMSPublic/WorkBookGlossary TWiki> CMSPublic Web>SWGuide>WorkBook>WorkBookGlossary (2022-12-16, TamasAlmosVami) Glossary and Index PV (physics) Primary Vertex (i.e. the interaction point)
>>366 補足 下記P15”In terms of Q we introduce the vertex operator corresponding to the external leg with momentum p:” とある。交点の方がイメージわくよね
https://arxiv.org/abs/0704.0101 https://arxiv.org/pdf/0704.0101.pdf The birth of string theory Paolo Di Vecchia1 [v1] Sun, 1 Apr 2007 Copenhagen, Denmark
Summary. In this contribution we go through the developments that in the years from 1968 to about 1974 led from the Veneziano model to the bosonic string theory. They include the construction of the N-point amplitude for scalar particles, its factorization through the introduction of an infinite number of oscillators and the proof that the physical subspace was a positive definite Hilbert space. We also discuss the zero slope limit and the calculation of loop diagrams. Lastly, we describe how it finally was recognized that a quantum relativistic string theory was the theory underlying the Veneziano model.
P15 In terms of Q we introduce the vertex operator corresponding to the external leg with momentum p:
https://academic.oup.com/ptep/article/2016/6/06A103/2330300 （PDFあり） Nambu, A Foreteller of Modern Physics I The birth of string theory H. Itoyama 2016 This is a brief summary of an introductory lecture for students and scholars in general given by the author at the Nambu Memorial Symposium which was held at Osaka City University on 29 September 2015. We review the invention of string theory by Professor Yoichiro Nambu following the discovery of the Veneziano amplitude. We also discuss Professor Nambu’s proposal on string theory in the Schild gauge in 1976, which is related to the matrix model of Yang?Mills type.
（こちらは本格的な本） https://www.アマゾン The Birth of String Theory Hardcover ? April 12, 2012 636 ページ English Edition by Andrea Cappelli (編集), Elena Castellani (編集), & 2 more
>>366 再録 https://en.wikipedia.org/wiki/Vertex Vertex Science and technology Physics ・Vertex (physics), the reconstructed location of an individual particle collision ・Vertex (optics), a point where the optical axis crosses an optical surface ・Vertex function, describing the interaction between a photon and an electron (引用終り)
>>376 >https://mathoverflow.net/questions/53988/what-is-the-motivation-for-a-vertex-algebra >What is the motivation for a vertex algebra?
追加引用 (回答のDavid Ben-Zvi氏は、Edward Frenkel氏からみの大物だね（後述）) 8 Answers 75 answered Feb 1, 2011 David Ben-Zvi Vertex algebras precisely model the structure of "holomorphic one-dimensional algebra" -- in other words, the algebraic structure that you get if you try to formalize the idea of operators (elements of your algebra) living at points of a Riemann surface, and get multiplied when you collide.
Our geometric understanding of how to formalize this idea has I think improved dramatically over the years with crucial steps being given by the point of view of "factorization algebras" by Beilinson and Drinfeld, which is explained (among other places :-) ) in the last chapter of my book with Edward Frenkel, "Vertex algebras and algebraic curves" (second edition only). This formalism gives a great way to understand the algebraic structure of local operators in general quantum field theories -- as is seen in the recent work of Kevin Costello -- or in topological field theory, where it appears eg in the work of Jacob Lurie (in particular the notion of "topological chiral homology").
In fact I now think the best way to understand a vertex algebra is to first really understand its topological analog, the structure of local operators in 2d topological field theory. If you check out any article about topological field theory it will explain that in a 2d TFT, we assign a vector space to the circle, it obtains a multiplication given by the pair of pants, and this multiplication is commutative and associative (and in fact a commutative Frobenius algebra, but I'll ignore that aspect here). It's very helpful to picture the pair of pants not traditionally but as a big disc with two small discs cut out -- that way you can see the commutativity easily, and also that if you think of those discs as small (after all everything is topologically invariant) you realize you're really describing operators labeled by points (local operators in physics, which we insert around a point) and the multiplication is given by their collision (ie zoom out the picture, the two small discs blend and look like one disc, so you've started with two operators and gotten a third).
Anyway this is getting long - to summarize, a vertex algebra is the holomorphic refinement of an E2 algebra, aka a "vector space with the algebraic structure inherent in a double loop space", where we allow holomorphic (rather than locally constant or up-to
AND we get perhaps the most important example of a vertex algebra--- take X in the above story to be BG, the classifying space of a group G. Then Ω^2X=ΩG is the "affine Grassmannian" for G, which we now realize "is" a vertex algebra.. by linearizing this space (taking delta functions supported at the identity) we recover the Kac-Moody vertex algebra (as is explained again in my book with Frenkel).
https://math.berkeley.edu/~frenkel/BOOK/ 本(my book with Frenkel)"Vertex Algebras and Algebraic Curves" by Edward Frenkel and David Ben-Zvi 2001
https://web.ma.utexas.edu/users/benzvi/ David Ben-Zvi https://en.wikipedia.org/wiki/David_Ben-Zvi David Dror Ben-Zvi is an American mathematician, currently the Joe B. and Louise Cook Professor of Mathematics at University of Texas at Austin.[1] Ben-Zvi earned his Ph.D. from Harvard University in 1999, with a dissertation entitled Spectral Curves, Opers And Integrable Systems supervised by Edward Frenkel.[2] In 2012, he became one of the inaugural Fellows of the American Mathematical Society.[3] (引用終り) 以上
例えば >>389より >https://mathoverflow.net/questions/53988/what-is-the-motivation-for-a-vertex-algebra >What is the motivation for a vertex algebra?
追加引用 回答1 You ask what physical problem vertex operators model but you actually give the answer yourself! :-) They can be used to answer questions about "two particles colliding in an infinite vacuum". A pair of strings coming from infinity, interacting "once", and going off to infinity, say, sweep out a surface that is topologically a sphere with 4 tubes sticking out out of it. String theory is (sort of) conformally invariant and this surface is conformally a Riemann sphere with 4 punctures in it. Vertex operators arise when studying quantum fields on Riemann spheres in the vicinity of these punctures. ? Dan Piponi Feb 1, 2011 at 19:13 (引用終り)
これ、”topologically a sphere with 4 tubes sticking out out of it.”は、下記のファインマンダイアグラムですね 英の.png図を、ご参照 おっと、”頂点(vertex): 線の分岐点”とありますね・・ｗ ファインマンダイアグラムが起源か・・ https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%B3%E3%83%9E%E3%83%B3%E3%83%BB%E3%83%80%E3%82%A4%E3%82%A2%E3%82%B0%E3%83%A9%E3%83%A0 ファインマンダイアグラムは、場の量子論において摂動展開の各項を図に示したものである。それぞれのダイアグラムは素粒子をはじめとする実際の粒子の反応過程を表現している。
https://upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/574px-Feynmann_Diagram_Gluon_Radiation.svg.png In this Feynman diagram, an electron (e-) and a positron (e+) annihilate, producing a photon (γ, represented by the blue sine wave) that becomes a quark?antiquark pair (quark q, antiquark q?), after which the antiquark radiates a gluon (g, represented by the green helix
458 名前：).
http://physnd.html.xdomain.jp/top.html 物理のぺーじ http://physnd.html.xdomain.jp/qed/qed.html QED http://physnd.html.xdomain.jp/qed/feyqed.pdf ファインマン図 QED でのファインマン図とファインマン則を簡単にまとめます。 両端の黒い丸の部分を頂点 (vertex) と呼び、相互作用部分を表し、QED では 2 本の電子の線と 1 本の光子の線 が常に出ているように書き (このように頂点から出ている線のことを足 (leg) と言ったりします)
>>371 >pantodon.jp/index.rb?body=VA_and_VOA#cite.0@Borcherds1986 >Vertex operator algebra の一つの起源は数理物理であり, 初期の段階では, 定義がはっきりしなかった。Vertex algebra の正確な定義を与えたのは, Borcherds [Bor86], vertex operator algebra の定義を与えたのは, Igor Frenkel と Lepowsky と Meurman らしい。 >[Bor86] >Richard E. Borcherds. “Vertex algebras, Kac-Moody algebras, and the Monster”. In: Proc. Nat. Acad. Sci. U.S.A. 83.10 (1986), pp. 3068?3071. url: dx.doi.org/10.1073/pnas.83.10.3068.
この[Bor86]の文献に下記3つあり 1. Frenkel, I. B., Lepowsky, J. & Meurman, A. (1984) Proc.Nati. Acad. Sci. USA 81, 3256-3260. 2. Frenkel, I. B. & Kac, V. G. (1980) Invent. Math. 62, 23-66. 3. Frenkel, I. B. (1985) Lect. Appl. Math. 21, 325-353.
このうちの1が下記です ここに、”These results were presented at the November 1983 workshop on Vertex Operators in Mathematics and Physics at the Mathematical Sciences Research Institute.” とあるので、1983年時点で、米国では”Vertex Operators”という用語は、Mathematics and Physicsで使われていた そして、この1984年の論文でも、”2. Vertex Operators and the Space V”の記載があります
https://www.pnas.org/doi/epdf/10.1073/pnas.81.10.3256 Proc.Nati.Acad.Sci.USAVol.81,pp.3256-3260,May1984 Mathematics Anatural representation of the Fischer-Griess Monster with the modular function J as character I.B.FRENKEL*,J.LEPOWSKY*,AND A.MEURMAN
P1 These results were presented at the November 1983 workshop on Vertex Operators in Mathematics and Physics at the Mathematical Sciences Research Institute. The details will appear elsewhere.
双対共鳴モデル：Dual resonance model（下記） ”Yoichiro Nambu,[2] Holger Bech Nielsen,[3] and Leonard Susskind[4] provided a physical interpretation in terms of an infinite number of simple harmonic oscillators describing the motion of an extended one-dimensional string, hence came the name "string theory."” ノーベル賞の南部先生ね
https://en.wikipedia.org/wiki/Dual_resonance_model Dual resonance model In theoretical physics, a dual resonance model arose during the early investigation (1968?1973) of string theory as an S-matrix theory of the strong interaction. Overview The dual resonance model was based upon the observation that the amplitudes for the s-channel scatterings matched exactly with the amplitudes for the t-channel scatterings among mesons and also the Regge trajectory. It began with the Euler beta function model of Gabriele Veneziano in 1968 for a 4-particle amplitude which has the property that it is explicitly s?t crossing symmetric, exhibits duality between the description in terms of Regge poles or of resonances, and provides a closed-form solution to non-linear finite-energy sum rules relating s- and t- channels.
The Veneziano formula was quickly generalized to an equally consistent N-particle amplitude[1] for which Yoichiro Nambu,[2] Holger Bech Nielsen,[3] and Leonard Susskind[4] provided a physical interpretation in terms of an infinite number of simple harmonic oscillators describing the motion of an extended one-dimensional string, hence came the name "string theory."
The study of dual resonance models was a relatively popular subject of study between 1968 and 1973.[5] It was even taught briefly as a graduate level course at MIT, by Sergio Fubini and Veneziano, who co-authored an early article.[6] It fell rapidly out of favor around 1973 when quantum chromodynamics became the main focus of theoretical research[7] (mainly due to the theoretical appeal of its asymptotic freedom).[8] See also QCD string Lund string model Notes 2. Nambu, Y. (1970). "Quark model and the factorization of the Veneziano amplitude." In R. Chand (ed.), Symmetries and quark models (pp. 269?277). Singapore: World Scientific. (引用終り) 以上
Crucial Computer Program for Particle Physics at Risk of Obsolescence | Quanta Magazine https://www.quantamagazine.org/crucial-computer-program-for-particle-physics-at-risk-of-obsolescence-20221201/
さらに用語vertex発掘した どうも、下記の ”F. J. Dyson, Phys. Rev. 75, 486 The Radiation Theories of Tomonaga, Schwinger,and Feynman” が、調べた範囲の起源みたい これより古いFeynmanの文献も当たったけど、Vertexという用語は見当たらなかった
そして P19 Through each point of a graph pass two electron lines, and therefore the electron lines together form one open polygon containing the vertices xk and x
529 名前：rk and possibly a number of closed polygons as well. とか P29 It will be found that in each graph there are at each vertex two electron lines and one photon line, with the exception of x0 at which there are two electron lines only; further, such graphs can exist only for even n. が、Vertexの説明になっています
（参考） https://en.wikipedia.org/wiki/Feynman_diagram Feynman_diagram より References 3. Feynman, Richard (1949). "The Theory of Positrons". Physical Review. 76 (6): 749?759. Bibcode:1949PhRv...76..749F. doi:10.1103/PhysRev.76.749. S2CID 120117564. In this solution, the 'negative energy states' appear in a form which may be pictured (as by Stuckelberg) in space-time as waves traveling away from the external potential backwards in time. Experimentally, such a wave corresponds to a positron approaching the potential and annihilating the electron. https://authors.library.caltech.edu/3520/ https://authors.library.caltech.edu/3520/1/FEYpr49b.pdf P2 注2)The equivalence of the entire procedure (including photoninteractions) with the work of Schwinger and Tomonaga has beendemonstrated by F. J. Dyson, Phys. Rev. 75, 486 (1949).
（上記より） web.ihep.su/dbserv/compas/src/dyson49/eng.pdf F. J. Dyson, Phys. Rev. 75, 486 The Radiation Theories of Tomonaga, Schwinger,and Feynman F.J. Dyson Institute for Advanced Study, Princeton, New Jersey P2 注：3 R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948); Phys. Rev. 74, 939, 1430 (1948); J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945). These articles describe early stages in the development of Feynman’s theory, little of which is yet published.
P19 7 GRAPHICAL REPRESENTATION OF MATRIX ELEMENTS P25 8 VACUUM POLARIZATION AND CHARGE RENORMALIZATION
＜vertex,vertices 抽出（下記”in Section VIII”は、上記のP25 8 のことか。自己言及ですねｗ）＞ P19 Through each point of a graph pass two electron lines, and therefore the electron lines together form one open polygon containing the vertices xk and xrk and possibly a number of closed polygons as well. P21 A “self-energy part” of a graph G is defined as follows; it is a set of one or more vertices not including x0, together with the lines joining them, which is connected with the remainder of G (or with the edge of the diagram) only by two electron lines or by one or two photon lines. P23 Ther
531 名前：e remains the case in which λ leads from one vertex x3 to another x4 of G0. In this case C(G0) contains in its integrand the function P29 All possible graphs G with (n + 1) vertices are now drawnas described in Section VIII omitting disconnected graphs, graphs with self-energy parts, and graphs with external vacuum polarization parts as definedin Section VIII. It will be found that in each graph there are at each vertex two electron lines and one photon line, with the exception of x0 at which there are two electron lines only; further, such graphs can exist only for even n. Kn is the sum of a contribution K(G) from each G.
Also, the integrand in Jn is a symmetrical function of x1, . . . , xn; therefore, graphs which differ only by a relabeling of the vertices x1, . . . , xn give identical contributions to Kn and need not be considered separately.
P31 Next, all admissable graphs with the three vertices x0, x1, x2 are to be drawn. It is easy to see that there are only two such graphs, that G shown in Fig.1, and the identical graph with x1 and x2 interchanged. (引用終り) 以上
https://en.wikipedia.org/wiki/Freeman_Dyson Career in the United States In 1949, Dyson demonstrated the equivalence of two formulations of quantum electrodynamics (QED): Richard Feynman's diagrams and the operator method developed by Julian Schwinger and Shin'ichir? Tomonaga. He was the first person after their creator to appreciate the power of Feynman diagrams and his paper written in 1948 and published in 1949 was the first to make use of them. He said in that paper that Feynman diagrams were not just a computational tool but a physical theory and developed rules for the diagrams that completely solved the renormalization problem. Dyson's paper and also his lectures presented Feynman's theories of QED in a form that other physicists could understand, facilitating the physics community's acceptance of Feynman's work. J. Robert Oppenheimer, in particular, was persuaded by Dyson that Feynman's new theory was as valid as Schwinger's and Tomonaga's. Also in 1949, in related work, Dyson invented the Dyson series. It was this paper that inspired John Ward to derive his celebrated Ward?Takahashi identity.[30]
関連追加 https://en.wikipedia.org/wiki/Quadratic_reciprocity Quadratic reciprocity より References ・Hilbert, David (1897), "Die Theorie der algebraischen Zahlkorper", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 4: 175?546, ISSN 0012-0456 （(どうも上記の英訳らしい） Hilbert, David (1998), The theory of algebraic number fields, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62779-1, MR 1646901） (独語) https://gdz.sub.uni-goettingen.de/download/pdf/PPN37721857X_0004/PPN37721857X_0004.pdf Werk Titel: Jahresbericht der Deutschen Mathematiker-Vereinigung Verlag: Georg Reimer Jahr: 1894/95 Kollektion: Mathematica Digitalisiert: Niedersachsische Staats- und Universitatsbibliothek Gottingen Werk Id: PPN37721857X_0004 PURL: resolver.sub.uni-goettingen.de/purl?PPN37721857X_0004 (PDFのP182) Die Theorie der algebraischen Zahlkorper David Hilbert. 目次 (以下のページは目次の通り) Cap. XXVII §122. Das Reciprocitatsgesets fur quqdratsche Reste ・・・384 Cap. XXVIII Das Reciprocitatsgesetz fur Ιte Potenzreste im regularen Kreiskorper. §154. Das Reciprocitatsgesetz fur Ιte Potenzreste und die Erganzungssatze ・・・470 §157. Ein besonderer Fall des Reciprocitatsgesetz fur zwei Primideale ・・・479 §158. Das Vorhandensein gewisser Hulfsprimideal, fur weiche Reciprocitatsgesetz gilt ・・・482 §159. Beweis des ersten Erganzungsatzes zum Reciprocitatsgesetz ・・・484 §160. Beweis des Reciprocitatsgesetzes zweishen zwei beliebigen Primidealen ・・・485 §161. Beweis des zweishen Erganzungsatzes zum Reciprocitatsgesetz ・・・488 (引用終り) 以上
＜関連追加引用＞ History and alternative statements The theorem was formulated in many ways before its modern form: Euler and Legendre did not have Gauss's congruence notation, nor did Gauss have the Legendre symbol. In this article p and q always refer to distinct positive odd primes, and x and y to unspecified integers.
There is no
599 名前：kind of reciprocity in the Hilbert reciprocity law; its name simply indicates the historical source of the result in quadratic reciprocity. Unlike quadratic reciprocity, which requires sign conditions (namely positivity of the primes involved) and a special treatment of the prime 2, the Hilbert reciprocity law treats all absolute values of the rationals on an equal footing. Therefore, it is a more natural way of expressing quadratic reciprocity with a view towards generalization: the Hilbert reciprocity law extends with very few changes to all global fields and this extension can rightly be considered a generalization of quadratic reciprocity to all global fields.
Connection with cyclotomic fields The early proofs of quadratic reciprocity are relatively unilluminating. The situation changed when Gauss used Gauss sums to show that quadratic fields are subfields of cyclotomic fields, and implicitly deduced quadratic reciprocity from a reciprocity theorem for cyclotomic fields. His proof was cast in modern form by later algebraic number theorists. This proof served as a template for class field theory, which can be viewed as a vast generalization of quadratic reciprocity.
Robert Langlands formulated the Langlands program, which gives a conjectural vast generalization of class field theory. He wrote:[27]
I confess that, as a student unaware of the history of the subject and unaware of the connection with cyclotomy, I did not find the law or its so-called elementary proofs appealing. I suppose, although I would not have (and could not have) expressed myself in this way that I saw it as little more than a mathematical curiosity, fit more for amateurs than for the attention of the serious mathematician that I then hoped to become. It was only in Hermann Weyl's book on the algebraic theory of numbers[28] that I appreciated it as anything more. (引用終り) 以上
"reciprocal"という言葉で想起されるのはオイラーの論文 Remarques sur un beau rapport entre les series des puissances tant directes que reciproques (Remarks on a beautiful relation between direct as well as reciprocal power series)
https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%89%E3%83%AA%E3%82%A2%E3%83%B3%EF%BC%9D%E3%83%9E%E3%83%AA%E3%83%BB%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB#CITEREF%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB2007 アドリアン＝マリ・ルジャンドル（仏: Adrien-Marie Legendre、1752年9月18日 - 1833年1月10日） 1798年の著書『数の理論に関する試作（Essai sur la Theorie des Nombres）』は、ドイツの天文学者、数学者、物理学者であるカール・フリードリヒ・ガウスの1801年の著書『整数論（Disquisitiones Arithmeticae）』の登場により、影に埋もれることとなった[2]。 (引用終り)
これ、下記PDFで ルジャンドル記号の導入があって ”§VI Theoreme contenant une loi reciprocite qui exite entre deux nombres premiers quelconques” とあるから ”reciprocite”（相互律）の用語は、ルジャンドルからだね
(参考) https://archive.org/details/essaisurlathor00lege/page/n1/mode/2up Essai sur la theorie des nombres by Legendre, A. M. (Adrien Marie), 1752-1833 Publication date 1798 Topics Number theory Publisher Paris, Duprat https://ia804700.us.archive.org/24/items/essaisurlathor00lege/essaisurlathor00lege.pdf
ここの §VI Theoreme contenant une loi reciprocite qui exite entre deux nombres premiers quelconques ・・・214 (引用終り) 以上
Shintani, Takuro (1976), “On evaluation of zeta functions of totally real algebraic number fields at non-positive integers”, Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics 23 (2): 393–417
（参考） https://en.wikipedia.org/wiki/Antoni_Zygmund Antoni Zygmund (December 25, 1900 ? May 30, 1992) In 1935 Zygmund published in Polish the original edition of what has become, in its English translation, the two-volume Trigonometric Series. It was described by Robert A. Fefferman as "one of the most influential books in the history of mathematical analysis" and "an extraordinarily comprehensive and masterful presentation of a ... vast field".[6] Jean-Pierre Kahane called the book "The Bible" of a harmonic analyst. The theory of trigonometric series had remained the largest component of Zygmund's mathematical investigations.[5]
1935 年、Zygmund はポーランド語で原版を出版し、その英語訳は 2 巻のTrigonometric Series になりました。ロバート A. フェファーマンは、「数学的分析の歴史の中で最も影響力のある本の 1 つ」であり、「非常に包括的で見事な ... 広大な分野のプレゼンテーション」と表現しました。[6] Jean-Pierre Kahane は、この本をハーモニック アナリストの「バイブル」と呼びました。三角級数の理論は、ジグムントの数学的研究の最大の構成要素であり続けました。[5]
References 5. Lorentz, G. G. (1993). "Antoni Zygmund and His Work" (PDF). Journal of Approximation Theory. 75: 1?7. doi:10.1006/jath.1993. 6. The 2nd edition of Zygmund's Trigonometric Series (Cambridge University Press, 1959) consists of 2 separate volumes. consists of the two volumes combined with a foreword by Robert A. Fefferman.
Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of L functions, proved by Lennart Carleson (1966). The name is also often used to refer to the extension of the result by Richard Hunt (1968) to L functions f []
https://de.wikipedia.org/wiki/K%C3%B6rper_(Algebra) Korper (Algebra) 6.Geschichte Wesentliche Ergebnisse der Korpertheorie sind Evariste Galois und Ernst Steinitz zu verdanken. Weitere Einzelheiten zur Genese des Begriffes liefert Wulf-Dieter Geyer in Kapitel Kapitel 2 seines Beitrages, in dem er u. a. auch die Rolle Richard Dedekinds hinweist (siehe Literatur).
6.歴史 体理論の重要な結果は、エヴァリスト ガロアとエルンスト シュタイニッツによるものです。Wulf-Dieter Geyer は、彼の寄稿の第 2 章で用語の起源に関する詳細を提供しています。また、 Richard Dedekindの役割も指摘しています(文献を参照)。
Literatur ・Wulf-Dieter Geyer: Field Theory. In: Volume I of the Proceedings of the Qinter School on Galois Theory, 15-24 February 2012, Universite du Luxembourg, Luxembourg. Juli 2013, abgerufen am 9. November 2022. siehe insbesondere Kapitel 2 (?Historical remarks about the concept of field“), Seite 29.
ありがとう ドイツ語は、独アルファベット程度は読めるが、検索すると 下記ですね ディリクレデデキント整数論講義 11. 代数的整数の理論で 下記独語版での §.160. Zahlenkorper ,452で 注** in meinen Gottinger Vorlesungen (1857 bis 1858), hatte ich denselben Begriff mit dem Namen eines rationalen Gebietes belegt, der aber weniger bequem ist. ↓（google訳 独→英） in my Gottingen lectures (1857-1858), I gave the same term the name of a rational domain, but this is less convenient. つまり 以前は、rationalen Gebietes＝rational domain＝「有理域」だった。（Korperにしたことへの注）
https://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlet Johann Peter Gustav Lejeune Dirichlet (German: [l????n di?i?kle?];[1] 13 February 1805 ? 5 May 1859) Gottingen (1855?1859) Dedekind, who felt that there were gaps in his mathematics education, considered that the occasion to study with Dirichlet made him "a new human being".[2] He later edited and published Dirichlet's lectures and other results in number theory under the title Vorlesungen uber Zahlentheorie (Lectures on Number Theory).
https://en.wikipedia.org/wiki/Vorlesungen_%C3%BCber_Zahlentheorie Vorlesungen uber Zahlentheorie (German for Lectures on Number Theory) is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Dirichlet and Dedekind's book This translation does not include Dedekind's Supplements X and XI in which he begins to develop the theory of ideals. The German titles of supplements X and XI are: Supplement X: Uber die Composition der binaren quadratische Formen (On the composition of binary quadratic forms) Supplement XI: Uber die Theorie der ganzen algebraischen Zahlen (On the theory of algebraic integers)
714 名前：. Hrsg. und mit Zusatzen versehen by Lejeune-Dirichlet, Peter Gustav, 1805-1859; Dedekind, Richard, 1831-1916 Publication date 1894 Topics Number theory Publisher Braunschweig F. Vieweg Collection gerstein; toronto Digitizing sponsor University of Toronto Contributor Gerstein - University of Toronto Language German (DOWNLOAD OPTIONS PDFより)
目次 XI. Ueber die Theorie der ganzen algebraischen Zahlen. §.160. Zahlenkorper ,452
Ein System A von reellen oder complexen Zahlen a soll ein Korper**) heissen, **) Vergl. §. 159 der zweiten Auflage dieses Werkes (1871). Dieser Name soll, ahnlich wie in den Naturwissenschaften, in der Geometrie und im Leben der menschlichen Gesellschaft, auch hier ein System bezeichnen,das eine gewisse Vollstandigkeit, Vollkommenheit, Abgeschlossenheit besitzt,wodurch es als ein organisches Ganzes, als eine naturliche Einheit erscheint. Anfangs, in meinen Gottinger Vorlesungen (1857 bis 1858), hatte ich denselben Begriff mit dem Namen eines rationalen Gebietes belegt, der aber weniger bequem ist. Der Begriff fallt im Wesentlichen zusammen mit Dem, was Kronecker einen Bationalitatsbereich genannt hat ( Grundzuge einer arithmetischen Theorie der algebraischen Grossen. 1882). Vergl. auch die von H. Weber und mir verfasste Theorie der algebraischen Functionen einer Veranderlichen. (Crelle's Journal, Bd. 92, 1882).
（google訳 独→英） A system A of real or complex numbers a shall be called a field**), **) cf. §. 159 of the second edition of this work (1871). As in the natural sciences, in geometry and in the life of human society, this name is also intended here to denote a system that has a certain completeness, perfection, closure, which makes it appear as an organic whole, as a natural unit. At first, in my Gottingen lectures (1857-1858), I gave the same term the name of a rational domain, but this is less convenient. The term essentially coincides with what Kronecker called a domain of batation (Basics of an arithmetic theory of algebraic magnitudes. 1882). compare also the theory of the algebraic functions of a variable written by H. Weber and myself. (Crelle's Journal, Vol. 92, 1882). (引用終り) 以上
（参考） https://www.iwanami.co.jp/book/b267429.html 数学入門 （上） 著者 遠山 啓 著 刊行日 1959/11/17 試し読み （冒頭からP14まで） https://www.iwanami.co.jp/moreinfo/tachiyomi/4160040.pdf 目次 はしがき I 数の幼年期 II 分離量と連続量 III 数の反意語 IV 代入─ずるい算数 V 図形の科学 VI 円の世界 VII 複素数─最後の楽章
https://www.iwanami.co.jp/book/b267430.html 数学入門 （下） 著者 遠山 啓 著 刊行日 1960/10/20 試し読み （冒頭からP16まで） https://www.iwanami.co.jp/moreinfo/tachiyomi/4160050.pdf 目次 VIII 数の魔術と科学 IX 変化の言語─関数 X 無限の算術─極限 XI 伸縮と回転 XII 分析の方法─微分 XIII 総合の方法─積分 XIV 微視の世界─微分方程式 あとがき (引用終り) 以上
（参考） https://ja.wikipedia.org/wiki/%E3%82%A2%E3%82%A4%E3%83%B3%E3%82%B7%E3%83%A5%E3%82%BF%E3%82%A4%E3%83%B3%E6%96%B9%E7%A8%8B%E5%BC%8F 一般相対性理論におけるアインシュタイン方程式（英: Einstein's equations, Einstein Field Equations）[注 1]は、万有引力・重力場を記述する場の方程式である。アルベルト・アインシュタインによって導入された。
https://ja.wikipedia.org/wiki/%E3%82%A2%E3%82%A4%E3%83%B3%E3%82%B7%E3%83%A5%E3%82%BF%E3%82%A4%E3%83%B3%E5%A4%9A%E6%A7%98%E4%BD%93 微分幾何と数理物理において、アインシュタイン多様体(Einstein m
https://ja.wikipedia.org/wiki/%E3%83%98%E3%83%AB%E3%83%9E%E3%83%B3%E3%83%BB%E3%83%AF%E3%82%A4%E3%83%AB ヘルマン・クラウス・フーゴー・ワイル（Hermann Klaus Hugo Weyl, 1885年11月9日 - 1955年12月8日） ワイルは空間、時間、物質、哲学、論理、対称性、数学史など、多岐に渡る分野について多くの論文と著書を残した。彼は一般相対性理論と電磁気学を結び付けようとした最初の人物の一人であり、アンリ・ポアンカレやヒルベルトの唱えた'普遍主義'について、同時代の誰よりも深く理解していた。特にマイケル・アティヤは、数学上の問題に取り組む際、常にワイルが先行する研究を行っていたと述懐している[1]。
https://ja.wikipedia.org/wiki/%E3%83%98%E3%83%AB%E3%83%9E%E3%83%B3%E3%83%BB%E3%83%AF%E3%82%A4%E3%83%AB ヘルマン・クラウス・フーゴー・ワイル（Hermann Klaus Hugo Weyl, 1885年11月9日 - 1955年12月8日） 参考文献 一次資料 ・1918. Raum, Zeit, Materie. 5 edns. to 1922 ed. with notes by Jurgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922 Space Time Matter, Methuen, rept. 1952 Dover. ISBN 0-486-60267-2.
（上記 Space Time Matter 英語版 下記にＰＤＦあり） https://archive.org/details/spacetimematter00weyluoft Space-time-matter by Weyl, Hermann, 1885-1955; Brose, Henry Herman Leopold Adolf, 1890- Publication date [1922] Topics Relativity (Physics), Space and time Publisher London, Methuen & co. ltd Collection gerstein; toronto Digitizing sponsor MSN Contributor Gerstein - University of Toronto Language English
（参考） https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory Yang?Mills theory History and theoretical description This eventually became the Yang?Mills theory, as Mills himself discussed: "During the academic year 1953-1954, Yang was a visitor to Brookhaven National Laboratory...I was at Brookhaven also...and was assigned to the same office as Yang. Yang, who has demonstrated on a number of occasions his generosity to physicists beginning their careers, told me about his idea of generalizing gauge invariance and we discussed it at some length...I was able to contribute something to the discussions, especially with regard to the quantization procedures, and to a small degree in working out the formalism; however, the key ideas were Yang's."[3] In early 1954, Yang and Mills extended the concept of gauge theory for abelian groups, e.g. quantum electrodynamics, to non-abelian groups to provide an explanation for strong interactions.[4] Similar work was done independently in January 1954 by Ronald Shaw, a graduate student at the University of Cambridge.[5]
Since no such massless particles were known at the time, Shaw and his supervisor Abdus Salam chose not to publish their work,[5] while Pauli criticized Yang's presentation of his work with Mills in February 1954.[6] Shortly after Yang and Mills published their paper in October 1954, Salam encouraged Shaw to publish his work to mark his contribution. Shaw declined, and instead it only forms a chapter of his PhD thesis published in 1956.[7][8] The idea was set aside until 1960, when the concept of particles acquiring mass through symmetry breaking in massless theories was put forward, initially by Jeffrey Goldstone, Yoichiro Nambu, and Giovanni Jona-Lasinio.
経歴 1916年8月28日 静岡県静岡市に生まれる 静岡県立静岡中学校、旧制静岡高等学校を経て大阪帝国大学に進学 1940年3月に大阪帝国大学理学部物理学科を卒業と同時に同大学副手に。その後、爾来助手、講師、助教授 1951年 理学博士（大阪大学）「On the covariant formalism of the quantum theory of fields(素粒子論、場の量子論の共変形式)」 1954年5～6月頃、楊振寧、ロバート・ミルズとは別に一般ゲージ理論の研究を完成させ、京大基礎物理学研究所で開催された小さな研究会で口頭発表していたが、1954年10月の楊（ノーベル物理学賞受賞者）とミルズの論文に対して発表が遅れたためにプライオリティは得られなかった 1954年8月からプリンストン高等研究所に研究員として渡米し、場の理論の発展に努めた (引用終り) 以上
Reflexions sur la resolution algebrique des equations, 1771. Lagrange https://fr.wikipedia.org/wiki/Joseph-Louis_Lagrange Principales publications Reflexions sur la resolution algebrique des equations, 1771. Ce memoire a inspire Abel et Galois.
sites.mathdoc.fr/cgi-bin/oeitem?id=OE_LAGRANGE__3_205_0 Gallica-Math: ?uvres completes Joseph Louis de Lagrange Reflexions sur la resolution algebrique des equations Document (Gallica) ?uvres completes, tome 3, 205-421 (volume) Nouveaux memoires de l'Academie royale des sciences et belles-lettres de Berlin, annees 1770 et 1771 ・Section premiere. De la resolution des equations du troisieme degre 207-254 | Document ・Section seconde. De la resolution des equations du quatrieme degre 254-304 | Document ・Section troisieme. De la resolution des equations du cinquieme degre et des degres ulterieurs 305-355 | Document ・Section quatrieme. Conclusion des reflexions precedentes, avec quelques remarques generales sur la transformation des equations, et sur leur reduction ou abaissement a un moindre degre 355-421 | Document gallica.bnf.fr/ark:/12148/bpt6k229222d/f208
History Lagrange himself did not prove the theorem in its general form. He stated, in his article Reflexions sur la resolution algebrique des equations,[3] that if a polynomial in n variables
768 名前：has its variables permuted in all n! ways, the number of different polynomials that are obtained is always a factor of n!. (For example, if the variables x, y, and z are permuted in all 6 possible ways in the polynomial x + y - z then we get a total of 3 different polynomials: x + y - z, x + z - y, and y + z - x. Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetric group Sn of the subgroup H of permutations that preserve the polynomial. (For the example of x + y - z, the subgroup H in S3 contains the identity and the transposition (x y).) So the size of H divides n!. With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.
In his Disquisitiones Arithmeticae in 1801, Carl Friedrich Gauss proved Lagrange's theorem for the special case of (Z/pZ)^*, the multiplicative group of nonzero integers modulo p, where p is a prime.[4] In 1844, Augustin-Louis Cauchy proved Lagrange's theorem for the symmetric group Sn.[5]
Camille Jordan finally proved Lagrange's theorem for the case of any permutation group in 1861.[6] []
https://www.アマゾン 量子力学 (物理学叢書 2) by シッフ, 井上健 (翻訳) 吉岡商店 1983
Review k野 4.0 out of 5 stars 個人的にはとても良い本で名著だと思う。 Reviewed in Japan on June 6, 2017 基礎がしっかり入ってこのサイズはなかなかの物だと思う。 学部レベルの量子論をしっかり流れるように読めるように成りたいと考えるならこの本を読めればまず間違いない。 しかし入門に適しているか？と聞かれたら不適と言わざるを得ない、 一通り他の入門書をやってからじゃないと読むのがキツイと思う、特に量子論で一般的な式の詳しい導出や解法は載っていないからだ。 予備知識が無いとそこで大幅に時間を取ってしまいモチベーションが保てなく成る。 私は[よくわかる量子力学]で一通り導出や基本的な解の捉え方を学んだ後に読み始めた、他にも色々量子論の本を漁ったが[よくわかる量子力学]が一番個人的な波長が合った、大学の教科書が役に立たなかったのでこの二冊でやりきった。
https://en.wikipedia.org/wiki/Leonard_I._Schiff Leonard Isaac Schiff was born in Fall River, Massachusetts, on March 29, 1915[1] and died on January 21, 1971, in Stanford, California. He was a physicist best known for his book Quantum Mechanics,[2][3] originally published in 1949 (a second edition appeared in 1955 and a third in 1968).
References 2. "Archived copy" (PDF). Archived from the original (PDF) on 2010-07-07. Retrieved 2009-10-30. 3. Seitz, Frederick (1950). "Review: L. I. Schiff, Quantum Mechanics". Bull. Amer. Math. Soc. 56 (2): 191?192. doi:10.1090/s0002-9904-1950-09377-x.
https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/55563/1/Kohei_Umeta_abstract.pdf Author(s) 梅田, 耕平 Citation 北海道大学. 博士(理学) 甲第11363号 Issue Date 2014-03-25 学位論文内容の要旨 (指数型正則関数の層に対する楔の刃の定理とラプラス超関数) 一変数ラプラス超関数の理論は、小松彦三郎氏により確立され常微分方程式及び偏微分 方程式の解法等に応用されている。ラプラス超関数とは無限遠方で高々指数増大する正則 関数の実軸の上下からの境界値の差として表わされる。元来、古典的なラプラス変換は無 限遠方で高々指数増大する関数に対して定義された。1987 年、小松彦三郎氏はラプラス超 関数を導入し、そのラプラス変換を構成する事によりすべての佐藤超関数はラプラス超関 数に拡張可能であることを示した。そのおかげで、我々は超関数の枠組みの中で任意の増 大度を持つ関数に対してもラプラス変換を扱うことが出来るようになった。この理論をさ らに発展させるには、ラプラス超関数の概念を局所化することでその代数的取り扱いを可 能とすることが望まれる。そこで、まずはじめに筆者は本多尚文氏との共著論文 ” On the sheaf of Laplace hyperfunctions with holomorphic parameters” の中で無限遠方で指数 型の増大度条件を持つ正則関数に対する擬凸領域上のコホモロジー群の消滅定理を示した。 その結果により、一変数ラプラス超関数のコホモロジー的な定義を与え代数的な取扱いを 可能とした。本論文では、無限遠方で指数型の増大度条件を持つ正則関数の層に対する楔 の刃定理について述べる。この定理は多変数ラプラス超関数の層を構成する上で本質的な 役割を果たす。以下、簡単に説明する。
https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/55565/1/Kohei_Umeta.pdf Title The edge of the wedge theorem for the s
779 名前：heaf of holomorphic functions of exponential type and Laplace hyperfunctions Author(s) 梅田, 耕平 Citation 北海道大学. 博士(理学) 甲第11363号 Issue Date 2014-03-25 []
https://arxiv.org/pdf/1604.00738.pdf Contemporary Mathematics Volume 703, 2018 dx.doi.org/10.1090/conm/703/14134 Inose’s construction and elliptic K3 surfaces with Mordell-Weil rank 15 revisited Abhinav Kumar and Masato Kuwata
References [I1] Hiroshi Inose, On certain Kummer surfaces which can be realized as non-singular quartic surfaces in P3, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 545?560.MR0429915 [I2] Hiroshi Inose, Defining equations of singular K3 surfaces and a notion of isogeny, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, 1978, 495?502. MR0578868 [SI] Tetsuji Shioda and Hiroshi Inose, On singular K3 surfaces, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 119?
792 名前：136. MR0441982
http://gcoe.math.kyoto-u.ac.jp/ 京都大学 グローバル COEプログラム http://gcoe.math.kyoto-u.ac.jp/product/chukanhokoku.html HOME >> 研究成果 >>中間報告書・外部評価報告書 http://gcoe.math.kyoto-u.ac.jp/docs/2010chukanhokoku.pdf 中間報告書2010 はじめに Global COE プロジェクト「数学のトップリーダーの育成」を開始してから，１年半が 経過した．国際交流などによるコア数学における研究者育成と多様な人材育成の２つの事業を行っている． 【講演】 Cohomologically trivial involutions of Enriques surfaces and Shioda-Inose correspondence, “Algebraic Geometry”, Lorentz Center, Leiden, the Netherland, July 2, 2008 (International conference) Homologically trivial involutions and Shioda-Inose correspondences for U+U(2), “Moduli and Discrete Groups”, Res. Inst. Math. Sci., Kyoto Univ., June 10, 2009 (International conference) []
文 献 [20] A. Kaneko, On continuation of regular soluti ons of partial differential equations to compact convex sets, J. Fac. Sci. Univ. Tokyo, 17 (1970), 567-580.-, Ibid. II. Ibid .,18(1971),416-433. [21) -,Fundamental principle and extension of solutions of partial differential equations with constant coefficients, Hyperfunctions and Pseudo- differential Equations, Part I, Proceedings of a Conference at Katada,1971, Springer, Lecture Notes in Mathematics, to appear.
https://www.アマゾン 百万人の数学 上 - December 18, 2015 by ランスロット・ホグベン (著), 久村 典子 (翻訳) レビュー Enriques_Castelnuovo 3.0 out of 5 stars ヨコのものをタテにしただけの新訳 Reviewed in Japan on June 20, 2019
https://www.アマゾン 百万人の数学〈上〉 (1969年) (筑摩叢書) Tankobon Hardcover by L.ホグベン (著), 今野 武雄 (翻訳) レビュー Enriques_Castelnuovo 4.0 out of 5 stars フィールズ賞のマンフォードも称賛 Reviewed in Japan on July 21, 2019 フィールズ賞（1974）受賞者である 英国生まれの米国の数学者 デーヴィッド・マンフォード氏 （1937ー）が ホグベン『百万人の数学』に対し 賛辞（tribute）を述べている由です。
https://en.wikipedia.org/wiki/Lancelot_Hogben Lancelot Thomas Hogben FRS[1] FRSE (9 December 1895 ? 22 August 1975) British experimental zoologist and medical statistician.
Popular science writing Hogben produced two best-selling works of popular science, Mathematics for the Million (1936) and Science for the Citizen (1938). Mathematics for the Million received widespread praise, with H. G. Wells saying that "Mathematics for the Million is a great book, a book of first-class importance".[20] The book was also lauded by Albert Einstein, Bertrand Russell and Julian Huxley.[20][21]Mathematics for the Million was reprinted after Hogben's death.[21]
References 21 "Mathematics for the Million...praised by Einstein, H. G. Wells and others, it was reprinted in paperback in 1993." De Smith, Michael John, Maths for the Mystified : An Exploration of the History of Mathematics and Its Relationship to Modern-Day Science and Computing.Leicester : Matador, 2006. (p.192) (引用終り) 以上
https://en.wikipedia.org/wiki/Platonism Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all doctrines of Plato.[1] Philosophy The primary concept is the Theory of Forms. The only true being is founded upon the forms, the eternal, unchangeable, perfect types, of which particular objects of moral and responsible sense are imperfect copies.
（参考：裏付けがとれなかったので、都市伝説かも） sasaki193.seesaa.net/article/422228729.html (有)佐々木石材工業 2015年07月12日 【名言】アンドリュー・カーネギーの墓碑に刻まれた文章 アンドリュー・カーネギーの墓碑に刻まれた文章はこちら 【己より賢き者を近づける術知りたる者、ここに眠る。】 【Here lies one who knew how to get around him men who were cleverer than himself.】
これ、良く纏まっているね 「1851年の論文でシルベスターは >I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered as from the womb of a common parent. >（以前の論文で、項を矩形状に並べた配列として定義した "Matrix" は、そのうちで異なる行列式の体系を生み出す共通の親としての母体である。） と説明している」 なるほどね
1851年の論文でシルベスターは >I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered as from the womb of a common parent. >（以前の論文で、項を矩形状に並べた配列として定義した "Matrix" は、そのうちで異なる行列式の体系を生み出す共通の親としての母体である。） と説明している。
本講義は微分方程式の実用的側面を毎年テーマを選んで解説するものであり, 本年度のテーマは有限要素法とする. 有限要素法とは，一言でいえば領域を三角形など簡単な形状を持った要素に分割して， 区分一次函数などの初等的な基底を用いた線型代数の計算で，難しい偏微分方程式の 問題をすいすい解いてしまおうというものである. 本講義ではおおむね C. Johnson 著 『Numerical solution of partial differential equations by the finite element method』(Cambridge University Press) に基づき, この理論の基礎的部分を解説する. だいたい同書の第７章くらいまでを目標とし, 楕円型の境界値問題については ほぼ一通りの知識を得ることを目指す. これに実際のプログラミングの解説を補って実習もしてもらう予定である.
History Matrices have a long history of application in solving linear equations but they were known as arrays until the 1800s. The Chinese text The Nine Chapters on the Mathematical Art written in 10th?2nd century BCE is the first example of the use of array methods to solve simultaneous equations,[103] including the concept of determinants. In 1545 Italian mathematician Gerolamo Cardano introduced the method to Europe when he published Ars Magna.[104] The Japanese mathematician Seki used the same array methods to solve simultaneous equations in 1683.[105]
References 18 C. Birkar, P. Cascini, C. Hacon, J. McKernan Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010)
https://arxiv.org/abs/math/0610203 [Submitted on 5 Oct 2006 (v1), last revised 14 Aug 2008 (this version, v2)] Existence of minimal models for varieties of log general type Caucher Birkar, Paolo Cascini, Christopher D. Hacon, James McKernan
https://www.ams.org/journals/jams/2010-23-02/S0894-0347-09-00649-3/S0894-0347-09-00649-3.pdf JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 23, Number 2, April 2010, Pages 405?468 EXISTENCE OF MINIMAL MODELS FOR VARIETIES OF LOG GENERAL TYPE CAUCHER BIRKAR, PAOLO CASCINI, CHRISTOPHER D. HACON,AND JAMES MCKERNAN Contents 1. Introduction 1.1. Minimal models 1.3. Fano varieties 1.4. Birational geometry 2. Description of the proof 2.2. Standard conjectures of the MMP 3. Preliminary results 4. Special finiteness 5. Log terminal models
>>787 追加 https://www.ams.org/journals/jams/2010-23-02/S0894-0347-09-00649-3/S0894-0347-09-00649-3.pdf JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 23, Number 2, April 2010, Pages 405?468 EXISTENCE OF MINIMAL MODELS FOR VARIETIES OF LOG GENERAL TYPE CAUCHER BIRKAR, PAOLO CASCINI, CHRISTOPHER D. HACON,AND JAMES MCKERNAN
P2 1. Introduction The purpose of this paper is to prove the following result in birational algebraic geometry: Theorem 1.1. Let (X, Δ) be a projective Kawamata log terminal pair. (引用終り)
https://ja.wikipedia.org/wiki/%E8%82%A5%E7%94%B0%E6%99%B4%E4%B8%89 肥田 晴三（はるぞう 1952年8月6日）は、日本の数学者で、数論・代数幾何学・モジュラー形式の研究で著名 経歴 肥田は、京都大学から1975年学士号を、1977年修士号を、1980年「志村曲線のヤコビアンの因子としての虚数乗法を持つアーベル多様体」(On Abelian Varieties with Complex Multiplication as Factors of the Jacobians of Shimura Curves)の論文により博士号を得た[1]。1987年からカリフォルニア大学ロサンゼルス校で教授を務めている。1979年から1981年までプリンストン高等研究所の訪問研究者だった。 1986年肥田は、バークレーの国際数学者会議の招待講演者だった。1991年、肥田はグッゲンハイム・フェローシップを受賞した[2]。1992年、代数群のp進L-関数とp進ヘッケ環に関する研究に対して、日本数学会の春季賞を肥田は受賞した。[3] アンドリュー・ワイルズによるフェルマーの最終定理の証明には、肥田理論が使用されている。
https://ja.wikipedia.org/wiki/%E6%9C%9B%E6%9C%88%E6%8B%93%E9%83%8E 望月 拓郎（1972年8月28日） 来歴 生い立ち 長野県長野市出身[2]。長野県長野高等学校を卒業し、京都大学に進学した[1]。理学部にて学んでいたが[1]、在学中にトポロジーの本を読み[3]、「計算で答えを出す高校までの数学からガラッと変わった」[3] と述懐している。大学院の理学研究科に飛び入学で進学するため、1994年（平成6年）に理学部を中途退学した[1]。1996年（平成8年）、京都大学の大学院における修士課程を修了した[1]。それに伴い、修士（理学）の学位を取得した。大学院在学中に「Gromov-Witten class and a perturbation theory in algebraic geometry」[4] と題した博士論文を執筆した
このことは、極小モデルの考え方を導く。各々の双有理同値類の中に一意に最も小さい代数多様体を見つけることは可能か？ 現代の定義は、射影的多様体 X が極小とは、標準ラインバンドル KX が X のすべての曲線で非負な次数を持つことである。言い換えると、KX はネフ（数値的正という意味だが、通常使用しているので、本文ではネフという用語を使用する。）[1]である。ブローアップした多様体が決して極小ではありえないことは、容易にチェックできる。
この考え方は、代数曲線（次元が 2 の多様体）に対しては完全に成り立つ。現代のことばでは、1890年から1910年までの代数幾何学のイタリア学派（英語版）の一つの中心的な結果は、曲面の分類の一部とあわせ、すべての曲面 X は、ある曲線 C が存在して積 P1 × C か、もしくは極小曲面 Y のどちらかに双有理同値である。[2] 2つの場合は互いに排他的であり、Y は存在するとしたら一意である。Y が存在すると、X の極小モデルと呼ばれる。
双有理幾何学の基本予想 双有理幾何学の基本予想とは、多重標準環は有限生成（英語版）であろうという予想である。このことは森プログラム（英語版）の大きな一つのステップと考えられている。 Caucher Birkar, Paolo Cascini, and Christopher D. Hacon et al. (2010) Yum-Tong Siu (2006) はこの証明をしたことをアナウンスした。
For Perelman's generalization of Thurston's geometrization theorem to all 3-manifolds, see Geometrization conjecture. In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.
Statement One form of Thurston's geometrization theorem states: If M is a compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of M has a complete hyperbolic structure of finite volume.
The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique.
https://en.wikipedia.org/wiki/Atoroidal Atoroidal
In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non-boundary parallel, incompressible torus, or it may be defined algebraically, as a subgroup Z ｘ Z of its fundamental group that is not conjugate to a peripheral subgroup (i.e., the image of the map on fundamental group induced by an inclusion of a boundary component). The terminology is not standardized, and different authors require atoroidal 3-manifolds to satisfy certain additional restrictions. For instance: 略 A 3-manifold that is not atoroidal is called toroidal. (引用終り) 以上
ここは下記の英語の翻訳らしい Siu (2002) proved the invariance of plurigenera under deformations for all smooth complex projective varieties. In particular, the Kodaira dimension does not change when the complex structure of the manifold is changed continuously.
https://en.wikipedia.org/wiki/Abundance_conjecture Abundance conjecture In algebraic geometry, the abundance conjecture is a conjecture in birational geometry, more precisely in the minimal model program, stating that for every projective variety X with Kawamata log terminal singularities over a field k if the canonical bundle K_{X} is nef, then K_{X} is semi-ample. Important cases of the abundance conjecture have been proven by Caucher Birkar.[1]
https://en.wikipedia.org/wiki/Canonical_singularity#Pairs Canonical singularity They were introduced by Reid (1980). Terminal singularities are important in the minimal model program because smooth minimal models do not always exist, and thus one must allow certain singularities, namely the terminal singularities. Pairs ・klt (Kawamata log terminal) if Discrep(X,Δ)>?1 and [Δ]<= 0
969 名前： for all i （下記 標準特異点関連） ・log resolution of D (e.g., Hironaka's resolution) ・下記FUJINOより log terminal singularities is divisorial log terminal (dlt, for short) Shokurov （Hironaka’s desingularization theorem suitably）
https://ja.wikipedia.org/wiki/%E6%A8%99%E6%BA%96%E7%89%B9%E7%95%B0%E7%82%B9 標準特異点 https://en.wikipedia.org/wiki/Canonical_singularity Canonical singularity In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal models. They were introduced by Reid (1980). Terminal singularities are important in the minimal model program because smooth minimal models do not always exist, and thus one must allow certain singularities, namely the terminal singularities. Definition Then the singularities of Y are called: terminal if ai > 0 for all i canonical if ai >= 0 for all i log terminal if ai > -1 for all i log canonical if ai >= -1 for all i. See also: multiplier ideal (algebraic geometry)
https://en.wikipedia.org/wiki/Multiplier_ideal Multiplier ideal Algebraic geometry In algebraic geometry, the multiplier ideal of an effective Q -divisor measures singularities coming from the fractional parts of D. Multiplier ideals are often applied in tandem with vanishing theorems such as the Kodaira vanishing theorem and the Kawamata?Viehweg vanishing theorem. Let X be a smooth complex variety and D an effective Q -divisor on it. Let μ :X'→ X be a log resolution of D (e.g., Hironaka's resolution).
下記FUJINOより抜粋 P6 5. Resolution Lemma We think that one of the most useful log terminal singularities is divisorial log terminal (dlt, for short), which was introduced by Shokurov (see [FA, (2.13.3)]). We defined it in Definition 4.1 above. By Szab´o’s work [Sz], the notion of dlt coincides with that of weakly Kawamata log terminal (wklt, for short). P7 By combining Theorem 5.1 with the usual desingularization arguments, we can recover the original Resolution Lemma without any difficulties. This means that, first, we use Hironaka’s desingularization theorem suitably, next, we apply Theorem 5.1 below, then we can recover Szab´o’s results.
www.math.nagoya-u.ac.jp/~fujino/what-HP.pdf WHAT IS LOG TERMINAL ? 2004/4/23 OSAMU FUJINO Abstract. In this paper, we explain the subtleties of various kinds of log terminal singularities. We focus on the notion of divisorial log terminal singularities, which seems to be the most useful one. We explain Szab´o’s resolution lemma, the notion of log resolution, adjunction formula for divisorial log terminal pairs, and so on. We also collect miscellaneous results and examples on singularities of pairs in the log MMP that help us understand log terminal singularities.
Contents 1. What is log terminal? 1 2. Preliminaries on Q-divisors 3 3. Singularities of pairs 5 4. Divisorial log terminal 6 5. Resolution Lemma 6 6. Whitney umbrella 8 7. What is a log resolution? 10 8. Examples 12 9. Adjunction for dlt pairs 14 10. Miscellaneous comments 15 References 16 (引用終り) 以上
Y.-T. Siu, Invariance of plurigenera, Invent.Math. 134 (1998), no. 3, 661?673. https://people.math.harvard.edu/~siu/siu_reprints/siu_plurigenera_invent1998.pdf Invent. math. 134, 661-673 (1998) DOI 10.1007/s002229800870 Invariance of plurigenera Yum-Tong Siu* Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
P2 multiplier ideal sheaf
>>857再録 https://en.wikipedia.org/wiki/Multiplier_ideal Multiplier ideal In commutative algebra, the multiplier ideal associated to a sheaf of ideals over a complex variety and a real number c consists (locally) of the functions h such that |h|^2/Σ|fi^2|^c is locally integrable, where the fi are a finite set of local generators of the ideal. Multiplier ideals were independently introduced by Nadel (1989) (who worked with sheaves over complex manifolds rather than ideals) and Lipman (1993), who called them adjoint ideals.
Multiplier ideals are discussed in the survey articles Blickle & Lazarsfeld (2004), Siu (2005), and Lazarsfeld (2009). Algebraic geometry In algebraic geometry, the multiplier ideal of an effective Q -divisor measures singularities coming from the fractional parts of D. Multiplier ideals are often applied in tandem with vanishing theorems such as the Kodaira vanishing theorem and the Kawamata?Viehweg vanishing theorem. Let X be a smooth complex variety and D an effective Q -divisor on it. Let μu :X'→ X be a log resolution of D (e.g., Hironaka's resolution). (引用終り) 以上
Extension of Twisted Pluricanonical Sections with Plurisubharmonic Weight and Invariance of Semipositively Twisted Plurigenera for Manifolds Not Necessarily of General Type January 2002 DOI: 10.1007/978-3-642-56202-0_15 Yum-Tong SiuYum-Tong Siu
>>880 下記は、論文でなく ”A short course on multiplier ideals”のレクチャーらしい ざっと読んだけど、ほとんど分からなかったｗ
けど、Introduction読むと、 ”The revolutionary work of Hacon-McKernan, Takayama and Birkar-Cascini-Hacon-McKernan ([14], [15], [28], [3]) ” とあるから、流れは合っているね
https://arxiv.org/abs/0901.0651 [Submitted on 6 Jan 2009]
1010 名前：A short course on multiplier ideals Robert Lazarsfeld These notes are the write-up of my 2008 PCMI lectures on multiplier ideals. They aim to give an introduction to the algebro-geometric side of the theory, with an emphasis on its global aspects. The focus is on concrete examples and applications. The lectures take into account a number of recent perspectives, including adjoint ideals and the resulting simplifications in Siu's theorem on plurigenera in the general type case. While the notes refer to my book [PAG] and other sources for some technical points, the conscientious reader should arrive at a reasonable grasp of the machinery after working through these lectures. https://arxiv.org/pdf/0901.0651.pdf
Introduction These notes are the write-up of my 2008 PCMI lectures on multiplier ideals. They aim to give an introduction to the algebro-geometric side of the theory, with an emphasis on its global aspects. Besides serving as warm-up for the lectures of Hacon, my hope was to convey to the audience a feeling for the sorts of problems for which multiplier ideals have proved useful. Thus I have focused on concrete examples and applications at the expense of general theory. While referring to [21] and other sources for some technical points, I have tried to include sufficient detail here so that the conscientious reader can arrive at a reasonable grasp of the machinery by working through these lectures.
The revolutionary work of Hacon-McKernan, Takayama and Birkar-Cascini-Hacon- McKernan ([14], [15], [28], [3]) appeared shortly after the publication of [21], and these papers have led to some changes of perspectives on multiplier ideals. In particular, the first three made clear the importance of adjoint ideals as a tool in proving extension theorems; these were not so clearly in focus at the time [21] was written. I have taken this new viewpoint into account in discussing the restriction theorem in Lecture 3. Adjoint ideals also open the door to an extremely transparent presentation of Siu’s theorem on deformation-invariance of plurigenera of varieties of general type, which appears in Lecture 5. (引用終り) 以上
1. Construction and Examples of Multiplier Ideals This preliminary lecture is devoted to the construction and first properties of multiplier ideals. We start by discussing the algebraic and analytic incarnations of these ideals. After giving the example of monomial ideals, we survey briefly some of the invariants of singularities that can be defined via multiplier ideals.
（monomial ideal） https://en.wikipedia.org/wiki/Monomial_ideal Monomial ideal In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. A toric ideal is an ideal generated by differences of monomials (provided the ideal is a prime ideal). An affine or projective algebraic variety defined by a toric ideal or a homogeneous toric ideal is an affine or projective toric variety, possibly non-normal.
University of Hong Kongか ヤウ先生と同じか・・ 1943生まれだと、ヤウ先生の先輩なんだ！ 知らなかったな、素人なので（苦笑）
https://en.wikipedia.org/wiki/Yum-Tong_Siu Yum-Tong Siu Yum-Tong Siu (Chinese: 蕭蔭堂; born May 6, 1943 in Guangzhou, China) is the William Elwood Byerly Professor of Mathematics at Harvard University. Siu is a prominent figure in the study of functions of several complex variables. His research interests involve the intersection of complex variables, differential geometry, and algebraic geometry. He has resolved various conjectures by applying estimates of the complex Neumann problem and the theory of multiplier ideal sheaves to algebraic geometry.[1][2] Education and career Siu obtained his B.A. in mathematics from the University of Hong Kong in 1963, his M.A. from the University of Minnesota, and his Ph.D. from Princeton University in 1966.[3]
1029 名前：Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds Jean-Pierre Demailly, János Kollár
https://doi.org/10.48550/arXiv.math/9910118
§4. Multiplier ideal sheaves and holomorphic approximations of psh singularities The most important concept relating psh functions to holomorphic objects is the concept of multiplier ideal sheaf, which was already considered implicitly in the work of Bombieri [Bom70], Skoda [Sko72] and Siu [Siu74]. The precise final formalization has been fixed by Nadel [Nad89]. []
Advances in Mathematics Volume 409, Part A, 19 November 2022, 108640 Advances in Mathematics Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic Tatsuro Kawakami
（参考） https://en.wikipedia.org/wiki/Vyacheslav_Shokurov Vyacheslav Shokurov born 18 May 1950
Work on birational geometry
Log flips
in 1985, Shokurov published a paper titled The nonvanishing theorem, which became a cornerstone for the whole MMP as it was used in the proofs of such fundamental theorems as the Cone theorem and the Semi-ampleness theorem. Also in this paper, Shokurov proved the termination of three-dimensional flips. And even though he proved this only for three-dimensional varieties, most of his techniques were later generalized by Yujiro Kawamata to obtain similar results for varieties of any dimension.
One of Shokurov's ideas formed a basis for a paper titled 3-fold log flips
1046 名前：where the existence of three-dimensional flips (first proved by Shigefumi Mori) was established in a more general log setting. The inductive method and the singularity theory of log pairs developed in the framework of that paper allowed most of the paper's results to be later generalized to arbitrary-dimensional varieties. Later on, in 2001, Shokurov announced the proof of the existence of 4-dimensional log flips, whose complete version appeared in two books: Flips for 3-folds and 4-folds and Birational geometry: linear systems and finitely-generated algebras. An application of Shokurov's ideas concerning the existence of log flips has led to the paper Existence of minimal models for varieties of log general type by Caucher Birkar, Paolo Cascini, Christopher Hacon and James McKernan.
P5 略はDeligneのHodge理論[1]において詳しく研究され,たと えば,これらの元はd閉型式であることが示され た。さて,われわれの双有理幾何では,もう少し 一般な層を考察する方がよい.すなわち,M= (m1,…,m%)を非負整数の組とし,Ω^1(1ogD)のm1 回Ov上テンソル積,Ω^2(logD)のm2回テンソル 積,…,…mn.回テンソル積を考え,これらのす べてのテンソル積をΩ(logD)^Mとかき, 略 文 献 [1 ] P. Deligne, Theorie de 1-lodge II, Publ. Math. de I. H. P. S. N., 40 (1973), 5-57. []
Logarithmic vanishing theorems on compact K¨ahler manifolds I
Chunle Huang, Kefeng Liu, Xueyuan Wan, and Xiaokui Yang
この論文によれば
The basic properties of the sheaf of logarithmic differential forms and of the sheaves with logarithmic integrable connections on smooth projective manifolds were developed by Deligne in [5].
X を複素多様体とし、D ⊂ X を因子、ω を X?D 上の正則 p-形式とする。ω と dω が D に沿って大きくとも 1 の位数の極を持つとき、ω を D に沿って対数的極を持つという。ω は対数的 p-形式とも呼ばれる。対数的 p-形式はD に沿った X 上の有理 p-形式の層をなし、次のように書く。 Ω^p X(log D). リーマン面の理論では、次の局所表現を持つ対数的 1-形式が存在する。
ここに g は 0 で正則で 0 とはならなく、m は f の 0 でのオーダーである。すなわち、ある開被覆が存在し、この微分形式の対数微分としての局所表現が存在する（通常の微分作用素 d/dz の中の外微分 d を少し変形する）。ω が整数の留数の単純極を持つだけであることに注意する。高次元の複素多様体では、ポアンカレ留数（英語版）(Poincare residue)は、極に沿った対数的微分形式の振る舞いを記述することに使われる。
https://en.wikipedia.org/wiki/Logarithmic_form Logarithmic form
In algebraic geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept was introduced by Pierre Deligne.[1] In short, logarithmic differentials have the mildest possible singularities needed in order to give information about an open submanifold (the complement of the divisor of poles). (This idea is made precise by several versions of de Rham's theorem discussed below.)
Historical terminology In the 19th-century theory of elliptic functions, 1-forms with logarithmic poles were sometimes called integrals of the second kind (and, with an unfortunate inconsistency, sometimes differentials of the third kind).
For example, the Weierstrass zeta function associated to a lattice Λ in C was called an "integral of the second kind" to mean that it could be written
ζ(z)=σ '(z)/σ(z) In modern terms, it follows that ζ(z)dz=σ(z)/σ is a 1-form on C with logarithmic poles on Λ , since Λ is the zero set of the Weierstrass sigma function σ(z).
Mixed Hodge theory for smooth varieties Over the complex numbers, Deligne proved a strengthening of Alexander Grothendieck's algebraic de Rham theorem, relating coherent sheaf cohomology with singular cohomology.
Notes [1] Deligne (1970), section II.3. References Deligne, Pierre (1970), Equations differentielles a points singuliers reguliers, Lecture Notes in Mathematics, vol. 163, Springer-Verlag, doi:10.1007/BFb0061194, ISBN 3540051902, MR 0417174, OCLC 169357
あんたの言い方ならば、下記 >>948-949 Logarithmic form Historical terminology In the 19th-century theory of elliptic functions, 略 ζ(z)=σ '(z)/σ(z) In modern terms, it follows that ζ(z)dz=σ(z)/σ is a 1-form on C with logarithmic poles on Λ , since Λ is the zero set of the Weierstrass sigma function σ(z).
なるほど、ありがとう 下記だね https://en.wikipedia.org/wiki/C._P._Ramanujam Chakravarthi Padmanabhan Ramanujam (9 January 1938 ? 27 October 1974) was an Indian mathematician who worked in the fields of number theory and algebraic geometry.
Like his namesake Srinivasa Ramanujan, Ramanujam also had a very short life.[1] (google訳一部修正) 彼の同名のスリニバサ・ラマヌジャンのように、ラマヌジャムも非常に短命でした。[1]
Early life and education
Career He proceeded to write his thesis in 1966 and took his doctoral examination in 1967. Dr. Siegel, who was one of the examiners, was highly impressed with the young man's depth of knowledge and his great mathematical abilities. Ramanujam was a scribe for Igor Shafarevich's course of lectures in 1965 on minimal models and birational transformation of two-dimensional schemes. Professor Shafarevich subsequently wrote to say that Ramanujam not only corrected his mistakes but complemented the proofs of many results. The same was the case with Mumford's lectures on abelian varieties, which were delivered at TIFR around 1967. Mumford wrote in the preface to his book that the notes improved upon his work and that his current work on abelian varieties was a joint effort between him and Ramanujam.
Illness and death In 1964, based on his participation in the International Colloquium on Differential Analysis, he earned the respect of Alexander Grothendieck and of David Mumford, who invited him to Paris and Harvard. He accepted the invitation and was in Paris, but for a brief period.
He was diagnosed in 1964 with schizophrenia with severe depression and lef
1102 名前：t Paris for Chennai.
During one of the attacks, he tried to take his life, but was rescued in time. However, late one evening on 27 October 1974, after a lively discussion with a visiting foreign professor he took his life with an overdose of barbiturates.
1）３次方程式、４次方程式の場合の解説は、ガロアの第一論文にガロアの理論の応用として簡単に記載がある 勿論、第一論文の解説本（彌永、倉田、守屋など）などでは、詳しい解説ある というか、それって石井本にもあったろうよｗ 2）”この過程で、群の固有分解（現代用語で正規部分群）の概念に到達する” の部分は、遺稿のChevallierへの手紙>>110において ”正規部分群について明記している （This is called proper decomposition：G = H + H S + H S' + ・・とG = H +TH +T'H +・・とが一致するとき）” の部分だよ 上記同様、第一論文の解説本（彌永、倉田、守屋など）などでは、詳しい解説ある 3）そのガロア分解式から成る 120次の方程式と、もとの5次方程式は、代数的には等価だよ 5次方程式を解くこと、即ち ガロア分解式の120次の方程式を解くってこと これが、ガロア第一論文のキモです
