（1.3.2）设 是拓扑空间， 和 是连通集，如果 对任意 成立，则 连通

定义有限交性质（简写为f.i.p.）为对任意闭集族 ，若满足下述性质1则满足性质2 ：
1. 对任意有限集 ，
2.

f.i.p.无非紧性的逆否形式，因而自然等价于紧。

前言

2024.9.27 xzq告诉我这个无非循环表示法，发现确实如此

在试图更清楚地表示出置换群的群结构时（例如画出更好看的乘法表等），我发现一种将 中置换 表为一个 点 边的有向图的表示法，有时可以极大便利和直观许多关于置换的初等技术。

正文

考虑将 中置换 表为一个 点 边的有向图， 给出这 条边，例如 将被表为更进一步的，置换乘法如 可以被表示为直观考虑的话，例如在1处放一个质点，被 作用可以看作沿 中 箭头被打出，再沿 中 箭头被打出，而得到 ，如此观察立即可得上图。有时更简洁地，无歧义时画图会不写数字，自环不会画出来或者有时会写作一个点。

带着这种观点，不难注意到结果等同于 沿 的推出，也等同于 沿 的拉回。也就是说，置换 左乘作用相当于沿 的推出作用，右乘则对应拉回。

许多熟知的置换恒等式可以变得更一目了然，例如对只需注意到

设 ，共轭 给出图同构 ，从而 交换即 当且仅当 给出图 的自同构

只需注意到 是 沿 推出而同时沿 拉回。直观来讲， 使 中每个点带着边同时沿 运动，也就是所谓

是单群

要证正规子群平凡相当于要证共轭作用和子群运算总能生成出三循环。现在设 ，按循环分解分类讨论：

1. 如果 中包含对换乘积，那么将 换为 再乘起来便能得到三循环（懒得画图）。
2. 如果 中包含五循环，现在考虑下式里左图（即 ，以最上为1，顺时针方向计数）被箭头上的置换（即 ）共轭作用至右图事实上这里只需注意到被标了彩色的“棚子”部分，被三循环的共轭作用，带着连着棚子的两条边整个翻倒了一下，便立即可得右图。进而将右图的逆乘在（左乘右乘均可）左图上，因为右图和左图有两条有向边完全相同，这一作用会抵消这两条边，得到的置换就会有两个不动点，是三循环。事实上

关于局部线性

我们熟知有柯西-黎曼方程从从形式上看，对 在作为 可微时有其中 ， 时 ，则将 展开 分别给出 的全微分，从而便有C-R方程。换种写法便是的存在性意味着

更进一步的， 作为 映射的Jacobian矩阵为 。另一方面， 作为 意义下的局部线性是指 ，其中 ， 时 ，感性理解为 ，其几何意义即旋转与伸缩，从而复数乘法的作用与复平面上的一些实线性变换有对应也就是复数的矩阵表示带入Jacobian矩阵立即可得C-R方程。

事实上因为 与 均受同一 作用，自然就有 ，而就一种更直观的几何视角而言，考虑下图 绿、紫、红、蓝部分分别为 在 方向的投影（投影或者说在分量上的贡献） 、 在 方向的投影 、 在 方向的投影 、 在 方向的投影 ，则C-R方程无非三角函数的诱导公式（注意实际上这些微分都是线性泛函！）。

Wirtinger导数

定义Wirtinger导数事实上 的线性组合 满足的唯一解便是上述定义。
这里 是 的对偶基 ， 是 的对偶基。

此时C-R方程可以被简单地表示为而同时

为了解释考虑一个有启发性的例子。

设 ，多项式 总可以表为 与 的多项式因为 满足Leibniz律，所以 则 意味着 ，即 时 ， 是 的多项式，感性一点讲就是依赖于 而不依赖于 。这种事实也可以推广到无穷收敛幂级数的类似情形。

共形映射

Reference

Ahlfors - Complex Analysis

Borcherds的网课第五节

安抚

禅定

题破山寺后禅院 常建
清晨入古寺，初日照高林。
曲径通幽处，禅房花木深。
山光悦鸟性，潭影空人心。
万籁此俱寂，但余钟磬音。
> 空、透明、宁静、寂静的声音、柔和天光

送僧归日本 钱起
上国随缘住，来途若梦行。
浮天沧海远，去世法舟轻。
水月通禅寂，鱼龙听梵声。
惟怜一灯影，万里眼中明。
> 安宁、夜晚、安静的窃窃私语、轻轻的浪花声、灯火

寻隐者不遇 贾岛
松下问童子，言师采药去。
只在此山中，云深不知处。
> 雾气

鹿柴 王维
空山不见人，但闻人语响。
返景入深林，复照青苔上。

自然

饮酒其五 陶渊明
结庐在人境，而无车马喧。
问君何能尔？心远地自偏。
采菊东篱下，悠然见南山。
山气日夕佳，飞鸟相与还。
此中有真意，欲辨已忘言。

归园田居其一 陶渊明
少无适俗韵，性本爱丘山。
误落尘网中，一去三十年。
羁鸟恋旧林，池鱼思故渊。
开荒南野际，守拙归园田。
方宅十余亩，草屋八九间。
榆柳荫后檐，桃李罗堂前。
暧暧远人村，依依墟里烟。
狗吠深巷中，鸡鸣桑树颠。
户庭无尘杂，虚室有余闲。
久在樊笼里，复得返自然。

辛丑岁七月赴假还江陵夜行涂口 陶渊明
闲居三十载，遂与尘事冥。
诗书敦宿好，林园无世情。
如何舍此去，遥遥至南荆！
叩枻新秋月，临流别友生。
凉风起将夕，夜景湛虚明。
昭昭天宇阔，皛皛川上平。
怀役不遑寐，中宵尚孤征。
商歌非吾事，依依在耦耕。
投冠旋旧墟，不为好爵萦。
养真衡茅下，庶以善自名。
> 星辰、晶莹、清澈、清凉、夜色、明亮

积雨辋川庄作 王维
积雨空林烟火迟，蒸藜炊黍饷东菑。
漠漠水田飞白鹭，阴阴夏木啭黄鹂。
山中习静观朝槿，松下清斋折露葵。
野老与人争席罢，海鸥何事更相疑。
> 闲适、清新、亲近、自由、空阔、晶莹

辋川闲居赠裴秀才迪 王维
寒山转苍翠，秋水日潺湲。
倚杖柴门外，临风听暮蝉。
渡头馀落日，墟里上孤烟。
复值接舆醉，狂歌五柳前。

下终南山过斛斯山人宿置酒 李白
暮从碧山下，山月随人归。
却顾所来径，苍苍横翠微。
相携及田家，童稚开荆扉。
绿竹入幽径，青萝拂行衣。
欢言得所憩，美酒聊共挥。
长歌吟松风，曲尽河星稀。
我醉君复乐，陶然共忘机。

声音或行动

归嵩山作 王维
清川带长薄，车马去闲闲。
流水如有意，暮禽相与还。
荒城临古渡，落日满秋山。
迢递嵩高下，归来且闭关。

溪居 柳宗元 （断章取义）
久为簪组累，幸此南夷谪。
闲依农圃邻，偶似山林客。
晓耕翻露草，夜榜响溪石。
来往不逢人，长歌楚天碧。

估客昼眠知浪静，舟人夜语觉潮生。 （断章取义×2）

春宿左省 杜甫 （断章取义×2）
花隐掖垣暮，啾啾栖鸟过。
星临万户动，月傍九霄多。
不寝听金钥，因风想玉珂。
明朝有封事，数问夜如何。

Terence Tao - https://terrytao.wordpress.com/
David Mumford - https://www.dam.brown.edu/people/mumford/
Vladimir Voevodsky - https://www.math.ias.edu/vladimir/ https://www.math.ias.edu/Voevodsky/
Curtis T. McMullen - https://people.math.harvard.edu/~ctm/
J.S. Milne - https://www.jmilne.org/math/
László Babai - https://people.cs.uchicago.edu/~laci/
Barry Mazur - https://sites.harvard.edu/barry-mazur/
Andrew Granville - https://dms.umontreal.ca/~andrew/expository.php
Misha Gromov - https://www.ihes.fr/~gromov/
Robert P. Langlands - https://www.ias.edu/math/people/faculty/rpl
Anthony W. Knapp - https://www.math.stonybrook.edu/~aknapp/
Ravi Vakil - https://virtualmath1.stanford.edu/~vakil/
Brian Conrad - Number Theory
Richard P. Stanley - https://math.mit.edu/~rstan/
Ronald Brown - https://groupoids.org.uk/
David Kazhdan - http://www.ma.huji.ac.il/~kazhdan/
George Lusztig - https://math.mit.edu/~gyuri/
Nigel Hitchin - http://people.maths.ox.ac.uk/~hitchin/
Ehud Hrushovski - http://www.ma.huji.ac.il/~ehud/
János Kollár - https://web.math.princeton.edu/~kollar/
Maxim Kontsevich - https://www.ihes.fr/~maxim/
David A. Cox - https://dacox.people.amherst.edu/
Shlomo Sternberg - https://people.math.harvard.edu/~shlomo/
Yiannis N. Moschovakis - https://www.math.ucla.edu/~ynm/
Tsit Yuen Lam - https://math.berkeley.edu/~lam
Jacob Lurie
Alexander Grothendieck - https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/
- https://agrothendieck.github.io/
Paul Erdős (The Erdos Project) -
Camillo De Lellis
Ben Green - additive combinatorics, analytic number theory, harmonic analysis, ergodic theory, discrete geometry and group theory, Papers, Notes, Reviews, Videos
Gian-Carlo Rota (The Forbidden City of Gian-Carlo Rota)
Martin Hairer - Probability and Stochastic Analysis, SPDE, Publications and Lectures
Ernst Hairer - Numerical analysis with Useful links
Victor Guillemin - Differential geometry, notes and books
Nicolai Reshetikhin - Topological Quantum Field Theory, Representation Theory and Mathematical Physics, Courses with booklists, some books and notes and other materials available

Notices of the AMS https://www.ams.org/cgi-bin/notices/amsnotices.pl?thispage=homenav
TRANSFORMING INSTRUCTION IN UNDERGRADUATE MATHEMATICS VIA PRIMARY HISTORICAL SOURCES https://digitalcommons.ursinus.edu/triumphs/
What's New https://terrytao.wordpress.com/
Ben Green (additive combinatorics, analytic number theory, harmonic analysis, ergodic theory, discrete geometry and group theory) - https://people.maths.ox.ac.uk/greenbj/
有用的资源 - https://hoanganhduc.github.io/misc/

百科各种
https://groupprops.subwiki.org/
https://oeis.org/?language=chineseS

网课 Borcherds/Gowers
https://www.youtube.com/watch?v=f3SJON86hcU&ab_channel=UConnMathematics (肖梁 代数数论)
https://space.bilibili.com/36114523
https://www.youtube.com/@UConnMath
Artem Chernikov
MIT OpenCourseWare

https://mathoverflow.net/questions/380933/great-graduate-courses-that-went-online-recently#:~:text=In%2009.2020%20by%20pure%20chance%20I

https://math.stackexchange.com/questions/4059864/are-there-any-famous-notable-mathematicians-who-have-their-own-youtube-channel#:~:text=Their%20levels%20of%20fame%20and%20notoriety

https://ocw.mit.edu/search/?d=Mathematics&s=department_course_numbers.sort_coursenum
Richard Melrose - Introduction To Functional Analysis
David Jerison - Fourier Analysis
Victor Guillemin - Topics in Several Complex Variables
Sigurdur Helgason - Functions of a Complex Variable
Tobias Colding - Introduction to Partial Differential Equations
Alexander Postnikov - Algebraic Combinatorics
Yufei Zhao - Probabilistic Method in Combinatorics
Pavel Etingof - Geometry and Quantum Field Theory
Richard Stanley - Combinatorial Analysis
https://ocw.mit.edu/search/?q=%22Prof.%20Pavel%20Etingof%22
https://ocw.mit.edu/search/?q=%22Prof.%20Richard%20Stanley%22

LaTeX
quiver - https://q.uiver.app/
mathcha - https://www.mathcha.io/editor

代数

Joseph H. Silverman - Abstract Algebra: An Integrated Approach
Martin Isaacs - Algebra A Graduate Course
Louis Halle Rowen - Graduate algebra
Saunders Mac Lane - Sheaves in Geometry and Logic: A First Introduction to Topos Theory
Michel Broué - From Rings and Modules to Hopf Algebras: One Flew Over the Algebraist's Nest

群论

Serre - Finite Group
J.S. Milne - Group Theory
H.S.M. Coxeter, W.O.J. Moser - Generators and Relations for Discrete Groups
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss - The symmetries of things
> 对称性与群论，前半部分分为面向普通人的几何物体的对称性概念、更进一步的关于有色物体对称性和轨道流形的讨论以及关于许多有限群和二维晶体群，后半部分关于许多空间群变换群和四维物体对称性等等

Roger C. Lyndon , Paul E. Schupp - Combinatorial Group Theory

表示论

J.-P. Serre - Linear Representation of Finite Group
Fulton, Harris - Representation Theory A First Course
Pavel Etingof - Introduction to Representation Theory
Barry Simon - Representations of Finite and Compact Groups
> 分析学家视角的表示论

Charles W. Curtis - Pioneers of Representation Theory Frobenius, Burnside, Schur, and Brauer
Walter Feit - The Representation Theory of Finite Groups
> 老书

Hermann Weyl - The Classical Groups : Their Invariants and Representations

交换代数/同调代数

Michael F. Atiyah, I.G. MacDonald - Introduction To Commutative Algebra
David Eisenbud - Commutative Algebra with a View Toward Algebraic Geometry
Masayoshi Nagata - Local Rings
> 有很多反例

Bourbaki - Commutative Algebra
Hideyuki Matsumura - Commutative Algebra
Sergei I. Gelfand, Yuri I. Manin - Methods of Homological Algebra
Charles A. Weibel - An Introduction to Homological Algebra
Alexandre Grothendieck - Some Aspects of Homological Algebra
Saunders Mac Lane - Homology
J.-P. Serre - Local Fields
J.-P. Serre - Local Algebra
Masaki Kashiwara, Pierre Schapira - Categories and Sheaves
P. J. Hilton, U. Stammbach - A course in homological algebra

代数几何

Michael Artin - Algebraic Geometry: Notes on a Course
Shafarevich - Basic Algebraic Geometry I/II
Griffiths, Harris - Principles of Algebraic Geometry
Griffiths - 代数曲线
William Fulton - Algebraic Curves, An Introduction to Algebraic Geometry
> 非常友善，学过基本的抽代就行

Ravi Vakil - The Rising Sea Foundations of Algebraic Geometry
Oscar Zariski - Algebraic Surfaces
> The second edition of the book under review appeared in 1971 enriched by updating appendices one for each chapter, which were written by S.S. Abhyankar, J. Lipman, and D. Mumford. These ten appendices gave outlines of the developments of algebraic surface theory during the period from 1935 to 1970, connecting in this way Zariski's original classic text to the methods and results of abstract modern algebraic geometry. The present edition is an unchanged reprint of the second edition from 1971.

David Eisenbud, Joe Harris - The Geometry of Schemes
J.-P. Serre - FAC (代数性凝聚层)
Peter Scholze - Algebraic Geometry (Notes)
Grothendieck - Pursuit Stacks https://thescrivener.github.io/PursuingStacks/ / https://arxiv.org/pdf/2111.01000.pdf
A. Beilinson, V. Drinfeld - Opers (Paper, arxiv)
Araceli Bonifant, John Milnor - Group Actions, Divisors, and Plane Curves (arxiv)
Araceli Bonifant, John Milnor - On Real and Complex Cubic Curves (arxiv)
Peter Scholze - Perfectoid Spaces A survey (arxiv)
Fundamental algebraic geometry: Grothendieck’s FGA explained
David Cox, John Little, Donal O’ Shea - Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
Claire Voisin - Hodge Theory and Complex Algebraic Geometry I

范畴论/泛代数

MacLane - Category Theory for Working Mathematicians
Emily Riehl - Category theory in context
George M. Bergman - An invitation to General Algebra and Universal Constructions

数论

Hardy, Wright - An Introduction to the Theory of Numbers
Gauss - Disquisitiones Arithmeticae
Dirichlet, Dedekind - Lectures on Number Theory
J.-P. Serre - A Course in Arithmetic
P. Erdos, Suranyi - Topics in the Theory of Numbers
华罗庚 - 数论导引
Joseph H. Silverman - A Friendly Introduction to Number Theory
Z. I. Borevich, I. R. Shafarevich - Number Theory
Allen Hatcher - Topology of Numbers
> This is an introduction to elementary number theory from a geometric point of view, in contrast to the usual strictly algebraic approach. A large part of the book is devoted to studying quadratic forms in two variables with integer coefficients, a very classical topic going back to Fermat, Euler, Lagrange, Legendre, and Gauss, but from a perspective that emphasizes Conway's much more recent notion of the topograph of a quadratic form. The book has been published by the AMS in 2022 as a paperback, ISBN 978-1-4704-5611-5. See the AMS webpage for the book.

John Horton Conway - The Sensual (Quadratic) Form
Yuri I. Manin, Alexei A. Panchishkin - Introduction to Modern Number Theory Fundamental Problems, Ideas and Theories

代数数论

J.-P. Serre - Local Fields
David Cox - Primes of the Form x² + ny²
Barry Mazur - On the passage from local to global in number theory (Paper, arxiv)
Ben Green - Algebraic Number Theory (2020, Notes, with sheets)
André Weil - Basic Number Theory
Emil Artin - Algebraic Numbers and Algebraic Functions
T.Y. Lam - Introduction to Quadratic Forms over Fields
Helmut Hasse - Number Theory
J.-P. Serre - Algebraic Groups and Class Fields
Franz Lemmermeyer - Quadratic Number Fields
> 串烧数学史

Pierre Samuel - Algebraic Number Theory
加藤和也, 黑川信重, 斋藤毅 - 数论I: Fermat的梦想和类域论

解析数论

Gerald Tenenbaum - Introduction to Analytic and Probabilistic Number Theory (解析和概率数论导引)
Henryk Iwaniec, Emmanuel Kowalski - Analytic Number Theory
Barry Mazur - Prime Numbers and the Riemann Hypothesis
H. L. Montgomery, R. C. Vaughan - Mulplicative Number Theory Ⅰ: Classical Theory

模形式

Zagier - The 1-2-3 of modular forms
Goro Shimura - Modular Forms
Goro Shimura - Elementary Dirichlet Series and Modular Forms

拓扑/几何

Nicolas Bourbaki - Elements of Mathematics General Topology
Steven G. Krantz - A Guide to Topology
Ronald Brown - Topology and groupoids
> With Exposition of part of Chapter 6

John L. Kelley - General Topology
Jacques Dixmier - General Topology
肖盖 - 拓扑学教程
汪林 - 拓扑空间与线性拓扑空间中的反例
C. McMullen - Topology (Notes)
O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov - Elementary Topology: Problem Textbook
> 包含对General Topology的详细介绍和对Algebraic Topology的介绍

John H. Conway, Peter G. Doyle, Jane Gilman, William P. Thurston - Geometry and the Imagination in Minneapolis (arxiv 1804.03055)
> 关于"Geometry and the Imagination"的讨论班的部分记录和补充文本
> 关于 The Shape of Space by Jeff Weeks 与 Introduction to Geometry by Coxeter，也推荐Flatland by Abbott 和 What is Mathematics by Courant and Robbins

Raoul Harry Bott - Differential Forms in Algebraic Topology
> 代数拓扑与微分拓扑

Glen Bredon - Topology and Geometry
> 代数拓扑与微分拓扑

Miles Reid, Balázs Szendrői - Geometry and Topology

David Hilbert, S. Cohn-Vossen - Geometry and the Imagination
Jeffrey R.Weeks - The Shape of Space
> 直观

Pavel Alexandroff - Elementary Concepts of Topology

K. Parthasarathy - Topology: An Invitation
> 结合数学史的入门教材

H. Edelsbrunner, J. L. Harer - Computational Topology: An Introduction
> 计算拓扑学

H.S.M. Coxeter - Introduction to Geometry
Marcel Berger - Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry
> 古典与现代几何，几何学史

B. A. Dubrovin, A. T. Fomenko, S. P. Novikov - Modern Geometry - Methods and Applications. Part I/II/III
> 各种现代几何

A. Mishchenko, A. Fomenko - A Course of Differential Geometry and Topology
> 包括但不限于黎曼几何和同调理论

Miles Reid, Balazs Szendroi - Geometry and Topology
S.P. Novikov, A.T. Fomenko - Basic Elements of Differential Geometry and Topology
S. Ramanan - Global Calculus
> Analysis, topology and algebra brought new power to geometry, revolutionizing the way geometers and physicists look at conceptual problems. Some of the key ingredients in this interplay are sheaves, cohomology, Lie groups, connections and differential operators. In Global Calculus, the appropriate formalism for these topics is laid out with numerous examples and applications by one of the experts in differential and algebraic geometry.

Ramanan has chosen an uncommon but natural path through the subject. In this almost completely self-contained account, these topics are developed from scratch. The basics of Fourier transforms, Sobolev theory and interior regularity are proved at the same time as symbol calculus, culminating in beautiful results in global analysis, real and complex. Many new perspectives on traditional and modern questions of differential analysis and geometry are the hallmarks of the book.

The book is suitable for a first year graduate course on global analysis.

李群李代数

Terence Tao - Hilbert's Fifth Problem and Related Topics
J.-P. Serre - Lie Algebras and Lie Groups
Nathan Jacobson - Lie Algebras
> 胡乱分类

微分几何

John Willard Milnor - Topology from the Differentiable Viewpoint
Lars Hörmander - Advanced Differential Calculus
Heinz Hopf - Differential Geometry in the Large
James Morrow、Kunihiko Kodaira - Complex Manifolds
Taubes - Differential Geometry
Charles W. Misner, Kip S. Thorne, John Archibald Wheeler - Gravitation
> 微分几何和广相

Peter Dombrowski - 150 Years After Gauss’ “Disquisitiones generales circa superficies curvas”
> Gauss与微分几何的诞生

Manfredo Perdigão do Carmo - Differential Forms and Applications
> 小书

Frank W. Warner - Foundations of Differentiable Manifolds and Lie Groups

Anatole Katok, Vaughn Climenhaga - Lectures on Surfaces
Tristan Needham - Visual Differential Geometry and Forms-A Mathematical Drama in Five Acts

陈省身 - Lectures on Differential Grometry
丘成桐 - 微分几何讲义
Surveys in Differential Geometry (好多卷)
Sigurdur Helgason - Differential Geometry, Lie Groups, and Symmetric Spaces
Wilhelm Klingenberg - a course in differential geometry
De Rham - Differentiable Manifolds

黎曼面

Simon Donaldson - Riemann Surfaces
Hermann Weyl - The Concept of a Riemann Surface
Jürgen Jost - Compact Riemann Surfaces
Lars Ahlfors - Riemann Surfaces

黎曼几何

Lars Hörmander - Riemann Geometry
Peter Peterson - Riemannian Geometry
Marcel Berger - A Panoramic View of Riemannian Geometry
> 参考、欣赏、兴趣，质量有待评估

Manfredo Perdigão do Carmo - Riemannian Geometry
Jürgen Jost - Riemannian Geometry and Geometric Analysis
Nail H. Ibragimov - Tensors and Riemannian Geometry
> 非常多物理，质量有待评估

David Dai-Wai Bao, Shiing-Shen Chern, Zhongmin Shen - An introduction to Riemann-Finsler geometry
Wilhelm P. A. Klingenberg - Riemann Geometry

低维几何

W. Thurston - Three-Dimensional Geometry and Topology
Marvin J. Greenberg - Euclidean and Non-Euclidean Geometries: Development and History

代数拓扑

Allen Hatcher - Algebraic Topology
> With extra chapter Chapter 5. Spectral Sequences and some extra exercises
> Revisions and Additions
> Corrections and Corrections to Algebraic Topology

Daniel Quillen - Homotopical Algebra
J. F. Adams - Algebraic Topology
> 很多数学史文献

William Fulton - Algebraic Topology: A First Course
Michael Atiyah - K-theory
J. P. May - A Concise Course in Algebraic Topology
Tammo tom Dieck - Algebraic Topology
> May的加细

J. P. May, K. Ponto - More Concise Algebraic Topology
Anatoly Fomenko, Dmitry Fuchs - Homotopical Topology
Samuel Eilenberg, Norman Steenrod - Foundations of Algebraic Topology
Friedl - Algebraic Topology I-V
> 用来查的巨型字典

复几何

Mark Green, Phillip Griffiths and Matt Kerr - Hodge Theory, Complex Geometry, and Representation Theory

微分拓扑

John Willard Milnor - Topology from the Differentiable Viewpoint
John Milnor - Differential Topology Forty-six Years Later
> In the 1965 Hedrick Lectures,1 I described the state of differential topology, a field that was then young but growing very rapidly. During the intervening years, many problems in differential and geometric topology that had seemed totally impossible have been solved, often using drastically new tools. The following is a brief survey, describing some of the highlights of these many developments.

C. T. C. Wall - Differential Topology
Victor Guillemin, Alan Pollack - Differential Topology
Morris W. Hirsch - Differential topology
R. C. Kirby, L. C. Siebenmann - Foundational essays on topological manifolds, smoothings and triangulations (1977, 355 p., Princeton University Press and University of Tokyo Press. V, Annals of Mathematics Studies)

低维拓扑

W. Thurston - Three-Dimensional Geometry and Topology
Vaughan F.R. Jones - The Jones Polynomial (exposition paper)

算子代数/非交换几何

不知道该怎么分类……

Operator Algebras, Quantization, and Noncommutative Geometry: A Centennial Celebration Honoring John von Neumann and Marshall H. Stone
> AMS举办的von Neumann和Stone百年诞辰庆典特别会议论文集及其它收录文章
> 关于算子代数、量子化和非交换几何

Alain Connes - Noncommutative Geometry
Kenneth R. Davidson - **C*-Algebras by Example
P. Halmos - What does the spectral theorem say?** American Mathematical Monthly, 70: 241-47.

分析

G. Wanner, E. Hairer - Analysis by its history
G.H. Hardy, J.E. Littlewood, G. Pólya - Inequalities
Lynn Harold Loomis, Shlomo Sternberg - Advanced Calculus
Elliott H. Lieb, Michael Loss - Analysis
Inequalities: Selecta of Elliott H. Lieb
> Brings together a host of inequalities otherwise scattered in the literature and hard to find

George Polya, Gabor Szegö, C.E. Billigheimer - Problems and theorems in analysis I/II
Jürgen Jost - Postmodern Analysis
Barry Simon - Real Analysis: A Comprehensive Course in Analysis, Part 1
Lars Hörmander - The Analysis of Linear Partial Differential Operators

复分析

Elias M. Stein - Complex Analysis
Tao - 246ABC Notes （Blog)
Lars Ahlfors - Complex Analysis
Kunihiko Kodaira - Complex Analysis
Henri Cartan - 解析函数论
Robert E. Greene, Steven G. Krantz - Function Theory of One Complex Variable
Barry Simon - Advanced Complex Analysis: A Comprehensive Course in Analysis, Part 2A/2B
C. L. Siegel - Topics in Complex Function Theory, Vol.1 : Elliptic Functions and Uniformization Theory

测度论

Elias M. Stein - Real Analysis
Folland - Real Analysis Modern Techniques and Their Applications
Richard Wheeden & Antoni Zygmumd - An Introduction to Real Analysis
A. N. Kolmogorov, S. V. Fomin - Elements of the Theory of Functions and Functional Analysis
汪林 - 实分析中的反例
Terence Tao - An Introduction to Measure Theory
Terence Tao - An Epsilon of Room I/II

泛函分析

Haim Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations
Peter D. Lax - Functional Analysis
Philippe G. Ciarlet - Linear and Nonlinear Functional Analysis with Applications
A. N. Kolmogorov, S. V. Fomin - Elements of the Theory of Functions and Functional Analysis
Elias M. Stein - Functional Analysis
Lars Hörmander - Linear Functional Analysis
Frigyes Riesz, Bela Sz.-Nagy - Functional Analysis
Jürgen Jost, Xianqing Li-Jost - Calculus of Variations
Alberto Bressan - Lecture Notes on Functional Analysis: With Applications to Linear Partial Differential Equations

调和分析

Elias M. Stein, Rami Shakarchi - Fourier Analysis An Introduction
Terence Tao - Higher Order Fourier Analysis
Barry Simon - Harmonic Analysis: A Comprehensive Course in Analysis, Part 3
Lars Hörmander - Lectures on Harmonic Analysis
Hugh L. Montgomery - Early Fourier Analysis

ODE/动力系统

Vladimir I. Arnold - Ordinary Differential Equations
Morris W. Hirsch, Stephen Smale, Robert L. Devaney - Differential Equations, Dynamical Systems, and an Introduction to Chaos
L. S. Pontryagin - Ordinary Differential Equations
Ernst Hairer, Syvert Paul Nørsett, Gerhard Wanner - Solving Ordinary Differential Equations I/II

PDE

Vladimir I. Arnold - Lectures on Partial Differential Equations
Lawrence C. Evans - Partial Differential Equations
Jürgen Jost - Partial Differential Equations
Lars Hörmander - Seminar Notes on Pseudo-Differential Operators and Boundary Problems

组合

Richard P. Stanley - Permutations
George Pólya, Robert E. Tarjan, Donald R. Woods - Notes on Introductory Combinatorics
A. Björner, R. P. Stanley - A Combinatorial Miscellany
Terence Tao - Additive Combinatorics
Bela Bollobás - Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Probabilistic Combinatorics
> Combinatorics is a book whose main theme is the study of subsets of a finite set. It gives a thorough grounding in the theories of set systems and hypergraphs, while providing an introduction to matroids, designs, combinatorial probability and Ramsey theory for infinite sets. The gems of the theory are emphasized: beautiful results with elegant proofs. The book developed from a course at Louisiana State University and combines a careful presentation with the informal style of those lectures. It should be an ideal text for senior undergraduates and beginning graduates.

图论

Béla Bollobás - Graph Theory: An Introductory Course

博弈论

John Horton Conway - On numbers and games
> 超现实数与组合博弈论

数学物理

V. I. Arnol’d - Mathematical Methods of Classical Mechanics
Michael Reed, Barry Simon - Methods of Modern Mathematical Physics
Michel Talagrand - What Is a Quantum Field Theory?
John Baez, Javier P. Muniain - Gauge Fields, Knots and Gravity
> 这是一本内容广泛、极具原创性的现代电磁学、规范场和引力的介绍，其中大部分内容是用微分形式语言表达的。在其众多优秀的特点中，包括对麦克斯韦方程中霍奇二元性的作用的深刻讨论。不要忽视这本书三部分中每一部分的注释：它们包含对进一步研究的注解建议（非常像这一部分！），它们还包含迷人的历史小故事和精辟的引文。作者以友好、非正式的方式直接与读者交谈，就像对坐在他们身边的聪明朋友说话一样，而不是对着虚空讲解干巴巴的定理，这让人感到非常新鲜和有益。(当然，这也正是我在 VDGF 中试图做的！）——Needham

Shlomo Sternberg - Curvature in Mathematics and Physics
> 形式是这本书的主要语言。正如书名所示，它包含了许多有趣的数学和物理学的应用。特别是，它深入处理了以下物理课题：霍奇对偶（Hodge duality）和电磁学，施瓦兹希尔德解（Schwarzschild solution）的几何和轨道的明确计算，以及极其重要的克尔解（Kerr solution）（代表一个旋转的黑洞）的几何，尽管他在实际计算曲率 2 形式方面止步不前。但这一列表没有完全列出对所涵盖的大量材料。警告：作者将这本书描述为适合高级本科生阅读-其实不然。但是，如果你已经掌握了我的第五幕，那么你就能从这本书里学到很多东西 ——Needham

Hagen Kleinert - Path Integrals
Charles W. Misner, Kip S. Thorne, John Archibald Wheeler - Gravitation
> 微分几何和广相

John Archibald Wheeler - A Journey into Gravity and Spacetime
Anthony Zee - Einstein Gravity in a Nutshell
Jürgen Jost - Geometry and Physics
> "Geometry and Physics" addresses mathematicians wanting to understand modern physics, and physicists wanting to learn geometry. It gives an introduction to modern quantum field theory and related areas of theoretical high-energy physics from the perspective of Riemannian geometry, and an introduction to modern geometry as needed and utilized in modern physics

Pierre Deligne et al. - Quantum Fields and Strings: A Course for Mathematicians
Michael Atiyah - The Geometry and Physics of Knots
Gerald B. Folland - Quantum Field Theory: A Tourist Guide for Mathematicians

概率/统计/信息论

Kai Lai Chung - Elementary Probability Theory
Rick Durrett - Probability: Theory and Examples
Kiyosi Itô - Introduction to probability theory (伊藤清概率论)
Jean Jacod, Philip Protter - Probability Essentials
S. R. S. Varadhan - Probability Theory
Albert N. Shiryaev - Probability 1/2
Albert N. Shiryaev, Andrew Lyasoff - Problems in Probability
> 概率论习题集

George Casella, Roger L. Berger - Statistical Inference
Alfred Renyi - Probability Theory

随机过程

Kiyosi Itô - Stochastic Processes(随机过程)

信息论

Imre Csiszár, János Körne - Information Theory: Coding Theorems for Discrete Memoryless Systems
Thomas M. Cover, Joy A. Thomas - Elements of Information Theory

数理逻辑

Stephen Cole Kleene - Mathematical logic
Kenneth Kunen - The Foundations of Mathematics
Kenneth Kunen - Set Theory
Yu. I. Manin - A Course in Mathematical Logic for Mathematicians
George S. Boolos, John P. Burgess - Computability and Logic (可计算性与数理逻辑)
Martin Davis - Applied Nonstandard Analysis
H. Jerome Keisler - Foundations of Infinitesimal Calculus
Thomas Jech - Set Theory
Akihiro Kanamori, Matthew Foreman, Akihiro Kanamori (eds.) - Handbook of Set Theory
Saharon Shelah - Proper & Improper Forcing
Gerald E. Sacks - higher recursion theory

计算数学/算法

Anne Greenbaum, Tim P. Chartier - Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms
Kenneth Lange - Optimization
Kenneth Lange - Algorithms from THE BOOK
Joachim von zur Gathen, Jürgen Gerhard - Modern Computer Algebra
Jeffrey Humpherys, Tyler J. Jarvis - Foundations of Applied Mathematics Volume 1: Mathematical Analysis and Volume 2: Algorithms, Approximation, Optimization
XIn-She Yang, Xing-Shi He - Mathematical Foundations of Nature-Inspired Algorithms
> The book begins with a short introduction that describes general principles of constrained and unconstrained optimization of univariate and multivariate functions. It then quickly summarizes several versions of gradient-based algorithms including the usual ones (steepest descent and conjugate gradient) as well as more advanced ones like stochastic gradient and subgradient methods.
Having set the stage with these more conventional algorithms, the authors describe a series of nature-inspired algorithms. They note that the “no-free-lunch” theorems proved in 1997 tell us that there is no best algorithm for solving all optimization problems because all algorithms are equally effective (or ineffective) when measured by average performance across all possible problems. Consequently, the authors consider nature-inspired algorithms that can be matched to specific kinds of applications. Algorithms they describe go by names such as particle swarm optimization, the bat algorithm, the firefly algorithm, and cuckoo search. Several of these algorithms are based on the idea of swarm intelligence. The aim of a swarming system is to allow the system to evolve and converge into stable states that include those with optimal performance.
The authors devote a couple of chapters to the analysis of algorithms. This has some general aspects (convergence, stability and robustness) as well as details that apply to the nature-inspired algorithms (determining and tuning of parameters and statistical characterization of performance). A final chapter describes some applications of nature-inspired algorithms that the authors have discovered. These include design optimization in engineering, image processing, vehicle routing and scheduling.
This is not a textbook and has no exercises. Most of the topics considered get very abbreviated treatments and many of the discussions of the algorithms are disappointingly sketchy. Even the algorithm analysis sections are short on detail. Critics have suggested that the elaborate metaphors of some nature-inspired algorithms have hidden their lack of novelty or effectiveness. There is just not enough detail in this book to allow any judgment in that direction.
The book is probably best suited as an inspiration for an independent project that might take one of the algorithms and fill out details of analysis and performance. The book’s level of sophistication varies, but most topics are accessible to upper level undergraduates.

数学史/传记/文集/综述

若干历史

Jean Dieudonné - A History of Algebraic and Differential Topology, 1900 - 1960
Jean Dieudonné - History of Algebraic Geometry
Jean Dieudonné - History of Functional Analysis

Bartel Leenert van der Waerden - A History of Algebra: From al-Khwārizmī to Emmy Noether

Nicolas Bourbaki - Elements of the History of Mathematics

James, I. M. ((eds.)) - History of Topology

Knoebel, A., Laubenbacher, R., Lodder, J. etc. - Mathematical Masterpieces Further Chronicles by the Explorers
> 四个主题：离散与连续；数值求解方程; 曲率和空间的概念；二次互反律

Marvin J. Greenberg - Euclidean and Non-Euclidean Geometries: Development and History
> 平面几何与双曲几何

Marcel Berger - Riemannian Geometry During the Second Half of the Twentieth Century

Peter Dombrowski - 150 Years After Gauss’ “Disquisitiones generales circa superficies curvas”
> Gauss与微分几何的诞生

David S. Richeson - Euler’s Gem
> 关于多面体欧拉公式的历史与思想

Gessel, Ira (ed.) Rota, Gian-Carlo (ed.) - Classic papers in combinatorics

Armand Borel - Essays in the History of Lie Groups and Algebraic Groups

Pesic, Peter (ed.) - Beyond geometry. Classic papers from Riemann to Einstein

Rodrigo A. Pérez - A Brief but Historic Article of Siegel
Luc Illusie, with Alexander Beilinson, Spencer Bloch, Vladimir Drinfeld, et al. - Reminiscences of Grothendieck and His School

一些综述

Vladimir I. Arnold - Huygens and Barrow, Newton and Hooke
Erdős Centennial
> Erdős成就的一个survey

The Legacy of John von Neumann
> 1988会议论文集，阐述了冯·诺依曼的观念和思想及它们对当代数学的影响，以及关于冯诺依曼的若干回忆
> 算子理论、博弈论、遍历理论、科学计算和数学史相关

Camillo De Lellis - The masterpieces of John Forbes Nash Jr. (arxiv 1606.02551)
> Nash成就的一个survey

The Legacy of Bernhard Riemann After One Hundred And Fifty Years Vol I/II
> 综述文集，黎曼的工作和思想在现代的发展

The Legacy of Norbert Wiener: A Centennial Symposium
> 1994年10月，在MIT的Wiener百年诞辰研讨会上发表的演讲合集

Felix Klein - Development of Mathematics in the Nineteenth Century (数学在十九世纪的发展)

传记/个人文集

Gregory Margulis - Autobiography

Paul Halmos - I Want to Be a Mathematician
> Halmos自传

Heinz Hopf - Selected Chapters of Geometry
> This is a write-up by Hans Samelson of lectures by Hopf in a course at ETH in the summer of 1940. The four chapters are:
Euler's Formula.
Graphs.
The Four Vertex Theorem and Related Matters.
The Isoperimetric Inequality.
These total just 41 pages. There is quite a bit of overlap with notes from another course of the same title taught by Hopf at New York University in 1946 and published as the first part of volume 1000 of the Springer Lecture Notes. The 1946 course seems to have covered slightly more material, but Samelson's write-up of the earlier course is more polished and has a more pleasing appearance, being in TEX with nice electronically-drawn figures.

黎曼全集

David Hilbert - Collected works
> 分卷：Number theory/Algebra, theory of invariants, geometry/Analysis

Collected Papers of John Milnor
> 分卷：Geometry ; The Fundamental Group; Differential Topology; Homotopy, Homology and Manifolds; Algebra; Dynamical Systems (1953-2000); and Dynamical Systems (1984-2012)

Collected Works of John Tate: Parts I and II
Collected Works of William P. Thurston with Commentary
> This four-part collection brings together in one place Thurston's major writings, many of which are appearing in publication for the first time. Volumes I–III contain commentaries by the Editors. Volume IV includes a preface by Steven P. Kerckhoff.

Selected Works of Eberhard Hopf with Commentaries
> The volume is presented in two main parts. The first section is dedicated to classical papers in analysis and fluid dynamics, and the second to ergodic theory. These works and all the others in the Selected Works carry commentaries by a stellar group of mathematicians who write of the origin of the problems, the important results that followed.

Selected Works of Phillip A. Griffiths with Commentary
> The four parts of Selected Works—Analytic Geometry, Algebraic Geometry, Variations of Hodge Structures, and Differential Systems—are organized according to the subject matter and are supplemented by Griffiths' brief, but extremely illuminating, personal reflections on the mathematical content and the times in which they were produced.

Gian-Carlo Rota - Indiscrete Thoughts
Gian-Carlo Rota - Discrete Thoughts
Gian-Carlo Rota on Combinatorics
> In this volume, the editor presents reprints of most of the fundamental papers of Gian-Carlo Rota in the classical core of cominatorics

Gian-Carlo Rota on Analysis and Probability

Felix Klein - Lectures on Mathematics (Klein数学讲座)
> 1893年芝加哥国际数学大会，F.Klein在美国西北大学作了为期两周的埃文斯顿学术报告会演讲，本书由他报告的讲义组成

Felix Klein, W. F. Sheppard, P. A. MacMahon, J. L. Mordell - Famous Problems and Other Monographs
> Klein: Famous Problems Of Elementary Geometry (初等几何的著名问题)
> Sheppard: From Determinant to Tensor
> MacMahon: Introduction to Combinatory Analysis
> Moderll: Three Lectures on Fermat's Last Theorem

Mikhail Gromov - Gromov的数学世界

Robert P. Langlands - langlands纲领和他的数学世界

John Milnor - Milnor眼中的数学和数学家

综合/数学哲学/其它

The Princeton Companion to Mathematics
The Princeton Companion to Applied Mathematics

Roger Penrose - The Road to Reality

Felix Klein - Lectures on Mathematics

Yuri I. Manin - Mathematics as Metaphor
> Manin文集

Vladimir I. Arnold - Real Algebraic Geometry

Shing-Tung Yau, Steve Nadis - The Shape of Inner Space (大宇之形)
> 科普，弦论与Calabi-Yau流形

Shing-Tung Yau, Steve Nadis - The Gravity of Math
> 科普，引力理论与数学物理

代数结构与拓扑结构 (Structures Algébriques et Structures Topologiques)
> 不知道该放哪……

Terence Tao - Poincaré’s Legacies Part I/II
> Part I of the second-year posts focuses on ergodic theory, combinatorics, and number theory. Chapter 2 consists of lecture notes from Tao's course on topological dynamics and ergodic theory. By means of various correspondence principles, recurrence theorems about dynamical systems are used to prove some deep theorems in combinatorics and other areas of mathematics. In addition to these lectures, a variety of other topics are discussed, ranging from recent developments in additive prime number theory to expository articles on individual mathematical topics such as the law of large numbers and the Lucas–Lehmer test for Mersenne primes. Some selected comments and feedback from blog readers have also been incorporated into the articles.
> Part I of the second-year posts focuses on geometry, topology, and partial differential equations. The major part of the book consists of lecture notes from Tao's course on the Poincaré conjecture and its recent spectacular solution by Perelman. The course incorporates a review of many of the basic concepts and results needed from Riemannian geometry and, to a lesser extent, from parabolic PDE. The aim is to cover in detail the high-level features of the argument, along with selected specific components of that argument, while sketching the remaining elements, with ample references to more complete treatments. In addition to these lectures, a variety of other topics are discussed, including expository articles on topics such as gauge theory, the Kakeya needle problem, and the Black–Scholes equation. Some selected comments and feedback from blog readers have also been incorporated into the articles.
> The lectures are as self-contained as possible, focusing more on the “big picture” than on technical details.

Terence Tao - Compactness and Contradiction

V. B. Alekseev - Abel's theorem in problems and solutions based on the lectures of professor V.I. Arnold
> 不知道该放哪×2

Arthur Jaffe, Frank Quinn - Theoretical Mathematics Toward a cultural synthesis of mathematics and theoretical physics (arxiv)
Thurston - On Proof and Progress in Mathematics
Michael Atiyah et al. - Responses to "Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics'', by A. Jaffe and F. Quinn (arxiv)
Arthur Jaffe, Frank Quinn - Response to comments on “theoretical mathematics”
Michael Atiyah - Reflections on geometry and physics
Yuri I. Manin - Truth, rigour, and common sense

Kurt Gödel - What is Cantor’s Continuum Problem?

Arnold, V. (ed.); Atiyah, M. (ed.); Lax, P. (ed.); Mazur, B. (ed.) Mathematics: frontiers and perspectives

杂项

Interview with Abel Laureate 2020 Gregory Margulis
Saunders MacLane - Duality for Groups
Saunders MacLane (1948) - Groups, categories and duality Proceedings of the Nat. Acad. of Sciences of the USA 34: 263–67.
Daniel E. Loeb, Gian-Carlo Rota - Recent contributions to the calculus of finite differences a survey (arxiv 9502210)
Green B, and Tao T. 2008. The primes contain arbitrarily long arithmetic progressions. Annals of Mathematics 167: 481-547.
J. H. Conway, S. Torquato - Packing, tiling, and covering with tetrahedra
Maryna Viazovska - The sphere packing problem in dimension 8
Maryna Viazovska - On discrete Fourier uniqueness sets in Euclidean space
Samuel Eilenberg, Saunders MacLane - General Theory of Natural Equivalences
> 范畴论的起源

Samuel Eilenberg, Saunders MacLane (1945,1950) - Relations Between Homology and Homotopy Groups of Spaces I/II
Samuel Eilenberg; John C. Moore (1962) - Limits and spectral sequences ", Topology 1 (1): 1–23, doi:10.1016/0040-9383(62)90093-9, ISSN 0040-9383
Samuel Eilenberg; Norman E. Steenrod (1945) - Axiomatic approach to homology theory Proceedings of the National Academy of Sciences of the United States of America 31 (4): 117–120. doi:10.1073/pnas.31.4.117. PMID 16578143
Armand Borel, Jean-Pierre Serre (1958) - Le théorème de Riemann-Roch (The Riemann–Roch theorem)
J. Michael Steele - The Cauchy-Schwarz Master Class
V. I. Arnold - Experimental Mathematics
Roger Penrose (1955) - A generalized inverse for matrices

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