资源链接:https://pan.baidu.com/s/1jQgmgTaaZnGTpGf-5w4gtg
国外的书,可以借鉴学习。目录如下:
第一学年
几何与拓扑:
1、James R. Munkres, Topology:较新的拓扑学的教材适用于本科高年级或研究生一年级;
2、Basic Topology by Armstrong:本科生拓扑学教材;
3、Kelley, General Topology:一般拓扑学的经典教材,不过观点较老;
4、Willard, General Topology:一般拓扑学新的经典教材;
5、Glen Bredon, Topology and geometry:研究生一年级的拓扑、几何教材;
6、Introduction to Topological Manifolds by John M. Lee:研究生一年级的拓扑、几何教材,是一本新书;
7、From calculus to cohomology by Madsen:很好的本科生代数拓扑、微分流形教材。
代数:
1、Abstract Algebra Dummit:最好的本科代数学参考书,标准的研究生一年级代数教材;
2、Algebra Lang:标准的研究生一、二年级代数教材,难度很高,适合作参考书GTM;
3、Algebra Hungerford:标准的研究生一年级代数教材,适合作参考书GTM;
4、Algebra M,Artin:标准的本科生代数教材;
5、Advanced Modern Algebra by Rotman:较新的研究生代数教材,很全面;
6、Algebra:a graduate course by Isaacs:较新的研究生代数教材;
7、Basic algebra Vol I&II by Jacobson:经典的代数学全面参考书,适合研究生参考。
分析基础:
1、Walter Rudin, Principles of mathematical analysis:本科数学分析的标准参考书;
2、Walter Rudin, Real and complex analysis:标准的研究生一年级分析教材;
3、Lars V. Ahlfors, Complex analysis:本科高年级和研究生一年级经典的复分析教材;
4、Functions of One Complex Variable I,J.B.Conway:研究生级别的单变量复分析经典GTM11;
5、Lang, Complex analysis:研究生级别的单变量复分析参考书;
6、Complex Analysis by Elias M. Stein:较新的研究生级别的单变量复分析教材;
7、Lang, Real and Functional analysis:研究生级别的分析参考书;
8、Royden, Real analysis:标准的研究生一年级实分析教材;
9、Folland, Real analysis:标准的研究生一年级实分析教材。
第二学年
代数:
1、Commutative ring theory, by H. Matsumura:较新的研究生交换代数标准教材;
2、Commutative Algebra I&II by Oscar Zariski , Pierre Samuel:经典的交换代数参考书GTM28-29;
3、An introduction to Commutative Algebra by Atiyah:标准的交换代数入门教材;
4、An introduction to homological algebra ,by weibel:较新的研究生二年级同调代数教材;
5、A Course in Homological Algebra by P.J.Hilton,U.Stammbach:经典全面的同调代数参考书GTM4;
6、Homological Algebra by Cartan:经典的同调代数参考书;
7、Methods of Homological Algebra by Sergei I. Gelfand, Yuri I. Manin:高级、经典的同调代数参考书;
8、Homology by Saunders Mac Lane:经典的同调代数系统介绍;
9、Commutative Algebra with a view toward Algebraic Geometry by Eisenbud:高级的代数几何、交换代数的参考书,最新的交换代数全面参考。
代数拓扑:
1、Algebraic Topology, A. Hatcher:最新的研究生代数拓扑标准教材;
2、Spaniers “Algebraic Topology”:经典的代数拓扑参考书;
3、Differential forms in algebraic topology, by Raoul Bott and Loring W. Tu:研究生代数拓扑标准教材;
4、Massey, A basic course in Algebraic topology:经典的研究生代数拓扑教材;
5、Fulton , Algebraic topology:a first course:很好本科生高年级和研究生一年级的代数拓扑参考书;
6、Glen Bredon, Topology and geometry:标准的研究生代数拓扑教材,有相当篇幅讲述光滑流形;
7、Algebraic Topology Homology and Homotopy:高级、经典的代数拓扑参考书;
8、A Concise Course in Algebraic Topology by J.P.May:研究生代数拓扑的入门教材,覆盖范围较广;
9、Elements of Homotopy Theory by G.W. Whitehead:高级、经典的代数拓扑参考书。
实分析、泛函分析:
1、Royden, Real analysis:标准研究生分析教材;
2、Walter Rudin, Real and complex analysis:标准研究生分析教材;
3、Halmos,”Measure Theory”:经典的研究生实分析教材,适合作参考书;
4、Walter Rudin, Functional analysis:标准的研究生泛函分析教材;
5、Conway,A course of Functional analysis:标准的研究生泛函分析教材;
6、Folland, Real analysis:标准研究生实分析教材;
7、Functional Analysis by Lax:高级的研究生泛函分析教材;
8、Functional Analysis by Yoshida:高级的研究生泛函分析参考书;
9、Measure Theory, Donald L. Cohn:经典的测度论参考书。
微分拓扑 李群、李代数
1、Hirsch, Differential topology:标准的研究生微分拓扑教材,有相当难度;
2、Lang, Differential and Riemannian manifolds:研究生微分流形的参考书,难度较高;
3、Warner,Foundations of Differentiable manifolds and Lie groups:标准研究生微分流形教材,有相当的篇幅讲述李群;
4、Representation theory: a first course, by W. Fulton and J. Harris:李群及其表示论标准教材;
5、Lie groups and algebraic groups, by A. L. Onishchik, E. B. Vinberg:李群的参考书;
6、Lectures on Lie Groups W.Y.Hsiang:李群的参考书;(项武义)
7、Introduction to Smooth Manifolds by John M. Lee:较新的关于光滑流形的标准教材;
8、Lie Groups, Lie Algebras, and Their Representation by V.S. Varadarajan:最重要的李群、李代数参考书;
9、Humphreys, Introduction to Lie Algebras and Representation Theory , SpringerVerlag, GTM9:标准的李代数入门教材。
第三学年
微分几何:
1、Peter Petersen, Riemannian Geometry:标准的黎曼几何教材;
2、Riemannian Manifolds: An Introduction to Curvature by John M. Lee:最新的黎曼几何教材;
3、doCarmo, Riemannian Geometry.:标准的黎曼几何教材;
4、M. Spivak, A Comprehensive Introduction to Differential Geometry I—V:全面的微分几何经典,适合作参考书;
5、Helgason , Differential Geometry,Lie groups,and symmetric spaces:标准的微分几何教材;
6、Lang, Fundamentals of Differential Geometry:最新的微分几何教材,很适合作参考书;
7、kobayashi/nomizu, Foundations of Differential Geometry:经典的微分几何参考书;
8、Boothby,Introduction to Differentiable manifolds and Riemannian Geometry:标准的微分几何入门教材,主要讲述微分流形;
9、Riemannian Geometry I.Chavel:经典的黎曼几何参考书;
10、Dubrovin, Fomenko, Novikov “Modern geometry-methods and applications”Vol 1—3:经典的现代几何学参考书。缺2
代数几何:
1、Harris,Algebraic Geometry: a first course:代数几何的入门教材;
2、Algebraic Geometry Robin Hartshorne :经典的代数几何教材,难度很高;
3、Basic Algebraic Geometry 1&2 2nd ed. I.R.Shafarevich.:非常好的代数几何入门教材;
4、Principles of Algebraic Geometry by giffiths/harris:全面、经典的代数几何参考书,偏复代数几何;
5、Commutative Algebra with a view toward Algebraic Geometry by Eisenbud:高级的代数几何、交换代数的参考书,最新的交换代数全面参考;
6、The Geometry of Schemes by Eisenbud:很好的研究生代数几何入门教材;
7、The Red Book of Varieties and Schemes by Mumford:标准的研究生代数几何入门教材;
8、Algebraic Geometry I : Complex Projective Varieties by David Mumford:复代数几何的经典。
调和分析 偏微分方程:
1、An Introduction to Harmonic Analysis,Third Edition Yitzhak Katznelson:调和分析的标准教材,很经典;
2、Evans, Partial differential equations:偏微分方程的经典教材;
3、Aleksei.A.Dezin,Partial differential equations,Springer-Verlag:偏微分方程的参考书;
4、L. Hormander “Linear Partial Differential Operators, ” I&II:偏微分方程的经典参考书;
5、A Course in Abstract Harmonic Analysis by Folland:高级的研究生调和分析教材;
6、Abstract Harmonic Analysis by Ross Hewitt:抽象调和分析的经典参考书;
7、Harmonic Analysis by Elias M. Stein:标准的研究生调和分析教材;
8、Elliptic Partial Differential Equations of Second Order by David Gilbarg:偏微分方程的经典参考书;
9、Partial Differential Equations ,by Jeffrey Rauch:标准的研究生偏微分方程教材。
复分析 多复分析导论:
1、Functions of One Complex Variable II,J.B.Conway:单复变的经典教材,第二卷较深入;
2、Lectures on Riemann Surfaces O.Forster:黎曼曲面的参考书;
3、Compact riemann surfaces Jost:黎曼曲面的参考书;
4、Compact riemann surfaces Narasimhan:黎曼曲面的参考书;
5、Hormander ” An introduction to Complex Analysis in Several Variables”:多复变的标准入门教材;
6、Riemann surfaces , Lang:黎曼曲面的参考书;
7、Riemann Surfaces by Hershel M. Farkas:标准的研究生黎曼曲面教材;
8、Function Theory of Several Complex Variables by Steven G. Krantz:高级的研究生多复变参考书;
9、Complex Analysis: The Geometric Viewpoint by Steven G. Krantz:高级的研究生复分析参考书。
数学基础:
1、halmos ,native set theory;
2、fraenkel ,abstract set theory;
3、ebbinghaus ,mathematical logic;
4、enderton ,a mathematical introduction to logic;
5、landau, foundations of analysis;
6、maclane ,categories for working mathematican。应该在核心课程学习的过程中穿插选修
假设本科应有的水平
分析:
Walter Rudin, Principles of mathematical analysis;
Apostol , mathematical analysis;
M.spivak , calculus on manifolds;
Munkres ,analysis on manifolds;
Kolmogorov/fomin , introductory real analysis;
Arnold ,ordinary differential equations。
代数:
linear algebra by Stephen H. Friedberg;
linear algebra by hoffman;
linear algebra done right by Axler;
advanced linear algebra by Roman;
algebra ,artin;
a first course in abstract algebra by rotman。
几何:
do carmo, differential geometry of curves and surfaces;
Differential topology by Pollack;
Hilbert ,foundations of geometry;
James R. Munkres, Topology。
原文地址:http://blog.51cto.com/14063572/2320754