Bibliography of Basic Courses for Undergraduate and Postgraduates in the United States: Geometry and Topology in the First Year: 1, James R. Munkres, Topology: Topology is a relatively new textbook, suitable for senior undergraduate or postgraduate freshmen; 2. Armstrong's "Basic Topology": a topology textbook for undergraduates: 3.Kelley, general topology: a classic textbook of general topology, but the viewpoint is outdated; 4. General Topology: A New Classic Textbook of General Topology; 5. Glen Braden, Topology and Geometry: A Textbook of Topology and Geometry for First-year Postgraduates; 6. Introduction to Topological Manifold of 6.John M. Lee: The textbook of Topology and Geometry for the first year of graduate students is a new book; 7. From Calculus to Homology Madsen: A good textbook for undergraduates of algebraic topology and differential manifolds. Algebra: 1, abstract algebra Dummit: the best algebra reference book for undergraduates and the standard algebra textbook for the first year of graduate students; 2. Algebra Lang: the standard algebra textbook for the first and second grades of graduate students is very difficult and suitable for reference books; 3. Algebra Hungerford: a standard algebra textbook for first-year graduate students, suitable for reference books; 4. Algebra M, Artin: standard algebra textbook for undergraduates; 5. rotman's Advanced Modern Algebra: a relatively new algebra textbook for graduate students, which is very comprehensive; 6. Algebra: Isaacs's postgraduate course: a relatively new algebra textbook for graduate students: 7. Basic algebra. Jacobson II: A comprehensive reference book of algebra classics, suitable for graduate students. Analysis Basis: 1, Walter Rudin, Principles of Mathematical Analysis: A Standard Reference Book for Undergraduate Mathematical Analysis; 2. Walter Rudin, Analysis of Real Part and Multiple Part: A Standard Analysis Textbook for First-year Postgraduates; 3.Lars V. Ahlfors, reanalysis: a classic reanalysis textbook for senior undergraduates and first-year postgraduates; 4. The function of a complex variable I, J.B.Conway: the classic of univariate complex analysis at graduate level: 5. Long, complex analysis: a reference book of univariate complex analysis at graduate level; 6. the complex analysis of 6.Elias M. Stein: a relatively new univariate complex analysis textbook for graduate students: 7. lang, real function analysis: a reference book for graduate students; 8. Royden, Real Change Analysis: A Standard Practical Analysis Textbook for First-year Postgraduates; 9. Forand, Practical Analysis: A standard practical analysis textbook for first-year graduate students. Algebra in the second academic year: 1, commutative ring theory, by H. Matsumura: a relatively new standard textbook for graduate students' commutative algebra; 2, commutative algebra I & II by oscar zariski, Pierre Samuel: a classic reference book of commutative algebra; 3.Atiyah's Introduction to Alternating Algebra: a standard introductory textbook for Alternating Algebra; 4. Veber's Introduction to Homology Algebra: a relatively new textbook of second-year algebra for graduate students; 5. P.J. Hilton's homology algebra course: a classic and comprehensive reference book on homology algebra; 6. Catan's homology algebra: a classic reference book on homology algebra; 7. Gelfand's method of homology algebra: an advanced and classic reference book of homology algebra; 8. Saunders Mac Lane's Homology: Introduction to Classical Homology Algebra System: 9. Alternating Algebra and Viewpoint Tower Algebra Geometry Eisenbard: A reference book of advanced algebraic geometry and commutative algebra, and a comprehensive reference of the latest commutative algebra. Algebraic Topology: 1, Algebraic Topology A. Hatcher: the latest standard textbook for postgraduate algebraic topology; 2.Spaniers Algebraic Topology: a classic reference book on algebraic topology; 3. Differential form in algebraic topology, Raoul Bott and Lorraine W. Tu: standard textbook of algebraic topology for graduate students; 4.Massey, Basic Course of Algebraic Topology: Classic Algebraic Topology Textbook for Postgraduates; 5. Fulton, Algebraic Topology: Lesson 1: A good reference book for senior undergraduates and first-year graduate students; 6.Glen Bredon, Topology and Geometry: A Textbook of Standard Algebraic Topology for Postgraduates, which has a considerable space on smooth manifolds; 7. Algebraic Topology Homology and Homotopy: Advanced and Classical Algebraic Topology Reference Books: A Concise Course of Algebraic Topology of 8.J.P.May: An Introduction to Algebraic Topology for Postgraduates, with a wide range of contents; 9. Elements of Homology by G.W. whitehead: Advanced and Classical Algebraic Topology Reference Books. Real analysis, functional analysis: 1, royden, real analysis: a standard graduate analysis textbook; 2. Walter Rudin, Real and Complex Analysis: A Standard Analysis Textbook for Postgraduates; 3. halmos's Theory of Measurement: a classic practical analysis textbook for graduate students, suitable for reference books; 4. Walter Rudin, Functional Analysis: A Standard Functional Analysis Textbook for Postgraduates; 5. Functional Analysis Course: Standard Functional Analysis Textbook for Postgraduates; 6. Forand, practical analysis: a standard practical analysis textbook for graduate students; 7. functional analysis of 7.Lax: advanced functional analysis textbook for graduate students: 8. functional analysis of Yoshida: reference book for advanced graduate students; 9. measurement theory, Donald L. Cohen: a classic reference book of measurement theory. Lie Groups of Differential Topology, Lie Algebra 1, Hirsch, Differential Topology: a standard differential topology textbook for graduate students, which is quite difficult; 2. Lang, Differential and Riemannian Manifolds: a reference book of differential manifolds for graduate students, which is more difficult; 3.Warner, "Differentiable Manifolds and the Basis of Lie Groups: a textbook for graduate students with standard differential manifolds", has a considerable space to talk about Lie Groups; 4. Presentation theory: lesson 1, W. Fulton and J. Harris: Lie group and its presentation theory standard textbook; 5. Lie groups and algebraic groups, A.L. Onishchik, E.B. Vinberg: Lie groups reference books; 6. Lie Group Reference Book "Lie Group Lecture": 7. John M. Lee's introduction to smooth manifolds: a relatively new standard textbook for smooth manifolds: 8. Lie groups, Lie algebras and their introduction: the most important reference books on Lie groups and Lie algebras; 9. Introduction to Lie Algebra and Representation Theory, springer Publishing House, GTM 9: A standard introductory textbook of Lie Algebra. Differential geometry in the third academic year: 1, peter peterson, Riemannian geometry: standard Riemannian geometry textbook; 2. Riemannian Manifold: Introduction to Curvature: The Latest Riemannian Geometry Textbook: 3.doCarmo, Riemannian Geometry. : standard Riemann geometry textbook; 4.M. spivak, Introduction to Differential Geometry Synthesis I-V: Classic of Differential Geometry Synthesis, suitable for reference books; 5. Helgason, differential geometry, Lie groups and symmetric spaces: a textbook of standard differential geometry; 6. Lang, Fundamentals of Differential Geometry: the latest textbook of differential geometry, which is very suitable for reference books; 7.Kobayashi/Nomizu, Fundamentals of Differential Geometry: A Classic Reference Book of Differential Geometry; 8. Introduction to Busby, Differential Manifolds and Riemannian Geometry: a standard introductory textbook for differential geometry, mainly focusing on differential manifolds; 9. Riemannian Geometry I.Chavel: Classic Riemannian Geometry Reference Book: 10, Modern Geometry-Methods and Applications by Dubrovin, Fomenko and Novikov Volume 1-3: Classic Reference Book of Modern Geometry. Algebraic Geometry: 1, Harris, Algebraic Geometry: Lesson 1: Introduction to Algebraic Geometry; 2. Algebraic Geometry Robin Harthorne: a classic textbook of algebraic geometry, which is very difficult; 3. Basic Algebraic Geometry1& Second edition. : Very good introductory textbook of Algebra Geometry; 4.Giffiths/Harris's Principles of Algebraic Geometry: a comprehensive and classic reference book of algebraic geometry, part of complex algebraic geometry; 5.Eisenbud's Alternating Algebra Oriented Algebra Geometry: A Reference Book of Higher Algebra Geometry and Commutative Algebra, and a Comprehensive Reference of the Latest Commutative Algebra; 6. Eisenbard's "Schema Geometry": an introductory textbook for postgraduate algebraic geometry: 7. Mountford's variables and schemes: a standard introductory textbook for postgraduate algebraic geometry: 8. Algebraic geometry I: complex projective changes. Partial differential equation of harmonic analysis 1, Introduction to harmonic analysis, third edition Yitzhak Katz Nelson: the standard textbook of harmonic analysis, very classic; 2. Evans, Partial Differential Equations: A Classic Textbook of Partial Differential Equations; 3. Alexei. A. Dejin, Partial Differential Equations, springer Publishing House: Reference Book of Partial Differential Equations; 4. L. Hormander's "Linear Partial Differential Operator", I & II: a classic reference book for partial differential equations; 5. Forand's course of abstract harmonic analysis: advanced graduate harmonic analysis textbook: 6. Ross Hewitt's abstract harmonic analysis: classic reference book of abstract harmonic analysis: 7. Harmonic analysis: standard graduate harmonic analysis textbook; 8. Second-order elliptic partial differential equations: a classic reference book for partial differential equations; 9. Partial differential equations, Jeffrey Lauch: a standard textbook for graduate students with partial differential equations. Introduction to Complex Variable Analysis and Multiple Repeated Variable Analysis 1, One-variable Complex Variable Function II, J.B.Conway: One-variable Complex Variable Classic Textbook, Volume II is more in-depth; 2. Lectures on Riemannian Surfaces O. Foster: Riemannian Surfaces Reference Books: 3. Compact Riemannian Surfaces: Riemannian Surfaces Reference Books: 4. Compact Riemannian Surfaces narasimhan: Riemannian Surfaces Reference Books: 5.Hormander "Introduction to Complex Analysis in Serious Variables": Standard Introductory Textbook of Multivariables: 6. Riemannian Surfaces: Riemannian Surfaces Reference Books: 7. Herschel M. Facas. 9. Complex analysis: geometric viewpoint: an advanced reference book for postgraduate reanalysis. Elective courses in professional direction: 1, multivariate complex analysis; 2. Complex geometry; 3. Geometric analysis; 4. Abstract harmonic analysis; 5. Algebraic geometry; 6. Algebraic number theory; 7. Differential geometry; 8. Algebraic groups, Lie algebras and quantum groups; 9. Functional analysis and operator algebra; 10, mathematical physics; 1 1, probability theory; 12, dynamic system and ergodic theory; 13, pan-algebra. Mathematical basis: 1, halmos, primary set theory; ; 2. fraenkel, abstract set theory; 3. Ebbinghaus, mathematical logic; 4. Enderton, a mathematical introduction to logic; 5. Landau, the basis of analysis; 6. McClane, the category of working mathematics. In the process of learning the core courses, we should intersperse the level analysis of elective hypothesis undergraduate courses: Walter Rudin, Principles of Mathematical Analysis; Apostol, mathematical analysis; M.spivak, calculus on manifold; Monkres, analysis on manifold; Kolmogorov/fomin, an introductory analysis; Arnold, ordinary differential equations. Algebra: linear algebra of Stephen H. Friedberg; Hoffman's linear algebra: Eksler correctly completed linear algebra; Roman's higher linear algebra: Algebra, an introductory course of abstract algebra by artin Rotman. Geometry: Docano, differential geometry of curves and surfaces; Differential topology of Pollack: Hilbert, geometric basis; Topology.