Recommendation of foreign mathematics books

Bibliography of basic courses for undergraduate and graduate students majoring in mathematics in the United States

First school year

Geometry and topology:

1, James R. Munkres, Topology: Topology is a relatively new textbook, suitable for senior undergraduates or first-year postgraduates;

2. Armstrong's Basic Topology: a textbook of topology 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3.Kelley, general topology: a classic textbook of general topology, but the viewpoint is outdated;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. The first volume of basic algebra. Jacobson II: A comprehensive reference book of algebra classics, suitable for graduate students.

Analysis basis:

1, Walter Rudin, Principles of Mathematical Analysis: 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 I, J.B.Conway of a complex variable: the classic of unary complex analysis at graduate level;

5. Long, complex analysis: a reference book for univariate complex analysis at the 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 analysis;

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.

Second year

Algebra:

1, theory of commutative rings, 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: An Introduction to Classical Homology Algebraic 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 Textbook of Algebraic Topology Standard for Postgraduates;

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;

8.J.P.May's concise course of algebraic topology: an introductory textbook of algebraic topology for graduate students, with a wide range of contents;

9. Elements of Homology by G.W. whitehead: Advanced and Classical Algebraic Topology Reference Books.

Real analysis and functional analysis:

1, Roden, practical 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. Yoshida's functional analysis: a reference book on functional analysis for senior graduate students;

9. measurement theory, Donald L. Cohen: a classic reference book of measurement theory.

Lie Groups and Lie Algebras of Differential Topology

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. Li Qun's reference book "Li Qun Lecture":

7. Introduction to Smooth Manifolds by John M. Lee: 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.

Third academic year

Differential geometry:

1, peter peterson, riemann geometry: a standard riemann geometry textbook;

2. Riemannian Manifold: Introduction to Curvature: The Latest Riemannian Geometry Textbook;

3.doCarmo, riemann 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: a 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 graduate students in algebraic geometry;

7. Mountford's red book on variables and schemes: a standard introductory textbook for graduate students in algebraic geometry;

8. Algebraic Geometry I: Complex Projective Change.

Harmonic analysis of partial differential equations

1, Introduction to Harmonic Analysis, third edition Yitzhak Katz Nelson: the standard textbook of harmonic analysis, which is 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: a textbook of harmonic analysis for senior graduate students;

6. Ross Hewitt's abstract harmonic analysis: a 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.

Brief introduction of complex analysis and multiple repetition analysis

1, unary complex variable function II, J.B.Conway: a classical textbook of unary complex variable, the second volume is more in-depth;

2. Lectures on Riemann Surfaces O. Foster: Riemann Surfaces Reference Books:

3. Compact Riemannian Surfaces: Riemannian Surfaces Reference Books:

4. Compact Riemannian Surfaces narasimhan: Riemannian Surfaces Reference Book:

5. Helmand's "Introduction to Complex Analysis in Serious Variables": a standard introductory textbook of multivariate;

6. Riemann Surfaces: Riemann Surfaces Reference Books:

7. Riemann Surfaces by Herschel M. Facas: a textbook of standard Riemann Surfaces for graduate students;

8.Steven G. Krantz's Theory of Severe Complex Variables: a multivariable advanced reference book for graduate students;

9. Complex analysis: geometric viewpoint: an advanced reference book for postgraduate reanalysis.

Elective courses in professional direction:

1, multiple repeated 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. Elective courses should be interspersed in the process of learning core courses.

Assume that the undergraduate course should have a level.

Analysis:

Walter Rudin, Principles of Mathematical Analysis;

Apostol, mathematical analysis;

M.spivak, calculus on manifold;