Abstract Algebra

代数学-1

Syllabuses on Algebra and Number Theory

Linear Algebra

Abstract vector spaces; subspaces; dimension; matrices and linear transformations; matrix algebras and groups; determinants and traces; eigenvectors and eigenvalues, characteristic and minimal polynomials; diagonalization and triangularization of operators; invariant subspaces and canonical forms; inner products and orthogonal bases; reduction of quadratic forms; hermitian and unitary operators, bilinear forms; dual spaces; adjoints. tensor products and tensor algebras;

Integers and polynomials

Integers, Euclidean algorithm, unique decomposition; congruence and the Chinese Remainder theorem; Quadratic reciprocity ; Indeterminate Equations. Polynomials, Euclidean algorithm, uniqueness decomposition, zeros; The fundamental theorem of algebra; Polynomials of integer coefficients, the Gauss lemma and the Eisenstein criterion; Polynomials of several variables, homogenous and symmetric polynomials, the fundamental theorem of symmetric polynomials.

Group

Groups and homomorphisms, Sylow theorem, finitely generated abelian groups. Examples: permutation groups, cyclic groups, dihedral groups, matrix groups, simple groups, Jordan-Holder theorem, linear groups (GL(n, F) and its subgroups), p-groups, solvable and nilpotent groups, group extensions, semi-direct products, free groups, amalgamated products and group presentations.

Ring

Basic properties of rings, units, ideals, homomorphisms, quotient rings, prime and maximal ideals, fields of fractions, Euclidean domains, principal ideal domains and unique factorization domains, polynomial and power series rings, Chinese Remainder Theorem, local rings and localization, Nakayama’s lemma, chain conditions and Noetherian rings, Hilbert basis theorem, Artin rings, integral ring extensions, Nullstellensatz, Dedekind domains,algebraic sets, Spec(A).

Module

Modules and algebra Free and projective; tensor products; irreducible modules and Schur’s lemma; semisimple, simple and primitive rings; density and Wederburn theorems; the structure of finitely generated modules over principal ideal domains, with application to abelian groups and canonical forms; categories and functors; complexes, injective modues, cohomology; Tor and Ext.

Field

Field extensions, algebraic extensions, transcendence bases; cyclic and cyclotomic extensions; solvability of polynomial equations; finite fields; separable and inseparable extensions; Galois theory, norms and traces, cyclic extensions, Galois theory of number fields, transcendence degree, function fields.

Group representation

Irreducible representations, Schur’s lemma, characters, Schur orthogonality, character tables, semisimple group rings, induced representations, Frobenius reciprocity, tensor products, symmetric and exterior powers, complex, real, and rational representations.

Lie Algebra

Basic concepts, semisimple Lie algebras, root systems, isomorphism and conjugacy theorems, representation theory.

Combinatorics (TBA)

References:

Strang, Linear algebra, Academic Press.
I.M. Gelfand, Linear Algebra

《整数与多项式》冯克勤余红兵著高等教育出版社
Jacobson, Nathan Basic algebra. I. Second edition. W. H. Freeman and Company, New York, 1985. xviii+499 pp.

Jacobson, Nathan Basic algebra. II. Second edition. W. H. Freeman and Company, New York, 1989. xviii+686 pp.

S. Lang, Algebra, Addison-Wesley
冯克勤，李尚志，查建国，章璞，《近世代数引论》
刘绍学，《近世代数基础》
J. P. Serre, Linear representations of finite groups
10.J. P. Serre: Complex semisimple Lie algebra and their representations

11.J. Humphreys: Introduction to Lie algebra and representation theory, GTM 009.

12.W. Fulton, Representation theory, a First Course, GTM 129.

1，代数学简史、线性方程组、矩阵，把线性方程组与矩阵化为阶梯形，Gauss消去法、低阶行列式、集合与映射、二元关系、等价关系、商映射、偏序集。

2，数学归纳法、置换、置换的循环结构、置换的符号、斜对称函数、数论的基本概念、算术基本定理。

3，向量与纯量、线性组合、线性相关与线性无关、直线的线性无关性，线性无关的基本引理，基与维数、线性方程组的基与秩，矩阵的秩、利用矩阵的秩研究线性方程组的相容性，线性方程组的可解性准则、线性映射、线性变换、线性函数、矩阵的运算、逆矩阵、矩阵的等价类、线性方程组的基本解系。线性方程组的解空间。

3。有限集合上的置换，置换的奇偶性，交换群，置换分解为轮换的积。

4，方阵的行列式，作为有向体积的行列式、行列式的基本性质、子式、余子式、行列式的展开。

5，非退化行列式的判定、行列式值为零的判据，矩阵按行（列）展开，伴随矩阵、线性方程组的Cramer法则、加边子式法、作为多

重线性规范反对称函数的行列式。矩阵的秩定理，范德蒙德行列式。

5。矩阵的运算及其性质，逆矩阵，求拟矩阵的方法，基本变换，矩阵的基本变换和基本矩阵的乘积的关系。

6。基本代数系统：群、环、域，剩余类环，域的特征。

6，二元运算、半群、幺半群、群、子群、循环群、群的同构、Cayley定理、群的同态与自同态、环、同余类、剩余类环、环的同态、整环、域、域的同构与自同构、域的特征、素域、复数域、复数的几何表示，本原根、复数的几何、交比。复数的三角和代数形式。

7，域上单变量多项式环，一元多项式环、多元多项式环、多项式分解成有余式的非平凡多项式之积的可能性与唯一性，唯一析因环、两个多项式的最大公约式，环中的最大公因与最小公倍、环中元素的互素、整除性的判定、欧几里德算法，Euclid环、既约多项式、本原多项式、Gauss引理、Eisentein判别法。

8。整数域与多项式域的分裂，实数域与复数域上的不可约多项式。

9。多项式的根，形式导数及其性质，多项式根的标准分解式，多项式根的范围，笛卡尔定理，计算多项式实根数量的斯图姆方法。

8，整环的分式域、有理函数域、最简分式、Bezout定理、多项式函数环、插值多项式，Lagrange多项式与Newton插值公式、多项式环的微分法、Vieta公式、对称与斜对称函数、Wilson定理。

9，对称多项式环、多称多项式的基本定理、待定系数法、等幂和、Newton公式、多项式的判别式、结式、复数域的代数封闭性、代数基本定理、计算多项式实根数量的Strum定理、多项式根的近似算法、整系数多项式的有理根。

11。有理分数域，有理分实式分解成部分分式之和，实数与复数域上的情况。

12。多变量多项式环，对称多项式，他的基本对称多项式展开，维特定理。

13。循环群，他的子群，循环群的同构，群的分解的拉格朗日定理。

14。多项式的判别式，多项式的判别式与他们的根的关系。

10，一般域上的线性空间、子空间、线性相关、线性无关、向量组的秩、基与维数、不同基之间的过渡矩阵、线性空间的同构、子空间的交与和、维数定理、直和、补空间、商空间、线性函数、对偶空间、线性无关的判别法。

11，线性映射、线性映射的矩阵表示、像与核、线性算子、线性算子代数、极小多项式、矩阵的相似、线性算子的行列式与迹。

12，不变子空间、特征值与特征向量、特征多项式、特征子空间、几何重数与代数重数、可对角化算子的判别法、不变子空间的存在性、共轭线性算子、商算子。

代数学-2

1，范畴、函子、Hamilton-Cayley定理、Jordan标准型、根子空间、循环子空间、循环矩阵、矩阵的有理标准型。

2，多项式矩阵、多项式矩阵的初等变换、多项式矩阵的相抵、Smith标准型、行列式因子、不变因子、初等因子组、特征方阵与Jordan标准型的关系、实方阵的实相似。

3，多重线性映射、双线性型、矩阵的相合变换、双线性型的秩、左根基、对称双线性型与斜对称双线性型、二次型、二次型的规范型、化二次型为规范型的方法、实二次型、惯性定理、正定二次型与正定矩阵、Jacobi方法、Sylvester定理、斜对称二次型的规范型、Pfaff型。

4， Euclid空间、内积、标准正交基、Gram-Schmidt正交化过程、Euclid 空间的同构、正交矩阵、正交群、辛空间、辛群、辛算子、酉空间、Hermite型、酉矩阵、酉群、赋范线性空间、按模收敛、绝对收敛。

5，内积空间上的线性算子、化二次型为主轴形式、把两个二次型同时化为规范型、保距算子的规范形式、极分解、奇异值分解、Schur定理、Witt扩张定理、复结构、复化线性空间、实化线性空间、实化线性算子、复化算子、最小二乘法、球面多项式、加权正交。

6，线性算子的范数、线性群的单参数子群、谱半径、仿射空间、仿射映射、仿射空间的同构、仿射子空间、仿射坐标系、仿射同构、Euclid度量、Gram行列式、有向体积。

7，仿射群、Euclid空间的运动群、保距变换群、凸集、Minkowski空间、伪欧氏空间、Lorenz群、仿射空间上的二次函数、化二次函数为规范型、Euclid空间上的二次函数。

8，二次曲面、二次曲面的中心、仿射空间中二次曲面的规范型、二次曲面

的分类、Euclid空间中的二次曲面、射影平面、高维射影空间、齐次坐标、仿射几何与射影几何的关系、代数簇、射影群、交比与重比、射影空间中二次曲面的分类、直线与射影二次曲面的相交。

9，张量的概念、张量的坐标、张量积、张量的卷积、对称与斜对称张量、张量空间、外代数。

1。群，子群，群的同构，群的阶，循环群，对称群的生成集，交错群与线性群，凯莱定理。

10，正规子群、左陪集与右陪集、代表元、Lagrange定理、循环群的结构、群在集合上的作用，群作用、轨道、稳定子群与轨道，稳定子群、群的共轭类。正规化子、可迁群、齐次空间。

11，典型群、满同态、四元数代数、置换群、对称。

12，商群、群的同态，同态基本定理、群的同构定理、

3。环的理想，商环，环的同太定理，多项式的分裂域，有限域。

换位群，换位子群、群的直积与半直积、生成元、自由群、可解群、有限R-群与对角矩阵群的可解性，有限阶交错群。单群。

1， Zassenhaus引理、Jordan-Holder定理、带算子的群、自同态环、自同构类群、Sylow定理、特征子群、Abel群、自由阿贝尔群，有限生成的Abel群、Frobenius-Stickelberger定理、有限Abel群的基本定理。有限生成阿贝尔群的结构定理。

7。域上的代数，实数域上的弗罗比尼斯定理，李代数。

8。有限群的表示论初步，特征标，可约表示与完全可约表示，有限群的完全可约表示的马舒克定理，群的正则表示理论，特征标的正则性，不可约表示的维数。

代数学-3

2，良序集、Zorn引理、选择公理、态射、自然变换、环的理想、商环、同态基本定理、环的同构定理、理想的运算、局部化、素理想。

3， Gauss整数、主理想环、

1。主理想环与Euclid环，主理想环与Euclid环上有限模的结构。

极大理想、分裂环上的多项式环的因式分解。唯一因子分解环的多项式扩张、环的直和、中国剩余定理、模、子模、模同态、商模、正合列、模的第一同构定理、循环模、直积与直和、自由模、环的整元素。

4，主理想环上的有限生成模、Noether环，Neother归纳原理、Noether环上的有限模，Hilbert定理与主理想定理。

4。Noether环的幂零根与Jacobson根。

Artin模、Neother模、Krull定理、模的同构定理、投射模、内射模、模的张量积。

5，域的扩张、有限扩张域，代数扩张、代数封闭。超越扩张、分裂域、Kronecker定理、可分多项式、有限域扩张、有限域的子域、有限域的自同构、Mobius反演公式、

分圆多项式。

6。有限扩张环，完全闭合子环。

7。正规环，有限扩张商域上的完全闭合正规Noether环。

8。有限生成代数，正规化的Noether引理，有限生成代数的有限扩张域与Jacobson根的同态。

6，代数闭域、域扩张的自同构、Galois群、Artin引理、Galois扩张、Galois理论主定理、尺规做图问题、三等分角问题、倍立方问题、分圆扩张、不可约性判别法、Brauer定理、Dedekind定理、Artin定理、正规基。

9。有限生成代数的超越次数。

10。仿射代数蔟，Hilbert零点定理，代数蔟的直积。

7，循环扩张、交换扩张、可解扩张、范数和迹、Speiser定理、Artin-Speiser定理、方程可用根式解的判别法、表示、表示空间、表示模。

8，酉表示、Maschke定理、多面体群、Schur定理、特征标、对称群的表示、Young图、Young表、不可约表示、交换群的表示、特征标群、Frobenius互反定理。

11。Zarisky拓扑，代数蔟的不可约性判别准则。不可约流形上直积的不可约性。

12。不可约代数流形的维数，直积与子流形的维数。

13。多项式域的扩张，他的存在性与唯一性，有限域。

14。Galois扩张，可分多项式的分裂域的Galois扩张理论，三次多项式的Galois群，一般多项式，分圆多项式与有限域。

15。Galois对应，代数方程的可解性。

9， SU(2)群和SU(3)群的表示、表示的张量积、特征标环、有限群中的刚性与有理性、结合代数、商代数、中心单代数、Wedderburn-Artin定理、可除代数、Wedderburn定理、代数的线性表示、表示的不可约与完全可约，完全不可约表示的子空间的不变性，Burnside定理。

10，矩阵Lie群、矩阵紧Lie群、矩阵Lie群的同态与同构、特殊线性群的极分解、Lie群、Lie代数、Lie代数的表示。

17。紧致线性群：完全不可约与不变轨道的分割，不变量的Hilbert定理。

18。代数闭域上的有限维结合代数的结构，线性表示的不可约性。

19。有限群的不可约线性表示的维数定理，特征标与矩阵元的正交性。

20。四元数代数及其与SO3和SO4群的关系，广义四元数代数。

21。Euclid平面与空间上的运动的有限群。

22。晶体群，Biberbach定理，晶体群的分类。

23。一维与二维上同调群，晶体群的抽象结构的Biberbach定理与Zassenhaus定理。

Groups &Abstract Algebra

1，A.I.Kostrikin，代数学引论，科学出版社，1977。

2，A.I.Kostrikin，代数学引论，物理数学文献出版社，2000。

3，E.B.Vinberg，代数学教程，法克斯特里亚出版社，2001。

4，A.G.Kurosh，高等代数教程，科学出版社，1971。

5，A.I.Kostrikin，代数学习题集，物理数学文献出版社，2001。

代数学

二年级，第一学期

参考书：

1，A.I.Kostrikin，代数学引论，科学出版社，1977。

2，A.I.Kostrikin，代数学引论，物理数学文献出版社，2000。

3，E.B.Vinberg，代数学教程，法克斯特里亚出版社，2001。

4，A.I.Kostrikin，代数学习题集，物理数学文献出版社，2001。

代数学的附加章节

第三学年全年课程

参考书目：

1，E.B.Vinberg，代数学教程，法克特里亚出版社，2002。

2，M.Atiyah、I.McDonald，Introduction To Commutative Algebra，Addison-Wesley，1969。

3，S.Lang，Algebra，Springer，2005。

Basic Algebra I&II, 2nd Edition by N. Jacobson
Algebra by Serge Lang
Dummit & Foote “Abstract Algebra” Wiley
Hungerford “Abstract Algebra: An Introduction” Brooks/Cole

Friedberg “Linear Algebra” 4th ed. Prentice Hall
Axler “Linear Algebra Done Right” 2nd ed. Springer-Verlag
Hoffman & Kunz , Linear Algebra

《抽象代数（Schaum’s题解精萃影印版）》

A.I. Kostrikin，Introduction to algebra，Springer-Verlag

N.Jacobson “Basic Algebra I,II”

N. Jacobson “Lectures on Abstract Algebra”(GTM.30,31,32) (中译本:抽象代数学,共三卷,理图里有)

Abstract Algebra Dummit：

Algebra Hungerford：

Algebra M,Artin：

a first course in abstract algebra by rotman。

Advanced Modern Algebra by Rotman：

Algebra：a graduate course by Isaacs：

《近世代数概论》Garrett Birkhoff;Saunders Mac Lane

Robinson “A course in the theory of Groups”(GTM 80)

10.E.Artin “伽罗华理论”

Edwards “Galois Theory”(GTM 101)

丁石孙,聂灵沼 “代数学引论”

徐诚浩 “抽象代数–方法导引”

莫宗坚《代数学》(上,下)” 北京大学出版社

熊全淹《近世代数》武汉大学出版社

库洛什 “群论”

冯克勤《近世代数引论》中国科学技术大学出版社

聂灵沼《代数学引论》高等教育出版社

《代数学》范德瓦尔登

《代数学引论》柯斯特利金

《近世代数基础》张禾瑞

《抽象代数基础》丘维声

《代数学引论》聂灵沼，丁石孙

《抽象代数》（1、2）赵春来等著

《近世代数》杨子胥

《近世代数习题解》杨子胥

此外还有一套《离散数学习题集》里面有

《抽象代数分册》张立昂

《抽象代数学》姚幕生（复旦）

《近世代数基础》刘绍学

《近世代数引论》章璞，李尚志，冯克勤（中科大）

《抽象代数》盛德成（西安）

《整数与多项式》冯克勤余红兵著高等教育出版社

冯克勤，李尚志，查建国，章璞，《近世代数引论》

刘绍学，《近世代数基础》

205《近世代数引论》冯克勤

206《代数学》(上,下) 莫宗坚

207 《近世代数》熊全淹

208《近世代数》盛德成

209《代数学引论》丁石孙，聂灵沼

【习题集】

210《抽象代数–方法导引》徐诚浩

【提高】

211《Algebra》S.Lang

212《伽罗华理论》E.Artin

Algebra -1

Syllabuses on Algebra and Number Theory

Linear Algebra

Integers and polynomials

Group

Ring

Module

Field

Group representation

Lie Algebra

Basic concepts, semisimple Lie algebras, root systems, isomorphism and conjugacy theorems, representation theory.

Combinatorics (TBA)

References:

1. Strang, Linear algebra, Academic Press.

2. I.M. Gelfand, Linear Algebra

3. Feng Keqin of the integers and polynomials Yu Hongbing with higher education press

4. Jacobson, Nathan Basic algebra. I. Second edition. W. H. Freeman and Company, New York, 1985. xviii+499 pp.

5. Jacobson, Nathan Basic algebra. II. Second edition. W. H. Freeman and Company, New York, 1989. xviii+686 pp.

6. S. Lang, Algebra, Addison-Wesley

7. Feng Keqin, Li Shangzhi, Cha Jianguo, Zhang Pu, an introduction to the modern algebra

8. Liu Shaoxue, the Foundation of modern algebra

9. J. P. Serre, Linear representations of finite groups

10. J. P. Serre: Complex semisimple Lie algebra and their representations

11. J. Humphreys: Introduction to Lie algebra and representation theory, GTM 009.

12. W. Fulton, Representation theory, a First Course, GTM 129.

1 , Brief history, algebra linear equations, matrix, stepped into linear equations and matrices,Gauss elimination method, low-order determinant, collection and mapping, binary relations, equivalence relations, business map, a partially ordered set.

2 , Mathematical induction, replacement, replacement of the loop structure, replacement of symbols, skew-symmetric functions, basic concepts of number theory, the fundamental theorem of arithmetic.

3 , Vector and scalar, linear combination, linear correlation and linear independence and the linear independence of the line, the fundamental lemma of the linearly independent, bases and dimensions and the base with the rank of linear equations, matrix of rank, the rank of the matrix compatibility study of linear equations, Linear equations by solvability criteria, linear mapping, linear transformation, linear function inverse matrix, matrix, matrix operations, the equivalent class, fundamental system of solutions of linear equations. The solution space of linear equations.

3 。 Replacement on a finite set, parity of the permutation, Abelian groups, product of permutation is decomposed into rotation.

4 , The determinant of square matrices, As to the volume of the determinant, Determinant of a basic nature, minor, minors, the determinant of a start.

5 , Non-degenerate type of judgement, the determinant criterion of zero matrices by row (column), with matrices, linear equations of theCramer law, jiabianzi law, as

Determinant of a linear specification antisymmetric function. Theorem of rank of a matrix, determinant fandemengde.

5 。 Operation and characteristics of the matrix, inverse matrix, methods of calculating the proposed matrix, fundamental transformation, product of the basic transformation of matrix and fundamental matrices.

6 。 Basic algebra: groups, rings, fields and residue class rings, domains feature.

6 , Two Yuan operation, and half group, and Yao half group, and group, and child group, and cycle group, and group of isomorphism, andCayley theorem, and group of with State and since with State, and ring, and with more than class, and remaining class ring, and ring of with State, and whole ring, and domain, and domain of isomorphism and since isomorphism, and domain of features, and pigment domain, and plural domain, and plural of geometry said, primitive root, and plural of geometry, and make than. Plural form of trigonometry and algebra.

7 , Univariate polynomial rings over fields, Monadic polynomial ring, a ring of multivariate polynomials, Polynomial decomposition product of the non-trivial polynomial potential and uniqueness, The only factorial rings, Largest Convention of two polynomials, Ring the least common multiple and greatest common, central elements of coprime, divisibility of the judgment, Euclid’s algorithm,Euclid Ring, irreducible polynomial and primitive polynomial, Gauss Lemmas, Eisentein Criterion.

8 。 Division of the field of integers and polynomials, the Reals and complex field of irreducible polynomials.

9 。 The roots of polynomials, derivatives and their properties, standard decomposition of roots of polynomials, polynomial root scope, Descartes ‘ theorem, calculate the number of real roots of a polynomial method of Sturm und drang.

8 , Integral field of fractions, rational function field, the most simple fractions,Bezout theorem, polynomial function loop, interpolation polynomial,Lagrange polynomial Newton Interpolation, polynomial rings of differential method,Vieta formula, symmetric and skew-symmetric functions,Wilson theorem.

9 , Symmetric more items type ring, and more said items type of basic theorem, and pending coefficient method, and, power and, andNewton formula, and more items type of discriminant type, and knot type, and plural domain of algebra closed, and algebra basic theorem, and calculation more items type real roots number of Strum theorem, and more items type root of approximate algorithm, and The rational roots of polynomials with integer coefficients.

11 。 Rational fraction field, rational real type is decomposed into partial fractions, and real and complex fields.

12 。 Multivariate polynomial ring, symmetric polynomials, his fundamental symmetric polynomials, Victor theorem.

13 。 Cyclic group, his subgroups, cyclic group isomorphism, cluster decomposition theorem of Lagrange.

14 。 The discriminant of the polynomial, the discriminant of the polynomial relation with their roots.

10 , General fields of linear spaces, subspaces, linear, linear independent vector group of rank, base and dimension, the transition between different matrices, linear space isomorphism, subspace, dimension theorem, direct and, space, commercial space, linear functions, independent of the dual space, linear discriminant method.

11 , Linear maps, matrix representation of linear mapping, such as nuclear, linear operator, linear algebras, minimal polynomials, matrices are similar, the determinant and trace of a linear operator.

12 , Invariant subspace, eigenvalues and eigenvectors, characteristic polynomial, characteristic subspaces, geometry, number and algebra and diagonalizable operators to distinguish law, the existence of invariant subspaces, conjugate linear operators and commercial operators.

Algebra -2

1 , Categories, functors and theHamilton-Cayley theorem, theJordan standard form, root space, cyclic subspace, matrix, matrix rational canonical form.

2 , Matrix polynomials, polynomial of matrix elementary transformation, polynomial matrix,Smith standard, determinant factor, invariant factors, primary group, feature matrix and Jordan standard form similar to the real, real square matrices.

3 , Multiple linear mapping, and double linear type, and matrix of consistency transform, and double linear type of rank, and left Foundation, and symmetric double linear type and oblique symmetric double linear type, and two times type, and two times type of specification type, and of two times type for specification type of method, and real two times type, and inertia theorem, and definite two times type and definite matrix, andJacobi method, andSylvesterTheorem, skew-symmetric quadratic norm type, Pfaff Type.

4 , Euclid spaces, inner product, the standard orthogonal basis,Gram-Schmidt orthogonalization process,Euclid space isomorphism, orthogonal matrices, orthogonal groups, and symplectic space and symplectic, unitary, Symplectic operator space,Hermite , Unitary matrices, the unitary group, normed linear spaces, according to the mode of convergence, absolute convergence.

5 , Within product space Shang of linear operator, and of two times type for spindle form, and put two a two times type while into specification type, and insurance from operator of specification form, and very decomposition, and singular value decomposition, andSchur theorem, andWitt expansion theorem, and complex structure, and complex of linear space, and real of linear space, and real of linear operator, and complex of operator, and Least squares, spheres, weighting orthogonal polynomials.

6 , Norm of a linear operator, a linear group of one-parameter subgroups, the spectral RADIUS, affine space, affine mapping, affine space isomorphism, affine subspace, affine affine coordinate system, isomorphism,Euclid metrics,Gramdeterminant, to the volume.

7 , Affine group,Euclid space group of motions, distance transformation groups, convex sets,Minkowski space, pseudo-Euclidean space,Lorenz Group, an affine space of quadratic function, quadratic function to standardize the type,Euclid Space of quadratic functions.

8 , Quadric, quadric surface Center, standardization of quadric surface in affine space, quadratic surfaces

The classification, Euclid Quadric surface in space, the projective plane, higher dimensional projective space, homogeneous coordinates and projective geometry, affine geometry, algebraic and projective groups, than with the weight ratio of quadric surface in projective space classification, linear and projective quadric surfaces intersect.

9 , Concepts of tensor, tensor coordinates, Zhang Liangji, tensor of convolution, symmetric and skew-symmetric tensor, tensor space, the exterior algebra.

1 。 Group, subgroup isomorphic to group, group order, cyclic groups, generated set of symmetric groups, linear groups and alternating groups, Cayley’s theorem.

10 , Normal subgroups, representatives of left cosets and right cosets, Yuan,Lagrange theorem, cyclic groups of structures, the role of groups in the collection, Group action, rail, stabilizers and orbit, stable subgroup, the conjugacy classes of the group. Normalizer, transitive groups and homogeneous spaces.

11 , Typical groups, homomorphism, the Quaternion algebra, permutation groups, symmetry.

12 , Quotient groups, and Group homomorphisms, Basic theorems, isomorphism theorems, a homomorphism

3 。 Ring ideal, quotient ring, ring too theorem, the splitting field of the polynomial, finite fields.

Transposition group Commutator direct product of groups, group and Semidirect products, generators, free groups, solvable groups, Limited R- Solvability of groups and matrix groups and alternating groups of finite order. Simple groups.

1 , Zassenhaus lemma and theJordan-Holder theorem, the operator group automorphism, endomorphism rings, groups and theSylow theorems, characteristic subgroup,Abel Group Free Abelian group, Finitely generated Abel Group, Frobenius-Stickelberger Theorem, finite Abel The fundamental theorem of group. Structure theorem for finitely generated Abelian groups.

7 。 Domain algebra, Florida binisi theorem on the real numbers, algebras.

8 。 The representation theory of finite groups initially, characters, representation and full representation, Mashooq full representation theorem for finite groups, regular representation of a group theory, regularity of the character, the dimension of the irreducible representation.

Algebra -3

2 , A well-ordered set,Zorn ‘s lemma, the axiom of choice and the morphisms, natural transformations, ideals and quotient ring of a ring, with the State the fundamental theorem, ring isomorphism theorems, ideal for operations, localization, a prime ideal.

3 , Gauss integral, a principal ideal ring,

1 。 Principal ideal rings and Euclid loop, principal ideal rings and Euclid limited model on the ring structure.

Great ideals, Factorization of polynomial rings over a Division ring. Unique factorization of polynomial expansion of the ring, ring, Chinese remainder theorem, die, die, die homomorphism, quotient module, right column, die the first isomorphism theorem, loop mode, direct product and direct sum, free mode, loops the entire element.

4 , Principal ideal rings of finitely generated modules, andNoether ring,Neother induction principle,Noether limited modules over rings,Hilbert Theorem and the principal ideal theorem.

4 。 Noether of nilpotent ring with Jacobson .

Artin Mold, Neother Mold, Krull Isomorphism theorem, theorem, die cast mold, injective tensor products, die.

5 , Domain expansion, finite extension fields, algebraic expansion, algebraically closed. Beyond expansion, splitting field,Kronecker theorem, can be divided into a number of children of the expansion, finite fields, finite fields, finite fields isomorphic andMobius inversion formula,

Cyclotomic polynomial.

6 。 Finite extension rings, completely closed loops.

7 。 Regular rings and finite extension field on a closed formal Noether ring.

8 。 Finitely generated algebra and regularization of Noether ‘s lemma, a finitely generated algebra of finite extension field Jacobson homomorphism.

6 , Algebraically closed fields, automorphisms of field extensions,Galois groups,Artin lemmas,Galois expansion,Galoistheory the master theorem, ruler diagram problems Trisection of angle problems, Doubling the cube problem, expanding circle, not about sexual discrimination law,Brauer theorem,Dedekind theorem,Artin theorem, normal basis.

9 。 Finitely generated algebra over the number.

10 。 Affine algebraic nest,Hilbert nullstellensatz, the direct product of algebraic mounting.

7 , Cycle solutions Exchange expansion, expansion, expansion, norm and trace andSpeiser theorem, theArtin-Speisertheorem, radical solutions to the equation of discriminant method, presentation, presentation spaces, said die.

8 , Unitary representation,Maschke theorem, polyhedral group,Schur theorem, character, representation of the symmetric group,Young diagrams,Young table, irreducible representations, Characters of group representation, Exchange groups,Frobenius reciprocity theorem.

11 。 Zarisky topology, algebraic cocooning irreducibility criterion. An irreducible manifold of a direct product of irreducible.

12 。 Dimension of an irreducible algebraic manifold, direct product and submanifolds of dimension.

13 。 Domain of polynomial expansion, his existence and uniqueness of a finite field.

14 。 Galois expansion can be divided into the splitting field of the polynomial’s Galois expansion theory, cubic polynomial’s Galois Group, generic polynomials, cyclotomic polynomial and finite fields.

15 。 Galois correspondence, solvability of algebraic equations.

9 , SU (2) and the SU (3) Group representations, tensor product, feature ring, rigid and rational in a finite group, the Union, Central simple algebra, algebra, algebraWedderburn-Artin theorem, division algebra, Wedderburn theorem, algebraic linear representation, irreducible and completely reducible, completely irreducible representations of subspaces invariant,Burnsidetheorem.

10 , Matrix Lie groups, matrix compact Lie Group, the matrix Lie Group homomorphism and isomorphism, special linear group of polar decomposition,Lie Group andLie Algebra,Lie algebra representations.

17 。 Compact linear group: altogether about as much as the same orbit segment invariant Hilbert theorem.

18 。 Finite-dimensional associative algebra over an algebraically closed field structure, linear representations of irreducibility.

19 。 Irreducible linear representations of finite dimension theorem, character and matrix element of orthogonality.

20 。 Quaternion algebra and its SO3 and SO4 groups and generalized Quaternion algebra.

21 。 Euclid the motion of plane and space of finite groups.

22 。 Crystallographic groups,Biberbach theorem, the classification of crystallographic groups.

23 。 One and two dimensional cohomology groups, Crystallographic groups the abstract structure of Biberbach theorem and the Zassenhaus theorem.

Groups &Abstract Algebra

1 , A.I.Kostrikin , Introduction to algebra, science press, 1977 。

2 , A.I.Kostrikin , Introduction to algebra, physics and mathematics literature Publishing House, 2000 。

3 , E.B.Vinberg , Algebra tutorials, Ashfaq, Asia Publishing House, 2001 。

4 , A.G.Kurosh , Advanced algebra tutorials, science press, 1971 。

5 , A.I.Kostrikin And algebra problem sets, and physical-mathematical literature Publishing House, 2001 。

Algebra

Second year first semester

Reference books:

1 , A.I.Kostrikin , Introduction to algebra, science press, 1977 。

2 , A.I.Kostrikin , Introduction to algebra, physics and mathematics literature Publishing House, 2000 。

3 , E.B.Vinberg , Algebra tutorials, Ashfaq, Asia Publishing House, 2001 。

4 , A.I.Kostrikin And algebra problem sets, and physical-mathematical literature Publishing House, 2001 。

Additional sections of the algebra

Third year year courses

Bibliography:

1 , E.B.Vinberg , Algebra tutorials, faketeliya Publishing House, 2002 。

2 , M.Atiyah 、 I.McDonald , Introduction To Commutative Algebra , Addison-Wesley , 1969 。

3 , S.Lang , Algebra , Springer , 2005 。

Basic Algebra I&II, 2nd Edition by N. Jacobson
Algebra by Serge Lang
Dummit & Foote “Abstract Algebra” Wiley
Hungerford “Abstract Algebra: An Introduction” Brooks/Cole

Friedberg “Linear Algebra” 4th ed. Prentice Hall
Axler “Linear Algebra Done Right” 2nd ed. Springer-Verlag
Hoffman & Kunz , Linear Algebra

Abstract algebra ( Schaum’s Solution highlights print version)

A.I. Kostrikin , Introduction to algebra , Springer-Verlag

N.Jacobson “Basic Algebra I,II”

N.Jacobson”LecturesonAbstractAlgebra”(GTM.30,31,32)( translation : abstract algebra , three volumes , )

Abstract Algebra Dummit :

Algebra Hungerford :

Algebra M,Artin :

a first course in abstract algebra by rotman 。

Advanced Modern Algebra by Rotman :

Algebra : a graduate course by Isaacs :

The introduction of modern algebra Garrett Birkhoff; Saunders Mac Lane

Robinson “A course in the theory of Groups”(GTM 80)

10.E.Artin ” Galois theory”

Edwards “Galois Theory”(GTM 101)

Ding shisun , Nie Ling Moor ” An introduction to algebra”

Xu Chenghao ” Abstract algebra — Method guidance”

Mo Zongjian algebra ( Shang , Xia )” Peking University Press

Xiong Quanyan modern algebra, Wuhan University Press

Kulos ” Group theory”

Feng Keqin The introduction to modern algebra University of science and technology of China Press

Nie Ling moor the introduction to algebra, higher education press

The algebra of van der waerden

Introduction to the algebra of kesitelijin

Rui Zhang Wo, the Foundation of modern algebra

The sound based on high-dimensional abstract algebra

The introduction to algebra Nie Ling Moor, Ding shisun

Of the abstract algebra ( 1 、 2 ) Zhao chunlai waiting

Of the modern algebra Yang Zixu

Of the solution to modern algebra learning Yang Zixu

In addition there is a set of discrete mathematics problem set

Of the abstract algebra fascicle Zhang Liang

The abstract algebra Yao (Fudan University)

The Foundation of modern algebra Liu Shaoxue

The introduction to modern algebra Zhang Pu, Li Shangzhi, Feng Keqin (Science)

The abstract algebra Cheng Decheng (XI an)

Feng Keqin Yu Hongbing the integers and polynomials with higher education press

Feng Keqin, Li Shangzhi, Cha Jianguo, Zhang Pu, an introduction to the modern algebra

Liu Shaoxue, of the Foundation of modern algebra

205 The modern algebra an introduction to Feng Keqin

206 The algebra ( Shang , Xia ) Mo Zongjian

207 The modern algebra Xiong Quanyan

208 The modern algebra Cheng Decheng

209 Introduction to the algebra of Ding shisun, Nie Ling Moor

“Onward”

210 The abstract algebra — Xu Chenghao method guidance

“Increase”

211 《 Algebra 》 S.Lang

212 Of the Galois theory E.Artin

Linear Algebra

Linear Algebra

1，A.I.Kostrikin、Yuri.I.Manin，线性代数与几何，莫斯科大学出版社，1980。

2，A.I.Maltsev，线性代数基础，科学出版社，1975。

3，A.G.Kurosh，高等代数教程，科学出版社，1968。

4，P.R.Halmos，Finite-Dimensional Vector Spaces，Springer，1974。

5，A.I.Kostrikin，代数学引论，物理数学文献出版社，2000。

6，I.V.Proskuryakov，线性代数习题集，科学出版社，1974。

7，A.I.Kostrikin，代数学习题集，物理数学文献出版社，2001。

甘特玛赫尔”矩阵论”

A.I. Kostrikin，Introduction to algebra，Exercises in Algebra。Springer-Verlag, 《代数学引论》

M. Postnikov，Linear algebra and differential geometry，Mir Publishers

普罗斯库列科夫《线性代数习题集》

法捷耶夫《高等代数习题集》

Lang. Serge，Linear algebra，Springer-Verlag

Friedberg “Linear Algebra” 4th ed. Prentice Hall
Axler “Linear Algebra Done Right” 2nd ed. Springer-Verlag
Hoffman & Kunz , Linear Algebra

Artin’s Algebra

Matrix theory

Advanced Linear Algebra” by Steven Roman.

[GSM], Dym, Linear Algebra in Action

Lax, Linear Algebra and its Applications

[UTX], Curtis, Abstract Linear Algebra

贾柯勃逊(N.Jacobson) Lectures on Abstract Algebra ,II:Linear Algebra

GTM(Graduate Texts in Mathematics)No.31 (“抽象代数学”第二卷:线性代数)

Greub Linear Algebra(GTM23)

Israel Gelfand，Lectures on Linear Algebra, “Linear Algebra”

“Introduction to Linear and Abstract Algebra”，http://www.math.miami.edu/~ec/book/

Strang, Linear algebra, Academic Press.

蒋尔雄,吴景琨等 “线性代数”

屠伯埙等 “高等代数” , “线性代数-方法导引”

许以超“代数学引论”, 《线性代数与矩阵论》高等教育出版社

华罗庚 “高等数学引论”

丘维声 “高等代数”(上,下)

李炯生,查建国《线性代数》中国科学技术大学出版社

李尚志老师的线性代数

张贤科《高等代数学》清华大学出版社

叶明训《线性空间引论》武汉大学出版社

林成森，盛松柏的“高等代数”，如Sturm序列，Shermon-Morrison公式，广义逆矩阵等

南大出版社出版，可惜好象并轨以后就没有再用了。

国内较好的高等代数教材还有清华计算机系用的那本，清华出版社出版

莫宗坚先生的“代数学”里，对此进行了深刻的讨论

王萼芳和丁石孙的《高等代数》。

《高等代数》北京大学数学系代数与几何教研室前代数小组王萼芳，石生明修订

《高等代数》张禾瑞，郝鈵新

《高等代数简明教程》蓝以中

《理解矩阵》孟岩

“理解矩阵（一）-（三）”

“读《理解矩阵》的一点心得及整理归类”(http://blog.csdn.net/shirley329/)

《高等代数习题解》, 《高等代数精选题解》杨子胥

《高等代数》（大学数学学习方法指导丛书）姚幕生

《高等代数与解析几何》孟道骥

高等代数与解析几何》,《高等代数与解析几何习题精解》陈志杰

此外还有西安交通大学的

《高等代数与几何》潘仲晏

中山大学的

《几何与代数导引》胡国权

50《代数学引论》柯斯特利金著

【习题集】

51《高等代数辅导与习题解答》或《高等代数(北大•第三版)导教•导学•导考》

52《高等代数习题集》第2版(修订本) 法杰耶夫,索明斯基著 ; 丁寿田原译，项观捷等修订

53《线性代数习题集》普罗斯库列柯夫编著

【辅导书】

54《高等代数:定理•问题•方法》胡适耕, 刘先忠编著

55《高等代数习题解》或者《高等代数精选题解》杨子胥著

杨子胥同宋宝和编著了一本《近世代数习题解》也可以作为今后参考。

56《高等代数解题方法》(第2版) 许甫华, 张贤科编著

【提高】

60《线性空间引论》叶明训编著

61《高等代数探究性课题集》邱森, 朱林生主编

62《矩阵分析及其应用》曾祥金，吴华安编著

63《近世代数观点下的高等代数》陈辉著

四、“线性代数”

【教材】

65《通俗线性代数讲义》李徐鸿编著

【习题集】

【辅导书】

66 《线性代数辅导》（第二版）胡金德、王飞燕编

67《线性代数典型题精讲》第2版许甫华编著

【提高】

《项武义基础数学讲义•基础代数学》可以课后翻翻看。

还有《线性代数五讲》龚升编著也可以看看。

Linear Algebra

1 , A.I.Kostrikin 、 Yuri.I.Manin Linear algebra and geometry, Moscow University Press, 1980 。

2 , A.I.Maltsev Linear algebra, science press, 1975 。

3 , A.G.Kurosh , Advanced algebra tutorials, science press, 1968 。

4 , P.R.Halmos , Finite-Dimensional Vector Spaces , Springer , 1974 。

5 , A.I.Kostrikin , Introduction to algebra, physics and mathematics literature Publishing House, 2000 。

6 , I.V.Proskuryakov , Linear algebra problem sets, science press, 1974 。

7 , A.I.Kostrikin And algebra problem sets, and physical-mathematical literature Publishing House, 2001 。

Gantemaheer ” Matrix theory”

A.I. Kostrikin , Introduction to algebra , Exercises in Algebra 。 Springer-Verlag, an introduction to algebra

M. Postnikov,Linear algebra and differential geometry,Mir Publishers

Puluosikuliekefu of the linear algebra problem sets

Fadeyev of the algebra problem sets

Lang. Serge , Linear algebra , Springer-Verlag

Friedberg “Linear Algebra” 4th ed. Prentice Hall
Axler “Linear Algebra Done Right” 2nd ed. Springer-Verlag
Hoffman & Kunz , Linear Algebra

Artin’s Algebra

Matrix theory

Advanced Linear Algebra” by Steven Roman.

[GSM], Dym, Linear Algebra in Action

Lax, Linear Algebra and its Applications

[UTX], Curtis, Abstract Linear Algebra

Gukeboxun (N.Jacobson) Lectures on Abstract Algebra ,II:Linear Algebra

GTM (Graduate Texts in Mathematics) No. 31 (” abstract algebra ” volume II : linear algebra)

Greub Linear Algebra(GTM23)

Israel Gelfand , Lectures on Linear Algebra, ” Linear Algebra “

“Introduction to Linear and Abstract Algebra” , http://www.math.miami.edu/~ec/book/

Strang, Linear algebra, Academic Press.

Jiang erxiong , Wu Jingkun ” Linear algebra”

Tu Boxun ” Advanced algebra ” , ” Linear algebra – Method guidance”

With Super ” An introduction to algebra “, Linear algebra and matrix theory, higher education press

Hua luogeng ” An introduction to higher mathematics”

High-dimensional ” Advanced algebra “( Shang , Xia)

Kenneth Lee Kwing-Chin , Cha Jianguo of the linear algebra of the University of science and technology of China Press

Linear algebra teacher Li Shangzhi

Zhang xianke higher algebra, Tsinghua University Press

Ye Mingxun introduction to the linear space of Wuhan University Press

Lin chengsen, Sheng pine “Advanced algebra”, such as Sturm Sequence, Shermon-Morrison Formulae of generalized inverse matrix

Nanjing University publishing, but seemed to be merged then there is never used again.

Domestic well higher algebra textbook and Tsinghua University Department of computer science, Tsinghua University Press

Mr Mo Zongjian “algebra”, which had a deep discussion

Wang Efang and Ding of the algebra.

The advanced algebra Department of mathematics of Peking University algebra and algebra before geometry teaching group Wang Efang, Shi Ming amendments

The higher algebra he Rui, Hao 鈵xin

The concise course of higher algebra blue

The understanding of matrix yan Meng

” Understanding the matrix (a) – (C) “

” Read the experiences and understanding of matrix categorized ” (http://blog.csdn.net/shirley329/)

Solution to the acquisition of higher algebra , The featured advanced algebra exercises Yang Zixu

The higher algebra (math study guide series) Yao

The higher algebra and analytic geometry Meng daoji

Higher algebra and analytic geometry , The exercises of higher algebra and analytic geometry explained Chan Chi Kit

In addition, XI ‘ an Jiaotong University

The higher algebra and geometry of Pan Zhongyan

Sun Yat-sen University

Of the geometry and algebra guide Hu guoquan

50 An introduction to the algebra kesitelijin of the

“Onward”

51 The guidance and solutions advanced algebra or the advanced algebra ( North-third edition ) • Teaching • learning exam

52 Of the high algebra problem sets 2 ( Revised edition ) Fajieyefu , Sominski with ; Dingshoutianyuan translation Outlook Czech Republic amendments

53 Written by puluosikuliekefu the linear algebra problem sets

“Books”

54 The higher algebra : Geng, theorems, problems and methods of Hu Shi , Written by Liu Xianzhong

55 The higher algebra learning exercises or the higher algebra of selected key Yang Zixu a

Yang Zixu Song Baohe compiled a book on modern algebra learning exercises also can be used as a future reference.

56 Of the high algebra problem solving method ( 2 ) Xu Fuhua , Written by Zhang xianke

“Increase”

60 The introduction to linear space Ye Mingxun authoring

61 The collection of inquiry advanced algebra topics Qiu Sen , Chief Editor Zhu linsheng

62 The matrix analysis and applications Andrew gold, written by Wu Huaan

63 Chen Hui of the modern algebra advanced algebra under the

D “linear algebra”

“Textbook”

65 Popular lectures on linear algebra written by Li Xuhong

“Onward”

“Books”

66 Of the linear algebra tutoring (Second Edition) Hu Jinde, Wang Feiyan series

67 Of the typical problems of linear algebra explains the 2 Written by Xu Fuhua

“Increase”

The Wuyi mathematics lecture elementary algebra lesson turned to look.

There is the linear algebra five talk written by Gong Sheng See also.

Geometry

几何学

1，点线面的相互关系、方向和角度与平行、恒等和叠合与对称、向量的加法和减法、向量与数量的乘法、内积、外积、混合积、向量对于给定基底的坐标。

1。线性向量空间，例子，子空间。

2。线性独立与相关，相关性的记号，与向量分解的关系，展开为线性无关向量之和。

3。多个向量的秩和他们的性质。

2，空间的维数、基与一般笛卡尔坐标。空间曲面和空间曲线的方程、坐标变换、平面方程、平面对于坐标系的位置、平面的相互位置。

4。向量空间的同构定理。

5。子空间的和与交，子空间的和的维数。

6。两个和多于两个子空间的直和，外直和。

3，直线方程、直线和平面的相互位置、两条直线的相互位置、二次曲面分类、椭圆面、双曲面、抛物面、锥面和柱面。

4，二次曲面的直母线、二次曲面的直径和直径平面、二次型的变换、不变量。

5，曲线直径、曲面和曲线的中心、曲线的对称轴、曲面的对称平面、双曲线的渐近线、双曲面的渐近锥面、曲线的切线、曲面的切平面。

6，正交变换、仿射变换、仿射变换的基本不变量、仿射变换下的二次曲线和二次曲面、射影变换、齐次坐标、无穷远点、射影变换下的二次曲线和二次曲面、极点和配极。

7， Euclid几何中的平面与直线、Euclid平面与复数、Euclid空间与仿射空间、仿射簇。

8，仿射直线与仿射平面的公理化模型、平面上的线性方程、凸几何、仿射几何的基本定理、仿射空间、他们的特性，仿射坐标系，向量和点的坐标。有限维凸几何、Caratheodory与Radon引理、Helly定理。

8。仿射与向量空间上的坐标变换，坐标变换的矩阵表示，新旧坐标上点与向量之间的联系。

9。仿射空间的子空间，参数方程，平行六面体。

10。作为线性方程组解集的仿射子空间。

11。仿射空间上两个子空间的相互联系。

12。仿射空间上多个点对线段的划分，分割点的坐标。

13。仿射空间的同构定理，仿射空间和向量空间上概念的等价性。

9，射影几何、射影直线与平面、Pappus与Desargues定理、n维射影空间简介、二次平面曲线的分类、四次方程、Pascal定理。

10，圆与球、球面几何、n维球的几何、Riemann椭圆几何、Lobachevsky几何的Klein模型、线性分式变换与球极投影、Lobachevsky几何的其它模型、初等双曲几何。

11， Euclid几何和Riemann椭圆几何及Lobachevsky几何的同构性、复射影空间、影变换的不动点、调和四重点与调和四重线。

14。标量积，欧氏向量——点空间，正交向量的线性无关性。

15。正交基与标架，正交化过程。

16。欧氏向量——点空间上的同构定理，两点间的有向距离，三角不等式，向量间的角度，正交向量的直角性，毕达哥拉斯定理，柯西——布尼雅科夫斯基不等式，例子。

17。子空间的正交完备性及其相关性质，子空间与向量所成角，子空间与向量所成距离。

18。子空间上向量的射影，傅利叶系数，超定方程组解的最小二乘法。

19。超平面上的正规向量，点到超平面的距离，平行超平面间的距离。

20。格拉姆行列式及其性质。

21。可测平行六面体的体积，体积与矩阵行列式之间的关系。

22。线性空间上的线性映射，矩阵与线性映射的解析形式。

23。线性映射的合成，线性空间合成的矩阵。

24。映射的核与像的维数，同构条件。

25。线性算子，他们的解析形式，基上算子的矩阵的独立性（包括张量情形）。

26。线性算子环，矩阵环的同构，算子的多项式，非退化算子（一般线性群）。

27。不变子空间，对算子的矩阵的影响，实域与复域上的不变子空间问题，算子的矩阵

的阶梯形。

28，算子的特征值与特征向量，特征子空间，特征子空间的和。

29。算子的特征多项式及其相关不变量，特征子空间的等价。

30，特征值的重数与特征子空间的维数，算子的矩阵的对角化条件。

31。多项式与它的零化算子，哈密顿——凯莱定理。

32。退化算子，其与算子的特征多项式系数的关系。

33。把空间分解成不变子空间的直和，其与讲特征多项式分解成基本因子的关系。

34。最小零化多项式，他和特征向量等的关系。

35。根子空间与算子的根向量。

36。根子空间上算子的矩阵的标准型，算子的矩阵的若尔当标准型，最小多项式与若尔当标准型的关系。

37。实空间的复化，线性映射与线性算子。

38，实算子的矩阵的标准型。

39。若尔当标准型的矩阵理论。

40。线性映射与仿射空间上的几何变换，三个点的相互关系，几何变换的解析形式，仿射表换与图形的仿射分类的概念。

41。线性函数空间，半线性函数，伴随基，变换的矩阵，线性与半线性函数空间上的坐标变换。

42。线性与半线性函数空间上的自共轭，自共轭基。

43。复空间上的算子与映射，对偶映射与基础空间上的算子，自对偶。

44。双线性与多线性函数及其坐标表示，向量函数空间，基本函数及其与基础空间的基的关系。

45。基上双线性函数的矩阵无关性，函数的秩。

46。双线性与多线性函数的核及其维数，非退化函数。

47。复空间上的双线性与多线性函数空间的自然同构。

48，对称、斜对称与埃尔米特函数。

49，子空间与向量的正交化及其与对称、反对称与埃尔米特函数的关系，正交补的维数与性质。

50。对称、斜对称与埃尔米特函数的正交形式。

51。函数的正交形式的唯一性，实域上的对称与埃尔米特函数的的惯性定理。

52。二次函数及其与二次性及规范型的关系。

53。规范型的雅可比定理与格拉姆方法。

54。正定系统与埃尔米特函数，西尔维斯特判据。

55。对称、斜对称与埃尔米特标量积，拟欧氏、埃尔米特与辛向量空间及其同构定理，正交化与对称基。

56。内积空间上的自然同构，线性函数的一般形式，零向量和拟欧氏、埃尔米特与辛向量空间的子空间，正交非迷向向量的线性无关性，格拉姆行列式，正交化过程，正交完备性。

57。辛向量空间，哈密顿基，迷向子空间。

58。酉空间，柯西——布尼雅科夫斯基不等式，三角不等式。

59。正交、拟正交、酉、拟酉与辛矩阵，特殊线性群。

60。标量积下不变的算子（正交、拟正交、酉、拟酉与辛）与他们的性质，不变子空间的性质，等距同构，算子群。

61，正交与酉算子的标准型及其唯一性，特征子空间，点空间上的正交变换。

62。群，平面上的伪标量，双曲三角，洛伦兹变换，三维拟欧氏空间。

63。复空间的实化，埃尔米特空间上的拟欧氏结构与辛结构。

64。算子与算子群的实化。

65。内积空间上的伴随算子的存在性与唯一性，与复空降上的对偶算子的联系。

66。欧氏空间与酉空间上的自伴算子及其标准型，特征值与子空间的性质。

67。自伴正交算子与自伴酉算子的极分解。

68。内积空间上的双线性与多线性函数，这些函数与算子空间的自然同构。

69。欧氏（酉）空间上的对称（埃尔米特）函数的标准型，点空间上的二阶超曲面方程的标准型。

70。一对其中之一为正定的二次型的不变量，在标准基上的讨论。71。张量，例子，张量与多线性函数，张量空间。

72。张量积，张量代数，张量空间上的基与坐标。

73。卷积算子及其性质，例子。

74。内积空间上的张量指标。

75。张量的对称与斜对称及其坐标，对称算子与交错张量及其性质。

76。斜张量，外积算子及其性质。

77。多重向量和斜对称函数及其坐标，例子，子空间的普吕克坐标。

78。多重向量和斜对称函数的简化。

79。多重向量和斜对称函数空间的基与维数。

解析几何

一，向量运算。

1。向量，向量的线性运算及其基本性质。

2。向量的线性相关及其几何意义。

3。基，向量子与点组成的坐标系，坐标系的几何意义。

4。线性相关，坐标系上的向量线性相关的判据。

5。数量积，它的主要性质与公式。

6。从一个基到另一个基的变换矩阵，点-向量坐标系的变换，变换矩阵的性质，标准正交基。

7。平面的定向，平行四边形的定向体积，它的基本性质和公式。

8。空间的定向，平行六面体的定向体积，它的基本性质和公式。

9。向量的向量积与混合积，它的基本性质与公式。

10。正交矩阵，正交坐标的变换，二阶正交矩阵的分类。

11。Euler角，用三阶正交矩阵表示Euler角。

12。极坐标系，空间上的柱坐标系与球坐标系，极坐标、柱坐标与球坐标与正交坐标的关系。

13。二重向量与三重向量，二重向量的线性运算与度量理论。

14。二维与三维线性算子，可逆线性算子，等距线性算子。

二，直线与平面。

15。直线与平面的参数方程，它们的集合意义。

16。作为直线或平面方程的一阶方程及其与参数方程的联系。

17。平面与空间中两条直线的相互关系。18。两个平面的相互关系，及其与一阶方程的联系。

19。向量、直线与平面形成的角及其计算方法。

20。从某一点到直线或者平面的距离，两条直线的距离。

21。平面与空间上一阶不等式的几何意义。

22。平面与空间上的线束，线束的线性无关。

23。空间上的平面束，平面束的线性无关。

24。平面上三条直线的相互关系，空间上三个平面的相互关系，他们的方程组的秩。

三，二次曲线与曲面。

25。代数曲线与曲面，代数曲线与曲面的次数，穿过直线的代数曲线，与平面相截的代数曲面。

26。可约曲线与可约曲面及其几何意义，包含直线的曲线的可约定理，包含平面的曲面的可约定理。

27。利用二阶正交变换化二次多项式为标准型，二次曲线的分类。

28。椭圆与双曲线、抛物线的焦点性质，双曲线的渐进线方程。

29。椭圆、双曲线、抛物线的准线性质，他们在极坐标下的方程。

30。圆柱面与圆锥面，旋转面。

31。利用三阶正交变换化二次多项式为标准型，二次曲面的分类。

32。椭球面、虚椭球面和双曲面，他们的基本性质，他们的图像的绘制。

33。椭圆抛物面，他的基本性质及其图像的绘制。

34。双曲抛物面，双曲抛物面的母线及其基本性质。

35。双叶双曲面与单叶双曲面，双叶双曲面与单叶双曲面及其基本性质。

四，正交不变量及二次曲面与曲面的分类。

36。多项式分类的正交不变量，正交不变量与多项式的系数与根。

37。二次多项式的正交不变量。

38。利用正交不变量对正交坐标系上的二次曲线进行分类。

39。利用正交不变量对正交坐标系上的二次曲面进行分类。

40。二次曲线与曲面的中心的坐标。

41。而曲面与曲线的二次方程的比例性。

42。二次曲线的渐进线，二次旋转面的渐进锥面，利用二次曲线与直线的相交对其进行分类。

43。圆锥曲线的直径与方程，旋转面与其中心线平面相交形成的曲线的直径及其方程。

44。曲线的共轭的方向及其直径，椭圆、双曲线与抛物线的共轭的方向及其直径。

45。对称轴，对称轴与直径的关系，对称轴向量的坐标，二次曲线的相互关系。

46。对称平面，对称平面与中心线平面的相互关系，对称平面的法向向量的坐标，二次曲面的相互关系。

47。二次曲线束，通过五个给定点的二次曲线，Sturm定理与Pascal定理。

五，仿射与正交变换。

48。仿射变换的定义及其性质，仿射变换群。

49。从一组基到另一组基的仿射变换的矩阵，它们的几何意义，仿射变换公式。

50。仿射变换群的等价关系，二次曲线的仿射分类。

51。二次曲面的仿射分类。

52。等度量变换的基本性质。

53。二次曲线的正交分类，二次曲线的正交不变量及其标准方程的系数。

54。二次曲面的正交分类，二次曲面的正交不变量及其标准方程的系数。

55。平面正交变换的结构。

56。空间正交变换的结构。

57。平面与空间仿射变换的结构。

六，射影几何，

58。作为直线与平面形成的把的射影平面，作为普通平面的推广的射影平面，点与直线在这两种情况下的射影坐标，对偶原理。

59。圆锥曲面模型上的曲线，二次曲线在扩充平面下的完备性，二次曲线在射影坐标下的方程。

60。射影变换，作为中心仿射变换推广的射影变换。

61。扩充平面上的仿射变换与射影变换的联系。

62。化二次型为标准性的变换的一致性，二次曲线的射影形式与射影分类及其与仿射分类的关系。

63。椭圆与双曲几何的基本概念，Erlangen纲领。

64。几何的公理系统，Euclid几何的公理系统，仿射几何的公理系统，复数域上的仿射几何。

Analytical Geometry

1，M.M.Postnikov，解析几何，科学出版社，1973。

2，P.S.Alexandrov，解析几何讲义，科学出版社，1967。

3，P.S.Alexandrov，解析几何与线性代数教程，科学出版社，1979。

4，B.N.Delone、D.A.Raykov，解析几何，第一卷，国家联合出版社，1948。

5，B.N.Delone、D.A.Raykov，解析几何，第二卷，国家联合出版社，1949.

6，Y.Smirnov，解析几何，俄罗斯教育与科学文献出版社，2004。

7，V.V.Prasolov、V.M.Tikhomirov，几何学，莫斯科不间断数学教育中心，1997。

8，P.S.Modenov、A.S.Parhomenko，解析几何习题集，科学出版社，1976。

Basic Topology by Armstrong
Differential Geometry of Curves and Surfaces by Manfredo Do Carmo
Hatcher “Algebraic Topology” Cambridge UP
Munkries “Topology” 2nd ed. Prentice Hall

A.T.Fomenko Differential geometry and topology，Consultants Bureau
Dubrovin, Fomenko, Novikov “Modern geometry-methods and applications”Vol 1—3
A Comprehensive Introduction to Differential Geometry vol 1-5 ,by Michael Spivak

Bogorelov，解析几何。

《解析几何习题集》巴赫瓦洛夫

狄隆涅 “(解析)几何学”

穆斯海里什维利 “解析几何学教程”

丘维生，《解析几何》，北大版
南开数学系，《空间解析几何》，高教版

陈(受鸟) “空间解析几何学”

朱鼎勋 “解析几何学”

吴光磊《解析几何简明教程》高等教育出版社

丘维声《解析几何》北京大学出版社

《解析几何》吕根林，许子道（有配套的辅导）

《解析几何》尤承业

《空间解析几何与微分几何》（大学数学学习方法指导丛书）黄宣国

《高等几何》梅向明等

《高等几何习题集》

《高等几何》朱德祥

《高等几何》周建伟

175《高等几何学习指导与习题选解》梅向明，刘增贤编

176《高等几何》第2版罗崇善, 庞朝阳, 田玉屏编著

Arithmetic Geometry

X Fulton, William. Algebraic Curves: An Introduction to Algebraic Geometry.

This book is available for free on Fulton’s website.

Buy at Amazon Milne, J. S. Elliptic Curves. BookSurge Publishers, 2006. ISBN: 9781419652578.

This book is also available for free on Milne’s website, along with addendum/erratum.

Buy at Amazon Serre, Jean-Pierre. A Course in Arithmetic. Springer-Verlag, 1996. ISBN: 9783540900405.

Buy at Amazon Shafarevich, I. R. (translated by Miles Reid). Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550.

Buy at Amazon Stichtenoth, H. Algebraic Function Fields and Codes. Berlin: Springer, 2008. ISBN 9783540768777. [Preview with Google Books]

Buy at Amazon Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer-Verlag, 2009. ISBN: 9780387094939. (errata) [Preview with Google Books]

Geometry

1 , Relationship between point, line, plane, directions and angles with parallel, identical and overlapping with symmetry, vector addition and subtraction, multiplication of a vector quantity, product, product, mixing plot coordinates, vectors for a given substrate.

1 。 Linear vector spaces, for example, the subspace.

2 。 Linear independence and correlation, correlation between signs and vector decomposition relationship of linearly independent vectors and.

3 。 Rank of multiple vectors and their properties.

2 , Spatial dimensions, base and General Cartesian Coordinates. Surface and space the equation of the curve, coordinate transformations, plane equation, plane, plane of the coordinate system relative positions.

4 。 Isomorphism theorems for vector spaces.

5 。 Subspace and make, and dimension of subspace.

6 。 Two or more than two spaces and straight, external direct sum.

3 , Linear equations, linear and Planar location, between two straight, quadratic surface classification, ellipsoid, hyperboloid, parabolic surfaces, conical and cylindrical.

4 , Ruled lines of quadratic surface, second surface diameter and diameter plane, transform and invariant of the quadratic form.

5 , Diameter of curves, surfaces, and curved Center, curved axis of symmetry, the symmetry of the surface plane, hyperbolic Asymptote, hyperboloid asymptotic cones, tangent to the curve, surface, tangent plane.

6 , Orthogonal transformations, affine transformations, affine transformation of the basic invariant, affine transformations of Conic Sections and quadric surfaces, projective transformations, homogeneous coordinates, the point at infinity, under the projective transformations of quadratic curve and quadratic surface, pole and polar.

7 , Euclid geometry in the plane and straight lines,Euclid plane with complex numbers,Euclid space and affine space, affine clusters.

8 , The affine line and axiomatic models of the affine plane, the plane of linear equations, convex geometry, affine geometry, the fundamental theorem, an affine space, their characteristics, the affine coordinate system, vectors and coordinates of the point. Finite-dimensional convex geometry,Caratheodory and Radon ‘s lemma, andHelly theorem.

8 。 Coordinates on a vector space and affine transform, coordinate transformation matrices, linkages between the new and old coordinates of points and vectors.

9 。 An affine subspace of the space, parametric equations, parallelepiped.

10 。 As an affine subspace of the set of solutions to linear equations.

11 。 Two subspaces of the affine space of interrelated.

12 。 Division of the affine space multiple points on the line, the coordinates of the point.

13 。 Isomorphism theorems of affine space, affine space and the concept of vector space equivalence.

9 , Projective geometry, lines and planes of projection,Pappus and Desargues theorem, then -dimensional projective space profile, secondary classification of Planar curve, quartic equation,Pascal Theorem.

10 , Round ball, spherical geometry,an n -dimensional spherical geometry,Riemann geometry, an ellipseLobachevskygeometry of the Klein model, Linear fractional transformations with the stereographic projection,Lobachevsky geometry model, elementary hyperbolic geometry.

11 , Euclid geometry or Riemann elliptic geometry or Lobachevsky geometry of homogeneous, complex projective space, shadow transform fixed points, to reconcile the four key and harmony Quartet.

14 。 Scalar product, Euclidean vectors – space, orthogonal vector linear independence.

15 。 Orthonormal frame, orthogonalization process.

16 。 Euclidean vectors – spatial isomorphism theorem, distance between two points, the triangle inequality, angle between the vectors, orthogonal vector at right angles to, the Pythagorean theorem, Cauchy – buniyakefusiji inequalities, examples.

17 。 Subspace properties of orthogonal complete set and its associated, subspaces and the angle of the vector, distance into subspace and vector.

18 。 The projection of the vector subspace, Fourier coefficients, solution of overdetermined equations of least squares.

19 。 Hyperplane normal vector, distance between points and Hyperplanes, the distance between two parallel hyperplanes.

20 。 The gram determinant and its nature.

21 。 Measured volume of the parallelepiped volume relationship with matrix determinant.

22 。 Linear map in linear space, analytic forms of matrices and linear maps.

23 。 Synthesis of linear maps, matrix synthesis of linear spaces.

24 。 Mapping the nuclear dimension, homogeneous conditions.

25 。 Linear operators and their analytical form, independence of the matrix of the base pay (including tensor).

26 。 Ring of linear operators, matrix ring isomorphisms of operator polynomials, non-degenerate operator (the General linear group).

27 。 Invariant subspace, matrix effects on operator, it is not on a domain and the domain invariant subspace problem, operator of matrix

A ladder.

28 , Operator of the characteristic value and the characteristic vector, subspace, subspace.

29 。 The characteristic polynomial and its related invariants of operator, equivalent to subspace.

30 , The Multiplicities of the eigenvalues and Eigen-subspace dimension, operator of matrix Diagonalization condition.

31 。 Polynomials and the annihilation operators, Hamilton – Cayley’s theorem.

32 。 Degradation of operators, and operator of its characteristic polynomial coefficient of relationship.

33 。 Decomposed into invariant subspaces of the space and, with characteristic polynomial is decomposed into basic factor speaking of relationships.

34 。 Minimal zero polynomial, his relations with the eigenvectors.

35 。 Root subspace with the vector operator roots.

36 。 Standard root spaces of matrices, operator of the Jordan canonical form of matrices, minimum polynomial relations with Jordan canonical form.

37 。 Real space complex, linear mapping and linear operators.

38 , Real operator matrix Standard.

39 。 Jordan canonical form of matrix theory.

40 。 Geometric transformation on linear map affine spaces, relations among the three points, analytic geometry transform, affine forms for affine classification and graphic concept.

41 。 Linear function spaces and linear functions, along with the base, transformation matrices, linear and semilinear function space coordinate transformation.

42 。 Linear and nonlinear function on the space of self-adjoint, since the conjugate base.

43 。 Operators of complex space and maps, dual mapping of operators on basic space, self-dual.

44 。 Double linear and multilinear functions and coordinates, the vector function space, basic functions and their relationship to basic space base.

45 。 Based on bilinear function is independent of the matrix, rank of a function.

46 。 Nuclear bilinear and linear functions and its dimension nondegenerate function.

47 。 Double linear and multilinear functions on complex spaces spaces naturally isomorphic.

48 Symmetric, hermitian and skew-symmetric functions.

49 The subspace orthogonal to the vector and its relationship with symmetrical, anti-symmetric hermitian function, dimension and nature of the orthogonal complement.

50 。 Symmetric forms, hermitian and skew-symmetric functions are orthogonal.

51 。 Functions of orthogonal form of uniqueness, real-symmetric hermitian function on the domain of the law of inertia.

52 。 Quadratic function and its relation with twice-sex and norms.

53 。 Normative methods of Jacobi’s theorem and Gramm.

54 。 System and hermitian positive definite function, Silvester criterion.

55 。 Symmetric, hermitian and skew-symmetric scalar product, to be Euclidean, hermitian and symplectic vector space isomorphism theorem and its orthogonal symmetric matrix.

56 。 Natural isomorphism on the inner product space, the General form of linear functions, zero-vectors of and quasi Euclidean, hermitian and symplectic subspace of a vector space, of orthogonal anisotropic linear independence of vectors, Gram determinant, orthogonalization, orthogonal complete.

57 。 Symplectic vector space, Hamiltonian matrix, isotropic subspaces.

58 。 Unitary spaces, Cauchy -the buniyakefusiji inequality, the triangle inequality.

59 。 To be orthogonal, unitary, orthogonal to be unitary symplectic matrix, the special linear group.

60 。 Dot product is invariant under the operator (orthogonal to be orthogonal, unitary, quasi unitary and spicy) with their nature, nature of the invariant subspace, isometric isomorphism, and operator groups.

61 , Orthogonal and unitary operator standard and unique, feature subspaces orthogonal transform on the space.

62 。 Group, pseudo scalars in the plane, hyperbolic trigonometry, the Lorentz transformation, three-dimensional quasi Euclidean space.

63 。 Complex space of real, hermitian structure of quasi-Euclidean space and symplectic structure.

64 。 Operators and operator group materialized.

65 。 On the inner product space of existence and uniqueness of the adjoint operator, with dual operator on a complex airborne contact.

66 。 And unitary space of self-adjoint operators on Euclidean space and its standard, characteristic value and the nature of the subspace.

67 。 Self-adjoint orthogonal operator self-adjoint polar decomposition of unitary operators.

68 。 Inner product on the space of bilinear and linear functions, these functions and operator spaces naturally isomorphic.

69 。 Euclid (unitary) spatially symmetric (hermitian) function of the standard type, spatial hypersurfaces of second order equation of the standard type.

70 。 One pair for positive definite quadratic forms invariant, in discussions on a standard basis.

71 。 Tensor examples of tensor and linear functions and tensor spaces.

72 。 Tensor and tensor algebra, tensor space based on the coordinates.

73 。 Convolution operators and their properties, examples.

74 。 On the inner product space of tensor indices.

75 。 Symmetric and skew-symmetric tensor and its coordinates, and alternating tensor of symmetric operators and their properties.

76 。 Diagonal tensor, the outer product operator and its properties.

77 。 Multiple vector and skew-symmetric functions and coordinates, for example, the Plücker coordinates of the subspace.

78 。 Multiple vector and skew-symmetric functions are simplified.

79 。 Multiple vector and the base and dimension of the space of skew-symmetric functions.

Analytic Geometry

A, vector operations.

1 。 Vector, vector linear operation and its basic properties.

2 。 Vector’s linear correlation and geometric significance.

3 。 Base to the coordinate system of the quantum dots, geometric meaning of the coordinate system.

4 。 Linear correlation, coordinates vector linear correlation criteria.

5 。 Scalar product, its main properties and formulas.

6 。 The transformation matrix from one base to another base, point – vector coordinate system transformation, the nature of transformation matrices, orthonormal basis.

7 。 Orientation of the plane, the orientation of the parallelogram volume, its basic properties and formulas.

8 。 Spatial orientation, the targeted volume of the parallelepiped, its basic properties and formulas.

9 。 Vector vector products and mixed products, its basic properties and formulas.

10 。 Orthogonal matrices, orthogonal coordinate transformation, classification of the erjiezheng matrix.

11 。 Euler angles, using sanjiezheng matrix representation of Euler angles.

12 。 The polar coordinate system, spatial cylindrical coordinates and spherical coordinates, polar coordinates, cylindrical coordinates and spherical coordinates to orthogonal coordinates.

13 。 Triple vector and bivector, dual vector a linear operation and measure theory.

14 。 Two-dimensional and three-dimensional linear operators and invertible linear operator, isometric linear operators.

Two, lines and planes.

15 。 The parametric equations of lines and planes, meaning their collection.

16 。 First-order equation as line or plane and its links with the parametric equations.

17 。 Relationship between the two lines in the plane and space. 18。 Relationship between two planes, and its links to first-order equations.

19 。 Angle of vectors, linear and Planar formation and its calculation method.

20 。 The distance from a point to a line or plane, the distance between two lines.

21 。 First-order geometric meaning of inequalities on the Planar and space.

22 。 Plane and space on the wiring harness, wiring harnesses are linearly independent.

23 。 Space plane pencil, pencil of planes are linearly independent.

24 。 The interrelationship of the three lines in the plane, spatial relationships among the three planes, the rank of their equations.

Third, the quadratic curves and surfaces.

25 。 Algebraic curves and surfaces, the number of algebraic curves and surfaces, through linear algebraic curve, algebraic surface, who with a planar phase.

26 。 Reducible curves and decomposed surface and geometric significance, contains a linear curve can be agreed, and includes flat surfaces can be agreed.

27 。 By erjiezheng transformation of quadratic polynomial for the standard type, the classification of quadratic curves.

28 。 The focus of the ellipse and Hyperbola, parabola, Hyperbola equation of Asymptote.

29 。 Alignment of the ellipse, Hyperbola, parabola, their equations in polar coordinates.

30 。 Cylindrical and conical surface, surface of revolution.

31 。 Using third-order orthogonal transformation of quadratic polynomial as standard, classification of quadric surfaces.

32 。 Ellipsoid, ellipsoid and hyperboloid deficiency, their basic properties, they draw their images.

33 。 Elliptic paraboloid, his basic properties and image rendering.

34 。 The hyperbolic paraboloid, hyperbolic paraboloid bus and its basic properties.

35 。 Double leaf bilateral curved surface and hyperboloid, double leaf bilateral curved surface and hyperboloid and its basic properties.

The four orthogonal invariants and classification of quadric surfaces and surfaces.

36 。 Classification of orthogonal polynomial invariants, orthogonal invariants with polynomial coefficients and roots.

37 。 Orthogonal invariants of the quadratic polynomial.

38 。 Orthogonal coordinate system using orthogonal invariants classify the conic section.

39 。 Orthogonal coordinate system using orthogonal invariants classify the quadric surfaces.

40 。 Quadratic curve and the coordinates of the center of the surface.

41 。 Proportion of quadratic equations and curves and surfaces.

42 。 Quadratic curve Asymptote, the asymptotic cone of quadratic rotating surfaces, use quadratic curve and the line of intersection of their classification.

43 。 Diameter and equations of Conic sections, surface of revolution with its centerline intersects the plane formed by the diameter of the curve and equation.

44 。 The conjugate direction and diameter of the curve, ellipse, Hyperbola and parabola Conjugate directions and diameters.

45 。 Symmetry, symmetry axis and the relationship of the diameter of symmetrical axis coordinate of the vector, Conic to each other.

46 。 The plane of symmetry, relationship between center line of the plane and the plane of symmetry, symmetry plane normal vector of coordinates, the interrelationship of quadric surface.

47 。 Conic bundle, quadratic curve through five given points,Sturm theorem and Pascal theorem.

Five, affine transforms.

48 。 Definition and properties of affine transformations, affine transformation group.

49 。 From one base to another base set of affine transformation matrices, their geometric meaning of affine transformation formula.

50 。 Affine transformation group equivalence, affine classification of quadratic curves.

51 。 Affine classification of quadric surfaces.

52 。 Isometric transformation properties.

53 。 Orthogonal classification of quadratic curve, Conic standards of orthogonal invariants and its coefficients of equations.

54 。 Orthogonal classification of quadric, quadric standards of orthogonal invariants and its coefficients of equations.

55 。 Planar orthogonal transform structure.

56 。 Spatial structure of orthogonal transformation.

57 。 Affine transformation of the plane and space structure.

Six, projective geometry,

58 。 As the formation of lines and planes of the projective plane, as a generalization of ordinary plane projective plane, points and lines in projective coordinates in both cases, the principle of duality.

59 。 Cone of curves on a surface model, the completeness of quadratic curves under the expanded flat, equation of Conic in projective coordinates.

60 。 Projective transformation, as generalized Centro-affine transformation projective transformations.

61 。 Extended affine and projective transformations on the plane of contact.

62 。 Transformation of quadratic forms as a standard of consistency, Conic projection form and classification of projective and affine classification of relationships.

63 。 Basic concept of elliptic and hyperbolic geometry,Erlangen programme.

64 。 Axioms of geometry,Euclid geometry axioms, axiom systems of affine geometry, affine geometry on the complex field.

Analytical Geometry

1 , M.M.Postnikov , Analytic geometry, science press, 1973 。

2 , P.S.Alexandrov , Lectures on analytic geometry, science press, 1967 。

3 , P.S.Alexandrov , Analytic geometry and linear algebra tutorials, science press, 1979 。

4 , B.N.Delone 、 D.A.Raykov , Analytic geometry, volume I, joint publishing house, 1948 。

5 , B.N.Delone 、 D.A.Raykov , Analytic geometry, volume II, National Union Publishing House, 1949.

6 , Y.Smirnov , Analytic geometry, Russian educational and scientific literature Publishing House, 2004 。

7 , V.V.Prasolov 、 V.M.Tikhomirov In geometry, continuous mathematics education centre in Moscow, 1997 。

8 , P.S.Modenov 、 A.S.Parhomenko And analytic geometry problem set, science press, 1976 。

Basic Topology by Armstrong
Differential Geometry of Curves and Surfaces by Manfredo Do Carmo
Hatcher “Algebraic Topology” Cambridge UP
Munkries “Topology” 2nd ed. Prentice Hall

A.T.Fomenko Differential geometry and topology ,Consultants Bureau
Dubrovin, Fomenko, Novikov “Modern geometry-methods and applications”Vol 1—3
A Comprehensive Introduction to Differential Geometry vol 1-5 ,by Michael Spivak

Bogorelov And analytic geometry.

The analytic geometry problem set, Bakhvalov

Di Long Nirvana “( Resolved ) Geometry”

Musihailishiweili ” Analytic geometry tutorials”

Qiuweisheng, analytic geometry, North Edition
Department of mathematics, Nankai, of the geometry of the higher education version

Chen ( Bird ) ” Space analytic geometry”

Zhu Dingxun ” Analytic geometry”

Wu Guanglei of analytic geometry is a simple tutorial by higher education press

Qiu weisheng analytic geometry, Peking University Press

The analytic geometry of Lv Genlin, Xu Zidao (supporting guidance)

The analytic geometry You Chengye

The space of analytic geometry and differential geometry (math study guide series) Huang Xuanguo

The higher geometry Mei Xiangming

Higher geometry problem set

Higher geometry of Zhu Dexiang

Higher geometry of zhoujianwei

175 Higher geometry study guide and exercises selected solutions Mei Xiangming, Liu Zengxian series

176 Higher geometry 2 Mr Chong Shan , Pang Chaoyang , written by Tian Yuping

Arithmetic Geometry

X Fulton, William. Algebraic Curves: An Introduction to Algebraic Geometry.

This book is available for free on Fulton’s website.

Buy at Amazon Milne, J. S. Elliptic Curves. BookSurge Publishers, 2006. ISBN: 9781419652578.

This book is also available for free on Milne’s website, along with addendum/erratum.

Buy at Amazon Serre, Jean-Pierre. A Course in Arithmetic. Springer-Verlag, 1996. ISBN: 9783540900405.

Buy at Amazon Shafarevich, I. R. (translated by Miles Reid). Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550.

Buy at Amazon Stichtenoth, H. Algebraic Function Fields and Codes. Berlin: Springer, 2008. ISBN 9783540768777. [Preview with Google Books]

Buy at Amazon Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer-Verlag, 2009. ISBN: 9780387094939. (errata) [Preview with Google Books]

Number Theory

数论

1。Euclid算法及其复杂性，Lame定理，扩充Euclid算法。

2。有限连分数，连分数的收敛性。

3。无限连分数，连分数表示的实数的唯一性，Lagrange定理。

4。同余定理，剩余环，Wilson定理，Euler函数，Fermat定理，Euler定理，中国剩余定理，同余方程的解。

5。素数的判别法，概率论与素数的关系，伪素数。

6。自乘的快速算法，密码学中的公钥的概念，RSA系统，电子签名。

7。整数的因式分解，Fermat、Dickson与Legendre方法。

8。二次剩余，Legendre与Jacobi符号的性质。

9。Soloveya-Shtrassena定理。

10。多项式的Euclid算法，同余方程的概率算法。

11。多项式的中国剩余定理，Berlekempa算法。

Number Theory

1，A.Akritas，Elements of Computer Algebra with Applications，John Wiley andSons，1989。

2，A.A.Bukhshtab，数论，教学法与教科书国家出版社，1960。

3，V.V.Yashchenko，密码学引论，莫斯科不间断数学教育中心，1998。

4，I.M.Vinogradov，数论基础，科学出版社，1953。

5，D.E.Knuth，Art of Computer Programming volume 2：Seminumerical Algorithms，Addison-Wesley，1997。

6，R.Lidl、H.Niederreiter，Finite Fields，Cambridge University Press。

7，P.Naudin、C.Quitté，Algorithmique Algébrique，Masson。

G.H.Hardy,An Introduction to the Theory of Numbers

Graham和Knuth 等人合著的经典“具体数学”吧，有翻译版，西电出的。

Bach的”Introduction to Algorithmic Number Theory”。

《离散数学》耿素云，屈婉玲

朱洪等 “算法设计和分析”

卢开澄”组合数学–算法与分析”

冯克勤《整数与多项式》高等教育出版社

潘承洞、潘承彪《初等数论》北京大学出版社

”数论导引“(华罗庚先生的名著，科学版，九章书店重印)。

(Advanced) Combinatorial Optimization

Cryptography

Quantum Computation

Number theory

1 。 Euclid algorithm and its complexity, theLame theorem extended Euclid algorithm.

2 。 Finite continued fraction, the convergence of the continued fraction.

3 。 An infinite continued fraction, the uniqueness of the continued fraction representation of a real number,Lagrange theorem.

4 。 Congruence theorem, the remaining rings,Wilson theorem andEuler function,Fermat theorem,Euler theorem, Chinese remainder theorem, the solution of the congruence equation.

5 。 Prime criterion, probability theory and Prime, Pseudoprime.

6 。 Fast algorithms for raising, the concept of public key cryptography,RSA system, electronic signatures.

7 。 Integer factorization,Fermat, andDickson and Legendre .

8 。 Quadratic residue,Legendre and Jacobi symbols of nature.

9 。 Soloveya-Shtrassena theorem.

10 。 Polynomial of the Euclid algorithms, probability of a congruence equation algorithm.

11 。 Chinese remainder theorem for polynomials,Berlekempa algorithm.

Number Theory

1 , A.Akritas , Elements of Computer Algebra with Applications , John Wiley andSons , 1989 。

2 , A.A.Bukhshtab , Number theory, teaching methods and textbooks the State Publishing House, 1960 。

3 , V.V.Yashchenko , An introduction to cryptography, continuous mathematics education centre in Moscow, 1998 。

4 , I.M.Vinogradov , Number theory Foundation, science press, 1953 。

5 , D.E.Knuth , Art of Computer Programming volume 2 : Seminumerical Algorithms , Addison-Wesley , 1997 。

6 , R.Lidl 、 H.Niederreiter , Finite Fields , Cambridge University Press 。

7 , P.Naudin 、 C.Quitt é, Algorithmique Alg é brique , Masson 。

G.H.Hardy,An Introduction to the Theory of Numbers

Graham Knuth Co-author of the classic “concrete Mathematics”, there are translations, from West.

Bach “Introduction to Algorithmic Number Theory” 。

The discrete mathematics Geng Suyun, Qu Wanling

Zhu Hong ” Design and analysis of algorithms”

Lu kaicheng ” Combinatorial mathematics — Algorithm and analysis”

Feng Keqin of the integers and polynomials of the higher education press

Pan Chengdong, Pan chenbiao of elementary number theory, Peking University Press

“Number theory-guided” ( Hua luogeng’s masterpiece, Science Edition, chapters Bookstore reprints ) 。

(Advanced)Combinatorial Optimization

Cryptography

Quantum Computation

Algorithms

Algorithm & its Design

X Cormen, Thomas, Charles Leiserson, et al. Introduction to Algorithms. 3rd ed. MIT Press, 2009. ISBN: 9780262033848. [Preview with Google Books]

Error-correct & Randomized and Distributed Alg.

X Fan, John L. Constrained Coding and Soft Iterative Decoding. Kluwer International Series in Engineering and Computer Science. Boston: Kluwer Academic Publishers, 2001, SECS 627.

Motwani, and Raghavan. Randomized Algorithms. Cambridge, UK: Cambridge University Press, 1995. ISBN: 0521474655.

Optional

Buy at Amazon Feller, William. An Introduction to Probability Theory and Its Applications.Vol. 1. New York, NY: John Wiley, 1968. ISBN: 0471257087. (This book is fairly old but adorns the shelves of most theoretical computer scientists.)

Nielsen, Michael A., and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge, UK: Cambridge University Press, September 2000. ISBN: 9780521635035.

Preskill, J. Notes on Quantum Computation.

Buy at Amazon Peres, Asher. Quantum Theory: Concepts and Methods. New York, NY: Springer, 1993. ISBN: 9780792325499.

Lynch, Nancy. Distributed Algorithms. Burlington, MA: Morgan Kaufmann, 1996. ISBN:9781558603486.

Attiya, Hagit, and Jennifer Welch. Distributed Computing: Fundamentals, Simulations, and Advanced Topics. 2nd ed. New York, NY: Wiley-Interscience, 2004. ISBN: 9780471453246.

Herlihy, Maurice, and Nir Shavit. The Art of Multiprocessor Programming. Burlington, MA: Morgan Kaufmann, 2008. ISBN: 9780123705914.

Guerraoui, Rachid, and Michal Kapalka. Transactional Memory: The Theory. San Rafael, CA: Morgan and Claypool, 2010. ISBN: 9781608450114.

Dolev, Shlomi. Self-Stabilization. Cambridge, MA: MIT Press, 2000. ISBN:9780262041782.

Kaynar, Disun, Nancy Lynch, Roberto Segala, and Frits Vaandrager. The Theory of Timed I/O Automata. 2nd ed. San Rafael, CA: Morgan and Claypool, 2010. ISBN:9781608450022.

Algorithm & its Design

X Cormen, Thomas, Charles Leiserson, et al. Introduction to Algorithms. 3rd ed. MIT Press, 2009. ISBN: 9780262033848. [Preview with Google Books]

Error-correct & Randomized and Distributed Alg.

X Fan, John L. Constrained Coding and Soft Iterative Decoding. Kluwer International Series in Engineering and Computer Science. Boston: Kluwer Academic Publishers, 2001, SECS 627.

Motwani, and Raghavan. Randomized Algorithms. Cambridge, UK: Cambridge University Press, 1995. ISBN: 0521474655.

Optional

Nielsen, Michael A., and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge, UK: Cambridge University Press, September 2000. ISBN: 9780521635035.

Preskill, J. Notes on Quantum Computation.

Buy at Amazon Peres, Asher. Quantum Theory: Concepts and Methods. New York, NY: Springer, 1993. ISBN: 9780792325499.

Lynch, Nancy. Distributed Algorithms. Burlington, MA: Morgan Kaufmann, 1996. ISBN:9781558603486.

Attiya, Hagit, and Jennifer Welch. Distributed Computing: Fundamentals, Simulations, and Advanced Topics. 2nd ed. New York, NY: Wiley-Interscience, 2004. ISBN: 9780471453246.

Herlihy, Maurice, and Nir Shavit. The Art of Multiprocessor Programming. Burlington, MA: Morgan Kaufmann, 2008. ISBN: 9780123705914.

Guerraoui, Rachid, and Michal Kapalka. Transactional Memory: The Theory. San Rafael, CA: Morgan and Claypool, 2010. ISBN: 9781608450114.

Dolev, Shlomi. Self-Stabilization. Cambridge, MA: MIT Press, 2000. ISBN:9780262041782.

Kaynar, Disun, Nancy Lynch, Roberto Segala, and Frits Vaandrager. The Theory of Timed I/O Automata. 2nd ed. San Rafael, CA: Morgan and Claypool, 2010. ISBN:9781608450022.

Discrete Mathematics

Discrete Mathematics

耿素云，离散数学，高教版
Discrete Mathematics and its Applications Kenneth H. Rosen

213《基础集合论》北师大

214《面向计算机科学的数理逻辑》陆钟万

215《图论及其算法》王树禾

216《图论及其应用》Bondy ，Murty

217《离散数学》耿素云，屈婉玲

218《具体数学》格拉厄姆，高德纳等

219《Introduction to Algorithms》 Corman

Algebraic Combinatorics & its Analysis

I.Tomescu “组合学引论” , “Problem in graph theory and combinatorics(???)”

Lovasz “Problems in Combinatorics(?)”

I. Anderson”Combinatorics of Finite Sets”

Bollobas”Combinatorics”

Ryser(赖瑟)”组合数学”

I. Anderson “A First Course in COmbinatorial Mathematics”

C.Berger “组合学原理”(上海科技)

Lovasz,et al.(ed.) “Handbook of Combinatorics”

李乔”组合数学基础”

魏万迪 “组合论”

C.L.Liu(刘炯朗,现新竹清华大学校长)

220《近代组合学》王天明编著

221《组合学笔记》康庆德著

Recursion Theory & Undecidability & Model

Automata & Computation & Complexity Theory

Moore, Cristopher, and Stephan Mertens. The Nature of Computation. Oxford University Press, 2011. ISBN: 9780199233212.

Sipser, Michael. Introduction to the Theory of Computation. Course Technology, 2005. ISBN: 9780534950972. Covers most material from the first half of the course.

Arora, Sanjeev, and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, 2009. ISBN: 9780521424264. Covers most material from the second half (as well as more advanced material that won’t be covered in this course).

Graph Theory

Graph theory and Discrete Mathematics

Hamiltonian, coloring, network flow, network algorithm, connectivity, spanning tree, connectivity testing, bipartite graphs, trees, breadth/depth first search.

Computational Number Theory

Primality, integer factorization; greatest common divisor; Chinese Remainder Theorem; modular arithmetic.

Computational geometry and discrete geometry

Convex hull, Delaunay triangulation, Voronoi diagram, arrangement, discrete curvature, discrete Ricci flow.

References:

A. Bondy and U. S. R. Murty: “Graph theory”, GTM, Springer, 1976.
T. H. Cormen, C. E. Leiserson, R. L. Rivest, & C. Stein, “Introduction to Algorithms”, MIT Press, 2009.
S. L. Devaloss and Joseph O’Rourke, “Discrete and Computational Geometry”, Princeton University Press, 2011.
Mark De Berg, “Computational Geometry: Algorithms and Applications”, Springer, 2008.
Xianfeng Gu and S. T. Yau, “Computational conformal geometry”, International Press, 2003.

Bondy,Murty “Graph Theoryand Applications(?)” “图论及其应用”习题解答, “图论和电路网络”

Harary(哈拉里) “Graph Theory”(图论)

B. Bollobas “Graph Theory”(GTM 63)

G.Chartrand,L. Lesniak”Graph and Digraphs”

C. Berger”Graph and Hypergraph”

Reinhard Diestel “Graph Theory”(GTM173)

世界图书引进有GTM系列的”Modern Graph Theory”。此书确实经典！

《图论及其算法》王树禾

Appel ,Haken “Every Planar Map is Four Colorable”

Steen(ed.) “mathematics today”

Discrete Mathematics

Geng Suyun, discrete mathematics, and higher education
Discrete Mathematics and its Applications Kenneth H. Rosen

213 The basic set theory, Beijing Normal University

214 The Lu Zhongwan of mathematical logic in computer science

215 Wang Shuhe the graph theory and algorithms

216 Of the graph theory and its applications Bondy , Murty

217 The discrete mathematics Geng Suyun, Qu Wanling

218 The specific mathematical Graham, Knuth

219 《 Introduction to Algorithms 》 Corman

Algebraic Combinatorics & its Analysis

I.Tomescu ” An introduction to Combinatorics ” , “Problem in graph theory and combinatorics(???)”

Lovasz “Problems in Combinatorics(?)”

I. Anderson”Combinatorics of Finite Sets”

Bollobas”Combinatorics”

Ryser ( Reiso )” Combinatorial mathematics”

I. Anderson “A First Course in COmbinatorial Mathematics”

C.Berger ” Principle of combination “( The Shanghai Science and technology)

Lovasz,et al.(ed.) “Handbook of Combinatorics”

Li ” Combinatorial mathematics Foundation”

Wei Wandi ” Combinatorial theory”

C. L. Liu ( Liu Chung-lang , President of National Tsing Hua University)

220 Written by Wang tianming of the modern Combinatorics

221 Kang qng-de of the Combinatorics notebook with

Recursion Theory & Undecidability & Model

Automata & Computation & Complexity Theory

Moore, Cristopher, and Stephan Mertens. The Nature of Computation. Oxford University Press, 2011. ISBN: 9780199233212.

Sipser, Michael. Introduction to the Theory of Computation. Course Technology, 2005. ISBN: 9780534950972. Covers most material from the first half of the course.

Graph Theory

Graph theory and Discrete Mathematics

Hamiltonian, coloring, network flow, network algorithm, connectivity, spanning tree, connectivity testing, bipartite graphs, trees, breadth/depth first search.

Computational Number Theory

Primality, integer factorization; greatest common divisor; Chinese Remainder Theorem; modular arithmetic.

Computational geometry and discrete geometry

Convex hull, Delaunay triangulation, Voronoi diagram, arrangement, discrete curvature, discrete Ricci flow.

References:

1. A. Bondy and U. S. R. Murty: “Graph theory“, GTM, Springer, 1976.

2. T. H. Cormen, C. E. Leiserson, R. L. Rivest, & C. Stein, “Introduction to Algorithms“, MIT Press, 2009.

3. S. L. Devaloss and Joseph O’Rourke, “Discrete and Computational Geometry“, Princeton University Press, 2011.

4. Mark De Berg, ” Computational Geometry: Algorithms and Applications “, Springer, 2008.

5. Xianfeng Gu and S. T. Yau, ” Computational conformal geometry “, International Press, 2003.

Bondy,Murty “Graph Theoryand Applications(?)” ” Graph theory and its applications ” Questions and problems , “Graph theory and network”

Harary ( Harare ) “Graph Theory” ( graph theory)

B. Bollobas “Graph Theory”(GTM 63)

G.Chartrand,L. Lesniak”Graph and Digraphs”

C. Berger”Graph and Hypergraph”

Reinhard Diestel “Graph Theory”(GTM173)

World Book introduction GTM Series “Modern Graph Theory” 。 This book does a classic!

The graph theory and algorithms of Wang Shuhe

Appel ,Haken “Every Planar Map is Four Colorable”

Steen(ed.) “mathematics today”

Other Numerical Simulations

Parallel Computing

Eigenvalues on Random Matrix

Non-linear Dynamics & Fluid & Interfacial

Acheson, D. J. Elementary Fluid Dynamics. Oxford University Press, 1990. ISBN: 9780198596608. [Preview with Google Books]

Gennes, Pierre-Gilles de, Françoise Brochard-Wyart, and David Quéré. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, 2003. ISBN: 9780387005928.

Homsy, G. M. Multimedia Fluid Mechanics. Cengage Learning, 2011. ISBN: 9780521721691.

Hydrodynamic Stability & Turbulence

Nanophotonics

Joannopoulos, John D., Steven G. Johnson, Robert D. Meade, and Joshua N. Winn. Photonic Crystals: Molding the Flow of Light. Princeton, NJ: Princeton University Press, 2008. ISBN: 9780691124568.

Inui, Tetsuro, Yukito Tanabe, and Y. Onodera. Group Theory and Its Applications in Physics (Springer Series in Solid-State Sciences). 2nd corrected ed. New York, NY: Springer-Verlag, February 1996. ISBN: 9783540604457.

Tinkham, Michael. Group Theory and Quantum Mechanics. Mineola, NY: Dover Publications, 2003. ISBN: 9780486432472.

Parallel Computing

Eigenvalues on Random Matrix

Non-linear Dynamics & Fluid & Interfacial

Acheson, D. J. Elementary Fluid Dynamics. Oxford University Press, 1990. ISBN: 9780198596608. [Preview with Google Books]

Gennes, Pierre-Gilles de, Françoise Brochard-Wyart, and David Quéré. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, 2003. ISBN: 9780387005928.

Homsy, G. M. Multimedia Fluid Mechanics. Cengage Learning, 2011. ISBN: 9780521721691.

Hydrodynamic Stability & Turbulence

Nanophotonics

Joannopoulos, John D., Steven G. Johnson, Robert D. Meade, and Joshua N. Winn. Photonic Crystals: Molding the Flow of Light. Princeton, NJ: Princeton University Press, 2008. ISBN: 9780691124568.

Tinkham, Michael. Group Theory and Quantum Mechanics. Mineola, NY: Dover Publications, 2003. ISBN: 9780486432472.

Numerical Analysis

Numerical Analysis

Syllabuses on Computational Mathematics
and Applied Mathematics

Interpolation and approximation

Trigonometric interpolation and approximation, fast Fourier transform; approximations by rational functions; polynomial and spline interpolations and approximation; least-squares approximation.

Nonlinear equation solvers

Convergence of iterative methods (bisection, Newton’s method, quasi-Newton’s methods and fixed-point methods)for both scalar equations and systems, finding roots of polynomials.

Linear systems and eigenvalue problems

Classical and modern iterative method for linear systems and eigenvalue problems, condition number and singular value decomposition.

Numerical solutions of ordinary differential equations

Single step methods and multi-step methods, stability, accuracy and convergence; absolute stability, long time behavior; numerical methods for

stiff ODE’s.

Numerical solutions of partial differential equations

Finite difference method, finite element method and spectral method: stability, accuracy

and convergence, Lax equivalence theorem.

References:

C. de Boor and S.D. Conte, Elementary Numerical Analysis, an algorithmic approach, McGraw-Hill, 2000.
G.H. Golub and C.F. van Loan, Matrix Computations, third edition, Johns Hopkins University Press, 1996.
E. Hairer, P. Syvert and G. Wanner, Solving Ordinary Differential Equations, Springer, 1993.
B. Gustafsson, H.-O. Kreiss and J. Oliger, Time Dependent Problems and Difference Methods, John Wiley Sons, 1995.
Lloyd N. Trefethen and David Bau, Numerical linear algebra, SIAM, 1997.
Susanne Brenner and Ridgway Scott, The Mathematical Theory of Finite Element Methods, Springer, 2010.

R.L. Burden and D. Faires, Numerical analysis, 7th edition, Thomson Learning。

J. Stoer and R. Bulirsch, An introduction to numerical analysis, Springer-Ver

lag,

《Introduction to Algorithms》Corman

C. de Boor and S.D. Conte, Elementary Numerical Analysis, an algorithmic approach, McGraw-Hill, 2000.

G.H. Golub and C.F. van Loan, Matrix Computations, third edition, Johns Hopkins University Press, 1996.

E. Hairer, P. Syvert and G. Wanner, Solving Ordinary Differential Equations, Springer, 1993.

B. Gustafsson, H.-O. Kreiss and J. Oliger, Time Dependent Problems and Difference Methods, John Wiley Sons, 1995.

G. Strang and G. Fix, An Analysis of the Finite Element Method, second edition, Wellesley-Cambridge Press, 2008.

228《数值分析:mathematics of scientific computing》(美)David Kincaid，Ward Cheney著

《数值逼近》李岳生，黄友谦

《数值分析方法》奚梅成

《数值计算方法》林成森

《数值逼近》王仁宏

《数值分析》李庆扬，王能超，易大义

《计算方法引论》徐萃薇，孙绳武

《数值分析基础》李庆扬，王能超，易大义

《数值逼近》蒋尔雄，赵风光

《数值分析引论》易大义

223《数值分析基础》关治，陆金甫著

225《数值方法》关治，陆金甫编著

229《计算方法典型例题分析》孙志忠编著

Numerical Linear Algebra & PDE

LeVeque, Randall J. Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2007. ISBN: 9780898716290.

Finite Volume Methods for Hyperbolic Problems. Cambridge texts in applied mathematics. Cambridge, UK: Cambridge University Press, 2002. ISBN: 9780521009249.

Fletcher, C. A. J. Computational Techniques for Fluid Dynamics. Fundamental and General Techniques Volume I. Springer series in computational physics. New York, NY: Springer-Verlag, 1996. ISBN: 9783540530589.

Buy at Amazon ———. Computational Techniques for Fluid Dynamics. Specific Techniques for Different Flow Categories Volume II. Springer series in computational physics. New York, NY: Springer-Verlag, 1991. ISBN: 9783540536017.

Buy at Amazon Canuto, Claudio S., M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics. New York, NY: Springer-Verlag, 2007. ISBN: 9783540307273.

Buy at Amazon Trefethen, Lloyd N. Spectral Methods in MATLAB (Software, Environments, Tools). Philadelphia, PA: Society for Industrial and Applied Mathematics, 2001. ISBN: 9780898714654.

Buy at Amazon Evans, Lawrence C. Partial Differential Equations. Vol. 19. Graduate studies in mathematics. Providence, RI: American Mathematical Society, 1998. ISBN: 9780821807729.

《矩阵计算和方程求根》曹志浩，张德玉，李瑞遐

《矩阵数值分析》邢志栋

《微分方程数值解法》李荣华，冯果忱

《微分方程数值解法》余德浩，汤华中

《微分方程数值解法》李立康，於崇华，朱政华

《非线性方程组解法与最优化方法》王德人

《最优化理论与算法》陈宝林

《最优化理论与方法》袁亚湘，孙文瑜

《信息论基础》叶中行

专门为数学系写的信息论

《信息论，编码与密码学》Ranjan Bose

Numerical Analysis

Syllabuses on Computational Mathematics
and Applied Mathematics

Interpolation and approximation

Trigonometric interpolation and approximation, fast Fourier transform; approximations by rational functions; polynomial and spline interpolations and approximation; least-squares approximation.

Nonlinear equation solvers

Convergence of iterative methods (bisection, Newton’s method, quasi-Newton’s methods and fixed-point methods)for both scalar equations and systems, finding roots of polynomials .

Linear systems and eigenvalue problems

Classical and modern iterative method for linear systems and eigenvalue problems, condition number and singular value decomposition.

Numerical solutions of ordinary differential equations

Single step methods and multi-step methods, stability, accuracy and convergence; absolute stability, long time behavior; numerical methods for

stiff ODE ‘s .

Numerical solutions of partial differential equations

Finite difference method, finite element method and spectral method: stability, accuracy

and convergence, Lax equivalence theorem.

References:

1. C. de Boor and S.D. Conte, Elementary Numerical Analysis, an algorithmic approach, McGraw-Hill, 2000.

2. G.H. Golub and C.F. van Loan, Matrix Computations, third edition , Johns Hopkins University Press, 1996.

3. E. Hairer, P. Syvert and G. Wanner, Solving Ordinary Differential Equations, Springer, 1993.

4. B. Gustafsson, H.-O. Kreiss and J. Oliger, Time Dependent Problems and Difference Methods, John Wiley Sons, 1995.

5. Lloyd N. Trefethen and David Bau, Numerical linear algebra , SIAM, 1997.

6. Susanne Brenner and Ridgway Scott, The Mathematical Theory of Finite Element Methods , Springer, 2010.

R.L. Burden and D. Faires, Numerical analysis, 7th edition, Thomson Learning 。

J. Stoer and R. Bulirsch, An introduction to numerical analysis, Springer-Ver

lag,

《 Introduction to Algorithms 》 Corman

C. de Boor and S.D. Conte, Elementary Numerical Analysis, an algorithmic approach, McGraw-Hill, 2000.

G.H. Golub and C.F. van Loan, Matrix Computations, third edition, Johns Hopkins University Press, 1996.

E. Hairer, P. Syvert and G. Wanner, Solving Ordinary Differential Equations, Springer, 1993.

B. Gustafsson, H.-O. Kreiss and J. Oliger, Time Dependent Problems and Difference Methods, John Wiley Sons, 1995.

G. Strang and G. Fix, An Analysis of the Finite Element Method, second edition, Wellesley-Cambridge Press, 2008.

228 The numerical analysis :mathematics of scientific computing 》 ( The United States ), David Kincaid , Ward Cheney The

The numerical approximation of Li Yuesheng, Huang Youqian

Of the numerical analysis method for Xi Meicheng

Of the numerical method of the forest forest

The numerical approximation of Wang Renhong

The numerical analysis of Li Qingyang, Wang nengchao, da Yi

Introduction to the calculation method of Xu Cui Wei, Sun Shengwu

Numerical analysis of Li Qing-Yang, Wang nengchao, da Yi

The numerical approximation of Jiang erxiong, Zhao scenery

An introduction to numerical analysis of righteousness

223 Numerical analysis on basic Kanji, Lu jinfu with

225 The numerical methods for Kanji, written by Lu jinfu

229 Written by Sun Zhizhong of the calculation method of typical analysis examples

Numerical Linear Algebra & PDE

Finite Volume Methods for Hyperbolic Problems. Cambridge texts in applied mathematics. Cambridge, UK: Cambridge University Press, 2002. ISBN: 9780521009249.

Buy at Amazon Trefethen, Lloyd N. Spectral Methods in MATLAB (Software, Environments, Tools). Philadelphia, PA: Society for Industrial and Applied Mathematics, 2001. ISBN: 9780898714654.

Buy at Amazon Evans, Lawrence C. Partial Differential Equations. Vol. 19. Graduate studies in mathematics. Providence, RI: American Mathematical Society, 1998. ISBN: 9780821807729.

The matrix calculations and equations Cao Zhihao, Zhang Deyu, Li Ruixia

Numerical analysis of the matrix of Xing Zhidong

The numerical solution of differential equations Li ronghua, Feng guochen

Galerkin approximations of the numerical solution of differential equations, soup Central

The numerical solution of differential equations Li likang, yuchonghua, Zhu Zhenghua

The solution of systems of nonlinear equations and optimization method of Wang Deren

Of the optimization theory and algorithms for Chen Baolin

Of the theory and method of optimal Yuan yaxiang, Sun wenyu

The characteristic of ye Zhong-XING

Specifically written for the Department of mathematics of information theory

The information theory, coding and Cryptography Ranjan Bose

Computer Science

Computing

奚梅成《数值分析方法》中国科学技术大学出版社

林成森《数值计算方法》科学出版社

谭浩强《C语言程序设计》清华大学出版社

黄刘生《数据结构》中国科学技术大学出版社

周佩玲《16位微机原理接口技术及其应用》中国科学技术大学出版社

李翰荪《电路分析》高等教育出版社

刘同怀《模拟电子线路》中国科学技术大学出版社

康华光《电子技术基础（数字部分）》高等教育出版社

德国Stoer的“数值分析引论”。

算法：Corman等著的”Introduction to Algorithms”

”现代计算机常用数据结构与算法“

形式语言与自动机。我们用过北邮的教材，

数据结构：初级算法课，另一种高级的程序设计课。北大的红皮书(许卓群等著，高教版)和

清华的绿皮书(严蔚敏等著，清华版)。

汇编预言和微机原理

模拟电路: 邱关源的“电路原理”，

教材：康华光的“电子技术基础”还是不错的。有兴趣也可以参考童诗白的书。

数字电路: 阎石的书也算一本好教材，遗憾的一点是集成电路讲少了些。

计算机系统结构：Stallings的”Computer Organization and Architecture:Designing for Performance”

“Computer architecture: a quantitative approach”, by Patterson & Hennessy。

操作系统：Tanenbaum的”Operating System Design and Implementation”和

“Modern Operating System”

形式语言: 编译原理中的前端我看只要学四个算法：１最容易实现的递归下降；２最好的自顶向下算法LL(k)；３最好的自底向上算法LR(k)；４LR(1)的简化SLR(也许还有另一简化LALR?)。５后端完全属于工程性质，自然又是another story。

推荐教材： Aho等人的著名的Dragon Book: “Compilers: Principles, Techniques and Tools”. 或者

Appel的”Modern Compiler Implementation in C”.

学数据库：Silberschatz, et al., “Database System Concepts”.

Tanenbaum:”Computer Networks”(清华影印本)。

集合论，数理逻辑与元数学。图论，算法图论；组合数学，组合算法。抽象代数。代数是无所不在的，

D.E.Knuth在Stanford开设了一门全新的课程Concrete Mathematics。有两层含义：

Computing

Xi Meicheng of the numerical analysis method of the University of science and technology of China press

Lin chengsen of the numerical method of the science press

Tan haoqiang C Programming language in Tsinghua University Press

Liusheng Huang of the data structure of the University of science and technology of China press

Zhou Peiling 16 Interface technique of microcomputer principle and application of University of science and technology of China Press

Li Han-sun of the circuit analysis of the higher education press

Liu Tonghuai of analog electronic circuit, China University of science and technology publishing house

Kang Hua Guang electronic technology Foundation (number of) higher education press

Germany Stoer “An introduction to numerical analysis”.

Algorithm: Corman Waiting for “Introduction to Algorithms”

“Modern computers commonly used data structures and algorithms”

Formal languages and automata. Bupt we used textbooks

Data structures: elementary algorithms class, another advanced programming class. North of the Red Book ( Xu Zhuoqun waiting, higher education ) and

Tsinghua’s Green Paper ( Yan Wei-min wait, Tsinghua University ) 。

Compilation of prophecy and of microcomputer principle

Analog circuits : Qiu Guanyuan “circuit”,

Book: Kang Hua Guang “electronic technology” is pretty good. Are interested can refer to the book Tong Shibai.

Digital circuit : Yan Shi’s book is a good textbook, it is regrettable that IC telling little some.

Computer architecture: Stallings “Computer Organization and Architecture:Designing for Performance”

” Computer architecture: a quantitative approach”, by Patterson & Hennessy 。

Operating system: Tanenbaum “Operating System Design and Implementation”

“Modern Operating System”

Formal languages : Compiler front end I do as long as the four algorithms: 1 is best achieved by recursive descent; 2 the best top-down method LL(k)And 3 best bottom-up algorithms LR(k) ;4 LR(1) Simplified SLR ( perhaps another simplified LALR?) 。 5 back-end belongs entirely to engineering properties, nature is another story.

Recommended book: Aho Famous people such as Dragon Book: “Compilers: Principles, Techniques and Tools”. Or

Appel “Modern Compiler Implementation in C”.

Database: Silberschatz, et al., “Database System Concepts”.

Tanenbaum: “Computer Networks” ( copy of Tsinghua University ) 。

Set theory, mathematical logic and metamathematics. Graph theory, algorithms, graph theory, combinatorics, and combinatorial algorithms. Abstract algebra. Algebra is omnipresent,

D.E.Knuth Stanford Opened a new course Concrete Mathematics 。 Has two meanings:

Operator Algebra

算子代数

1，B(H)代数及其上的拓扑，理想，Calkin定理。

2，双交换定理及其推论。

3，正则泛函，Dixmier-Sakai定理。

4，C*代数的正泛函，GNS构造。

5，交换C*代数，他的特征标，Gelfand-Naimark定理。

6，因子，零除数引理，I类因子。

7，因子的维数理论，因子的分类。

8，迹定理。

9，分解II ∞=II1.I∞。

10，代数与空间共构的比率，因子的标准型。

11，因子的无穷张量形式构造，幂因子。

12，因子的群结构。

13，叉积，因子的度量结构。

14，AF代数与分支图。

15，Hilbert空间上的路径积分，算子的对角化与分解。

16，可分解性判据。

17，Hilbert空间的分解。

18，von Neumann代数的分解，中心分解。

19，极大交换子代数的展开。

20，C*代数的分类。

21，竹崎正道-富山淳理论。

23，子因子的指标。

1，M.A.Naimark，赋范环，科学出版社，1968，第7、8章。

2，J.Dixmier，Les C*-Algèbres et Leurs Représentations，Gauthier-Villars，1969，§§1-9。

3，J.Dixmier，Les Algebras d’Opérateurs dans l’espace Hilbertien，Gauthier-Villars，1957。

4，S.Sakai，C*-algebras and W*-algebras，Springer-Verlag，1971。

5，M.Takesaki，Theory of operator algebras I-III，Springer-Verlag，2002-2003。

Operator algebras

1 , B(H) On algebraic topology, ideal, Calkin Theorem.

2 Double switching theorem and its corollaries.

3 Are functional, Dixmier-Sakai Theorem.

4 , C* Algebra are functional, GNS Structure.

5 To Exchange C* Algebra, his character, Gelfand-Naimark Theorem.

6 Factor, zero divisor lemma I Such factors.

7 , Dimension theory of factor, factor category.

8 , Trace theorems.

9 , Decomposition II ∞=II1. I∞。

10 , Algebra and space total frame rate factor of standard type.

11 And infinite tensor structure of factor, power factor.

12 Factor structure.

13 , The cross product, factor metric structure.

14 , AF Branch of algebra and diagram.

15 , Hilbert Space path integral, Diagonalization and decomposition of the operator.

16 , Criterion of decomposability.

17 , Hilbert The decomposition of space.

18 , von Neumann Algebraic decomposition, decomposition of the Centre.

19 And maximal Abelian subalgebras.

20 , C* Algebraic classifications.

21 , Chuchi path – Fu Shanchun theory.

23 , Factor indicator.

1 , M.A.Naimark , Normed ring, science press, 1968 , 7 、 8 Chapters.

2 , J.Dixmier , Les C*-Alg è bres et Leurs Repr é sentations , Gauthier-Villars , 1969 ,§§ 1-9 。

3 , J.Dixmier , Les Algebras d’Op é rateurs dans l’espace Hilbertien , Gauthier-Villars , 1957 。

4 , S.Sakai , C*-algebras and W*-algebras , Springer-Verlag , 1971 。

5 , M.Takesaki , Theory of operator algebras I-III , Springer-Verlag , 2002-2003 。