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当代著名数学家丘成桐教授最近提议，在中国（包括中国大陆、香港和台湾）的
高校开展大学生数学竞赛，以全面测试大学生的数学知识、修养与能力，促进中
国的大学数学教育改革。丘先生领衔的学术与命题委员会将基础数学与应用数学
凝练成分析与微分方程，几何与拓扑，代数、组合与数论，计算、统计与应用数
学等四个方面，并将提供详细的大纲与参考书。测试范围和国外知名大学的数学
资格考试相当。测试成绩将为国内外录取研究生提供重要的参考，这将为年轻学
子进入现代数学大门提供有利的条件。
中国是泱泱大国，可造就之人才不可胜数，关键是全面提高中国数学人才的修养
与能力，尤其是正在成长中的新生力量能得到有效、及时的指导。近现代数学硕
果累累，大师不断，是现代文明的瑰宝。继承、发展现代数学也是中国贡献于世
界的重要方面。这一竞赛将对中国的大学数学教育起良好的促进作用，在课程设
置和思维方法上与国际一流的数学系接轨，通过测试的学子有望通过国内外的数
学博士资格考试。此举将为中国和世界数学提供源源不断的优异的后备力量，其
中将涌现出许多优秀的数学家。本竞赛将全面提升中国的大学数学教育，为把中
国的人口大国变成人才大国做贡献。
就举办中国大学生数学竞赛致数学同仁的一封公开信
尊敬的同仁，
我们最近倡议举办中国大学生数学竞赛。特写此信向您说明为何要举办这样的竞
赛。我们希望竞赛能够达到如下的目标。
1．培养一位好的数学家，一个重要的步骤是在他们开展研究工作前具备数学的
基础知识和技能。为此，世界主要的数学机构都为研究生设置资格考试。中国好
的学生应该能通过世界上主要数学机构的资格考试，例如哈佛、斯坦福、伯克利、
普林斯顿、哥伦比亚等。可是近年来，这些学校来自中国的学生常常有通不过资
格考试的情况，而来自东欧的好学生则无问题。中国大学数学竞赛的竞赛内容将
足够广泛与基本。这样将有助于为中国和世界的数学机构提供良好的数学生源，
为中国和世界的数学发展培植深厚的土壤。
2．应当为没能进入到中国顶尖大学的学生提供公平的竞争机会。这一点很重要。
一个高中生可能有很多原因使得他（她）没能进入到顶尖大学，例如运气不好、
身体不适、贫穷或者不成熟等。可是这些情况可能会发生变化，而这些学生突然
决定要好好干。可在目前中国的情况下他们到世界和中国一流机构深造的机会较
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少。我们应当给他们以机会。我本人在香港就有类似的经历。我曾经很难进入到
一流大学，经过艰苦的努力与奋斗才得到承认。
3．要让在顶尖学校学习的学生感到在数学上经常有压力和挑战，使得他们刻苦
工作，保持一个良好的工作状态。现在经常有来自中国大陆的学生大谈数学的哲
学，而不能坐下来做扎实的计算。这个竞赛的内容就是数学的基本知识与基本功，
通过这个竞赛将能够非常有效地改变这种状况。
4．我们将邀请世界著名的数学家口试我们竞赛优秀的学生（暂定前 15 名）。
这些数学家将亲自鉴定这些学生的水平与能力，为他们进入世界一流的研究院提
供帮助。我相信许多一流数学家将会乐意面试我们的优秀学生。
总之，我相信这个竞赛将把中国大学生的数学教育提高到一个新的水平。我们诚
请您的学生参加中国大学数学竞赛。详情见：http://www.cms.zju.edu.cn
致礼
丘成桐
哈佛大学数学系主任
中国大学数学竞赛学术与命题委员会主席
--------------------------竞赛规则
1 本竞赛面向中国大陆、香港、台湾高校的尚未取得学士学位的本科在校生。
2 各高校可组队参加。每队由四位队员组成。每校不超过两个队。各高校个人参
赛的名额没有限制。团体赛与个人赛分别举行。比赛共两天。
3 第一天早晨举行团体赛。在团体比赛中，每队队员先用半小时时间讨论分题目。
而后的两个半小时各位队员独立做题。
4 在余下的一天半举行个人赛。每个科目的做题时间为两个半小时。
5 在比赛中，不允许队员讨论或与外界联系。
竞赛内容
竞赛将全面测试学生的知识与技能。测试将分成 4 个科目：
1 分析与微分方程
2 几何与拓扑
3 代数、数论与组合
4 应用、计算和概率统计
竞赛学术与出题委员会将提供每个科目的详细大纲和参考书。每个科目将有五道
题目。每个参加者可选其中的三个科目。
竞赛前 15 名优胜者将参加 2010 年 10 月 14 日的口试。
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打分
每题 20 分。全部正确解决问题的得满分。解题中有一定想法和进展的可得部分
分数。
竞赛时间
首次竞赛在 2010 年 10 月 12 日举行。
竞赛地点
竞赛将在中国大陆、香港、台湾的主要大学举行。
注册
可从浙江杭州浙江大学数学中心及网页索取注册材料。每位参加者应在八月中之
前完成注册。
参考文献
Geometry and Topology (preliminary draft)
Differential geometry of curves and surfaces
Differentia geometry of curves and surfaces.
M. do Carmo: Prentice- Hall, 1976 (25th printing)
Table of Contents
1. Curves: Parametrized Curves.
2. Regular Surfaces: Regular Surfaces; Inverse Images of Regular Values.
3. Geometry of the Gauss Map: Definition of the Gauss Map and Its
Fundamental Properties.
4. Intrinsic Geometry of Surfaces: Isometrics; Conformal Maps.
5. Global Differential Geometry: Rigidity of the Sphere.
Differential Manifolds and Riemannian Geometry
An Introduction to Differentiable Manifolds and Riemannian Geometry.
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W.M. Boothby: Academic Press, Inc., Orlando, FL, 1986
Contents
I. Introduction to Manifolds
1. Preliminary Comments on Rn
2. Rn and Euclidean Space 4
3. Topological Manifolds 6
4. Further Examples of Manifolds. Cutting and Pasting Abstract Manifolds.
Some Examples 14
II. Functions of Several Variables and Mappings
1. Differentiability for Functions of Several Variables 20
2. Differentiability of Mappings and Jacobians 25
3. The Space of Tangent Vectors at a Point of Rn 29
4. Another Definition of Ta(Rn) 32
5. Vector Fields on Open Subsets of Rn
6. The Inverse Function Theorem 41
7. The Rank of a Mapping 46
III. Differentiable Manifolds and Submanifolds
1. The Definition of a Differentiable Manifold
2. Further Examples 59
3. Differentiable Functions and Mappings
4. Rank of a Mapping, Immersions 68
5. Submanifolds 74
6. Lie Groups 80 (*)
7. The Action of a Lie Group on a Manifold. Transformation Groups (*)
8. The Action of a Discrete Group on a Manifold 93 (*)
9. Covering Manifolds 98 (*)
IV. Vector Fields on a Manifold
1. The Tangent Space at a Point of a Manifold
2. Vector Field 113
3. One-Parameter and Local One-Parameter Groups Acting on a Manifold
4. The Existence Theorem for Ordinary Differential Equations 127
5. Some Examples of One-Parameter Groups Acting on a Manifold (*)
6. One-Parameter Subgroups of Lie Groups 142 (*)
7. The Lie Algebra of Vector Fields on a Manifold (*)
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8. Frobenius's Theorem 153 (*)
9. Homogeneous Spaces 160 (*)
V. Tensors and Tensor Fields on Manifolds
1. Tangent Covectors 171
Covectors on Manifolds 172
Covector Fields and Mappings 174
2. Bilinear Forms. The Riemannian Metric
3. Riemannian Manifolds as Metric Spaces
4. Partitions of Unity 186
Some Applications of the Partition of Unity
5. Tensor Fields
192
Tensors on a Vector Space
Tensor Fields
194
Mappings and Covariant Tensors 195
The Symmetrizing and Alternating Transformations
6. Multiplication of Tensors 199
Multiplication of Tensors on a Vector Space
Multiplication of Tensor Fields 201
Exterior Multiplication of Alternating Tensors
The Exterior Algebra on Manifolds 206
7. Orientation of Manifolds and the Volume Element
8. Exterior Differentiation
212
An Application to Frobenius's Theorem 177
VI. Integration on Manifolds
1. Integration in Rn Domains of Integration 223
Basic Properties of the Riemann Integral 224
2. A Generalization to Manifolds
229
Integration on Riemannian Manifolds
3. Integration on Lie Groups 237
4. Manifolds with Boundary 243
5. Stokes's Theorem for Manifolds 251
6. Homotopy of Mappings. The Fundamental Group 258 (*)
Homotopy of Paths and Loops. The Fundamental Group (*)
7. Some Applications of Differential Forms.
The de Rham Groups
The Homotopy Operator 268 (*)
8. Some Further Applications of de Rham Groups (*)
The de Rham Groups of Lie Groups 276 (*)
9. Covering Spaces and Fundamental Group 280 (*)
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VII. Differentiation on Riemannian Manifolds
1. Differentiation of Vector Fields along Curves in Rn
The Geometry of Space Curves 292
Curvature of Plane Curves 296
2. Differentiation of Vector Fields on Submanifolds of Rn
Formulas for Covariant Derivatives 303
Differentiation of Vector Fields 305
3. Differentiation on Riemannian Manifolds 308
Constant Vector Fields and Parallel Displacement
4. Addenda to the Theory of Differentiation on a Manifold
The Curvature Tensor 316 (*)
The Riemannian Connection and Exterior Differential Forms (*)
5. Geodesic Curves on Riemannian Manifolds 321 (*)
6. The Tangent Bundle and Exponential Mapping. Normal Coordinates (*)
7. Some Further Properties of Geodesics 332 (*)
8. Symmetric Riemannian Manifolds 340 (*)
9. Some Examples 346 (*)
VIII. Curvature
1. The Geometry of Surfaces in E^3
The Principal Curvatures at a Point of a Surface 359
2. The Gaussian and Mean Curvatures of a Surface 363
The Theorema Egregium of Gauss 366
3. Basic Properties of the Riemann Curvature Tensor
4. Curvature Forms and the Equations of Structure
5. Differentiation of Covariant Tensor Fields 384 (*)
6. Manifolds of Constant Curvature 391 (*)
Spaces of Positive Curvature 394
Spaces of Zero Curvature 396
Spaces of Constant Negative Curvature
Other references:
M. Spivak, A comprehensive introduction to differential geometry
N. Hicks, Notes on differential geometry, Van Nostrand.
J. Milnor, Morse Theory
Basic topology
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M. Armstrong, Springer, Undergraduate texts in mathematics
Table of contents:
Preface. 1: Introduction. 2: Continuity. 3: Compactness and connectedness. 4: Identification
spaces. 5: The fundamental group. 6: Triangulations. 7: Surfaces. 8: Simplicial homology. 9:
Degree and Lefschetz number. 10: Knots and covering spaces. Appendix: Generators and
relations.
Algebraic topology
Algebraic topology
By A Hatcher (http://www.math.cornell.edu/~hatcher/AT/ATpage.html)
Chapter 0. Some Underlying Geometric Notions
Homotopy and Homotopy Type. Cell Complexes. Operations on Spaces. Two
Criteria for Homotopy Equivalence. The Homotopy Extension Property.
Chapter 1. Fundamental Group and Covering Spaces
1. Basic Constructions.
Paths and Homotopy. The Fundamental Group of the Circle. Induced
Homomorphisms.
2. Van Kampen's Theorem
Free Products of Groups. The van Kampen Theorem. Applications to Cell
Complexes.
3. Covering Spaces
Lifting Properties. The Classification of Covering Spaces. Deck
Transformations and Group Actions.
4. Additional Topics (*)
Graphs and Free Groups. K(G,1) Spaces and Graphs of Groups.
Chapter 2. Homology
1. Simplicial and Singular Homology
Delta-Complexes. Simplicial Homology. Singular Homology. Homotopy
Invariance. Exact Sequences and Excision. The Equivalence of Simplicial and
Singular Homology.
2. Computations and Applications
Degree. Cellular Homology. Mayer-Vietoris Sequences. Homology with
Coefficients.
3. The Formal Viewpoint (*)
Axioms for Homology. Categories and Functors.
4. Additional Topics (*)
Homology and Fundamental Group. Classical Applications. Simplicial
Approximation.
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Chapter 3. Cohomology
1. Cohomology Groups
The Universal Coefficient Theorem. Cohomology of Spaces.
2. Cup Product
The Cohomology Ring. A Kunneth Formula. Spaces with Polynomial
Cohomology.
3. Poincare Duality
Orientations and Homology. The Duality Theorem. Cup Product and Duality.
Other Forms of Duality.
4. Additional Topics (*)
The Universal Coefficient Theorem for Homology. The General Kunneth
Formula. H-Spaces and Hopf Algebras. The Cohomology of SO(n). Bockstein
Homomorphisms. Limits. More About Ext. Transfer Homomorphisms. Local
Coefficients.
Chapter 4. Homotopy Theory (*)
1. Homotopy Groups
Definitions and Basic Constructions. Whitehead's Theorem. Cellular
Approximation. CW Approximation.
2. Elementary Methods of Calculation
Excision for Homotopy Groups. The Hurewicz Theorem. Fiber Bundles. Stable
Homotopy Groups.
3. Connections with Cohomology
The Homotopy Construction of Cohomology. Fibrations. Postnikov Towers.
Obstruction Theory.
4. Additional Topics
Basepoints and Homotopy. The Hopf Invariant. Minimal Cell Structures.
Cohomology of Fiber Bundles. The Brown Representability Theorem. Spectra
and Homology Theories. Gluing Constructions. Eckmann-Hilton Duality.
Stable Splittings of Spaces. The Loopspace of a Suspension. Symmetric
Products and the Dold-Thom Theorem. Steenrod Squares and Powers.
Appendix
Topology of Cell Complexes. The Compact-Open Topology.
Other references:
J. Milnor, Topology from the differentiable viewpoint
R. Bott and L. Tu, Differential forms in algebraic topology
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V. Guillemin, A. Pollack, Differential topology
Note: Chapters and sections with (*) shall not be covered in the contests.
----------------------------------------------
Algebra, Number Theory and
Combinatorics (preliminary draft)
Linear algebra
Linear algebra. Matrices, linear transformations, change of basis; nullity-rank theorem.
Eigenvalues and eigenvectors; determinants, characteristic and minimal polynomials,
Cayley-Hamilton theorem; diagonalization and triangularization of operators; Jordan normal form,
rational canonical form; invariant subspaces and canonical forms; inner product spaces, hermitian
and unitary operators, adjoints. Multilinear algebra, bilinear forms, orthogonal groups, spectral
theorem (real and complex cases).
Texts:
Strang, Linear algebra, Academic Press.
Noble-Daniel, Applied linear algebra, Prentice-Hall;
Hoffman-Kunze, Linear algebra, Prentice-Hall
Algebra
I. GROUP THEORY. Groups, subgroups, normal subgroups, homomorphisms, quotient groups,
automorphisms, groups acting on sets, Sylow theorems and applications, 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.
II. COMMUTATIVE RINGS. 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, Noetherian rings, Hilbert basis theorem,
Artin rings, integral ring extensions, Nullstellensatz, Dedekind domains(*),algebraic sets(*),
Spec(A)(*).
III. MODULES. Finitely generated modules over PID's and applications, chain conditions
(especially Noetherian modules), projective and injective modules, tensor products, symmetric and exterior powers, Hom , Tor and Ext(*) .
IV. FIELD THEORY. Degree of an extension, algebraic and transcendental field extensions,
splitting fields and algebraic closure, finite fields, normal and separable extensions, theorem of the
primitive element, inseparable extensions, field embeddings and automorphisms, Galois theory,
norms and traces, cyclic extensions, Galois theory of number fields, transcendence degree,
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function fields.
V. GROUP REPRESENTATIONS and Non-commutative rings. 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, division rings(*), matrix rings(*), semisimple rings and
modules(*).
References:
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.
Dummit and Foote: Abstract Algebra
S. Lang, Algebra, Addison-Wesley;
M. Artin, Algebra
J. P. Serre, Linear representations of finite groups.
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》
Elementary Number Theory
1. Integers: a) Divisibility of integers, Euclidean algorithm, unique decomposition
theorem of integers; b) congruence and congruence classes, solving congruence
equations, the Chinese Remainder theorem; c) Prime roots and indexes; d) Quadratic
reciprocity ; e)Indeterminate Equations.
2. Polynomials： a) Polynomials of one variable, Euclidean algorithm, uniqueness
decomposition theorem, zeros; b) The fundamental theorem of algebra, unit roots ; c)
Polynomials of integer coefficients, the Gauss lemma and the Eisenstein criterion; d)
Polynomials of several variables, homogenous and symmetric polynomials, the
fundamental theorem of symmetric polynomials.
References:
Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Sixth edition. Revised by D.
R. Heath-Brown and J. H. Silverman. Oxford University Press, Oxford, 2008. xxii+621 pp.
Borevich, A. I.; Shafarevich, I. R. Number theory. Translated from the Russian by Newcomb
Greenleaf. Pure and Applied Mathematics, Vol. 20 Academic Press, New York-London 1966
x+435 pp.
J. P. Serre, A course in arithmetics.
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《整数与多项式》冯克勤 余红兵著 高等教育出版社
Combinatorics (*)
van Lint, J. H.(NL-EIND); Wilson, R. M.(1-CAIT) A course in combinatorics. (English
summary) Cambridge University Press, Cambridge, 1992. xii+530 pp.
Table of Contents
Preface
1. Graphs
2. Trees
3. Colorings of graphs and Ramsey's theorem
4. Turan's theorem and extremal graphs
5. Systems of distinct representatives
6. Dilworth's theorem and extremal set theory
7. Flows in networks
8. De Bruijn sequences
9. The addressing problem for graphs
10. The principle of inclusion and exclusion inversion formulae
11. Permanents
12. The Van der Waerden conjecture
13. Elementary counting Stirling numbers
14. Recursions and generating functions
15. Partitions
16. (0,1)-matrices
17. Latin squares
18. Hadamard matrices, Reed-Muller codes
19. Designs
20. Codes and designs
21. Strongly regular graphs and partial geometries
22. Orthogonal Latin squares
23. Projective and combinatorial geometries
24. Gaussian numbers and q-analogues
25. Lattices and Mobius inversion
26. Combinatorial designs and projective geometries
27. Difference sets and automorphisms
28. Difference sets and the group ring
29. Codes and symmetric designs
30. Association schemes
31. Algebraic graph theory: eigenvalue techniques
32. Graphs: planarity and duality
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33. Graphs: colorings and embeddings
34. Electrical networks and squared squares
35. Polya theory of counting
36. Baranyai's theorem
Note: Chapters and sections with (*) shall not be covered in the contests.
Analysis and differential equations (preliminary draft)
Mathematical analysis (including ODE)
1. Basic Calculus. Basic first- and second-year calculus. Derivatives of maps from
Rm to Rn, gradient, chain rule; maxima and minima, Lagrange multipliers; line and
surface integrals of scalar and vector functions; Gauss’s, Green’s and Stokes’
theorems. Ordinary differential equations; explicit solutions of simple equations.
2. Classical analysis. Topology of Rn and metric spaces; properties of continuous
functions, compactness, connectedness, limit points; least upper bound property of R.
Sequences and series, Cauchy sequences, uniform convergence and its relation to
derivatives and integrals; power series, radius of convergence, Weierstrass M-test;
convergence of improper integrals. Compactness in functions spaces. Inverse and
implicit function theorems and applications; the derivative as a linear map; existence
and uniqueness theorems for solutions of ordinary differential equations; elementary
Fourier series.
References:
Rudin, Principles of mathematical analysis, McGraw-Hill.
Courant, Richard; John, Fritz Introduction to calculus and analysis. Vol. I. Reprint of
the 1989 edition.Classics in Mathematics. Springer-Verlag, Berlin, 1999. xxiv+661
pp.
Courant, Richard; John, Fritz Introduction to calculus and analysis. Vol. II. With the
assistance of Albert A. Blank and Alan Solomon. Reprint of the 1974
edition. Springer-Verlag, New York, 1989. xxvi+954 pp.
Hardy, G. H.; Littlewood, J. E.; Pólya, G. Inequalities. Reprint of the 1952
edition. Cambridge
Mathematical
Library. Cambridge
University
Press,
Cambridge, 1988. xii+324 pp.
Pólya, George; Szegő, Gabor Problems and theorems in analysis. I. Series, integral
calculus, theory of functions. II. Theory of functions, zeros, polynomials,
determinants, number theory, geometry. Translated from the German by C. E.
Billigheimer. Reprint
of the 1976 English
translation. Classics in
Mathematics. Springer-Verlag, Berlin, 1998.
Coddington,
Earl
A.; Levinson,
Norman Theory
of
ordinary
differential
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equations. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
xii+429 pp.
V. I. Arnold,
Ordinary Differential Equations, Springer-Verlag, Berlin, 2006.
Real analysis and functional analysis
Set Theory.
Countable and uncountable sets, the axiom of choice, Zorn's lemma.
Metric spaces. Completeness; separability; compactness; Baire category; uniform
continuity; connectedness; continuous mappings of compact spaces.
Functions on topological spaces. Equicontinuity and Ascoli's theorem; the
Stone-Weierstrass theorem; topologies on function spaces; compactness in function
spaces.
Measure.
Measures and outer measures; measurability and _-algebras; Borel sets;
extension of measures; Lebesgue and Lebesgue-Stieltjes measures; signed measures;
absolute continuity and singularity; product measures.
Measurable functions. Properties of measurable functions; approximation by
simple functions and by continuous functions; convergence in measure; Egorov's
theorem; Lusin's theorem; Jensen's inequality.
Integration.
Construction and properties of the integral; convergence theorems;
Radon-Nykodym theorem; Fubini's theorem; mean convergence.
Special properties of functions on the real line. Monotone functions; functions
of bounded variation and Borel measures; absolute continuity; differentiation
and integration; convex functions; semicontinuity; Borel sets; properties of the
Cantor set.
Elementary properties of Banach and Hilbert spaces.
Lp spaces; C(X); completeness and the Riesz-Fischer theorem; orthonormal bases;
linear functionals; Riesz representation theorem; linear transformations and dual
spaces; interpolation
of linear operators; Hahn-Banach theorem; open mapping theorem; uniform
boundedness
(or Banach-Steinhaus) theorem; closed graph theorem.
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Basic harmonic analysis.
Basic properties of Fourier series and the Fourier transform; Poission summation
formula; convolution.
Compact operators
Basic properties of compact operators, Riesz- Fredholm theory, spectrum of compact
operators.
References:
Royden, Real Analysis, except chapters 8, 13, 15.
E.M. Stein and R. Shakarchi， Real Analysis: Measure Theory, Integration, and
Hilbert Spaces, Princeton University Press, 2005
周民强, 实变函数论, 北京大学出版社, 2001
夏道行等，《实变函数论与泛函分析》，人民教育出版社.
Peter D. Lax, Functional Analysis, Wiley-Interscience, 2002.
吉田耕作, 泛函分析.
Complex analysis
An Introduction to the Theory of Analytic Functions of
One Complex Variable
By Lars Valerian Ahlfors, McGraw-Hill College, August 1979
Preface
Chapter 1: Complex numbers
1. The algebra of complex numbers
2. The geometric Representation of Complex Numbers
Chapter 2: Complex functions
1. Introduction to the concept of Analytic function
2. Elementary Theory of Power Series
3. The exponential and trigonometric Functions
Chapter 3: Analytic functions as mappings
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1. Elementary Point Set Topology
2. Conformality
3. Linear Transformations
4. Elementary Conformal Mappings
Chapter 4: Complex Integration
1. Fundamental Theorems
2. Cauchy's Integral Formula
3. Local Properties of Analytic Functions
4. The General Form of Cauchy's Theorem
5. The Calculus of Residues
6. Harmonic Functions
Chapter 5: Series and Product Developments
1. Power Series Expansions
2. Partial Fractions and Factorization
3. Entire Functions
4. Normal Families
Chapter 6: Conformal Mapping, Dirichlet’s Problem
1. The Riemann Mapping Theorem
2. Conformal Mapping of Polygons
3. A closer Look at Harmonic Functions
4. The Dirichlet Problem
5. Canonical Mappings of Multiply Connected Regions
Chapter 7: Elliptic Functions
1. Simply Periodic Functions
2. Doubly Periodic Functions
3. The Weierstrass Theory
Chapter 8: Global Analytic functions
1. Analytic Continuation
2. Algebraic Functions
3. Picard's Theorem
4. Linear Differential Equations
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Other references:
K. Kodaira, Complex Analysis
Rudin, Real and complex analysis
龚升，简明复分析
Basic partial differential equations
First order partial differential equations, linear and quasi-linear PDE
Wave equations: initial condition and boundary condition, well-poseness,
Sturn-Liouville eigen-value problem, energy functional method, uniqueness and
stability of solutions
Heat equations: initial conditions, maximal principle and uniqueness and stability
Potential equations: Green functions and existence of solutions of Dirichlet problem,
harmonic functions, Hopf’s maximal principle and existence of solutions of
Neumann’s problem, weak solutions, eigen-value problem of the Laplace operator
Generalized functions and fundamental solutions of PDE
References:
《Basic Partial Differential Equations》, D. Bleecker, G. Csordas 著, 李俊杰 译，高
等教育出版社，2008.
《数学物理方法》
，柯朗、希尔伯特著。
Gilbarg and Trudinger, Elliptic Partial Differential Equations of Second Order,
Chapter 7.
Computational
Mathematics,
Applied
Mathematics,
Probability and Statistics (preliminary draft)
Computational Mathematics
1. Interpolation and approximation
--- polynomial interpolation and least square approximation
--- trigonometric interpolation and approximation,
fast Fourier transform
--- approximations by rational functions
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--- splines
2. Nonlinear equation solvers
--- convergence of iterative methods (bisection,
secant method, Newton method, other iterative
methods) for both scalar equations and systems
--- finding roots of polynomials
3. Linear systems and eigenvalue problems
--- direct solvers (Gauss elimination, LU decomposition,
pivoting, operation count, banded matrices,
round-off error accumulation)
--- iterative solvers (Jacobi, Gauss-Seidel, successive
over-relaxation, conjugate gradient method, multi-grid
method, Krylov methods)
--- numerical solutions for eigenvalues and eigenvectors
4. Numerical solutions of ordinary differential equations
--- one step methods (Taylor series method and Runge-Kutta
method)
--- stability, accuracy and convergence
--- absolute stability, long time behavior
--- multi-step methods
5. Numerical solutions of partial differential equations
--- finite difference method
--- stability, accuracy and convergence, Lax equivalence theorem
--- finite element method, boundary value problems
References:
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[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] G. Strang and G. Fix,
An
Analysis
of
the
Finite
Element
Method,
second
edition,
Wellesley-Cambridge Press, 2008.
Probability
Introduction to Probability Models
by Sheldon M. Ross
1 Introduction to Probability Theory
2 Random Variables
3 Conditional Probability and Conditional Expectation
4 Markov Chains
5 The Exponential Distribution and the Poisson Process
6 Continuous-Time Markov Chains
7 Renewal Theory and Its Applications
8 Queueing Theory
9 Reliability Theory
10 Brownian Motion and Stationary Processes
11 Simulation
Statistics
Overview
The statistics part contains seven areas: (1) distribution theory, (2) likelihood principle,
(3) testing, (4) estimation, (5) Bayesian statistics, (6) nonparametric statistics, (7)
regression and (8) large sample theory.
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1. Distribution Theory and Basic Statistics
Families of continuous distributions: Chi-sq, t, F, gamma, beta
Families of discrete distributions: Multinomial, Poisson, negative binomial
Basic statistics: Mean, median, quantiles, order statistics
2. Likelihood principle
Likelihood function, parametric inference, sufficiency, factorization theorem,
ancillary statistic, conditional likelihood, marginal likelihood.
3. Testing
Neyman-Pearson paradigm, null and alternative hypotheses, simple and composite
hypotheses, type I and type II errors, power, most powerful test, likelihood ratio test,
Neyman-Pearson Theorem, monotone likelihood ratio, uniformly most powerful test,
generalized likelihood ratio test.
4. Estimation
Parameter estimation, method of moments, maximum likelihood estimation,
unbiasedness, quadratic and other criterion functions, Rao-Blackwell Theorem, Fisher
information, Cramer-Rao bound, confidence interval, duality between confidence
interval and hypothesis testing.
5. Bayesian Statistics
Prior, posterior, conjugate priors, Bayesian loss
6. Nonparametric statistics
Permutation test, permutation distribution, rank statistics, Wilcoxon-Mann-Whitney
test, log-rank test, Kolmogorov-Smirnov-type tests.
7. Regression
Linear regression, least squares method, Gauss-Markov Theorem, logistic regression,
maximum likelihood
8. Large sample theory
Consistency, asymptotic normality, chi-sq approximation to likelihood ratio statistic,
large-sample based confidence interval, asymptotic properties of empirical
distribution.
References
Casella, G. and Berger, R.L. (2002). Statistical Inference (2nd Ed.) Duxbury Press.
茆诗松，程依明，濮晓龙，概率论与数理统计教程（第二版）
，高等教育出版社，2008.
陈家鼎，孙山泽，李东风，刘力平，数理统计学讲义，高等教育出版社，2006.
郑明，陈子毅，汪嘉冈，数理统计讲义，复旦大学出版社，2006.
陈希孺，倪国熙，数理统计学教程，中国科学技术大学出版社，2009.
Applied Mathematics
1. Laplace method
2. Stationary phase method
3. Regular perturbation method
4. Singular perturbation method
5. Matched asymptotic method
6. Two time scale asymptotic expansions for ODEs
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7. Variational methods
8. Method of averaging for ODEs
9. Homogenization method for elliptic PDEs
10. Homogenization method for linear hyperbolic PDEs
11. Homogenization and front propagation of Hamilton-Jacobi equations
12. Front propagation of reaction-diffusion-advection equations in
heterogeneous media.
13. Geometric optics for dispersive wave equations
14. Slowly varying amplitude expansions for dispersive PDEs (sine-Gordon,
nonlinear Schroedinger)
Reference Books:
1. N. De Bruijn, “Asymptotic Methods in Analysis”, Dover, NY, 1981.
2. J. Kervorkian, J. Cole, “Perturbation Methods in Applied Mathematics”, Applied
Math Sciences, 34, Springer, 1981.
3. G. Whitham, “Linear and Nonlinear Waves”, John-Wiley and Sons, 1974.
4. J. Keener, “Principles of Applied Mathematics”, Addison-Wesley, 1988.
5. A. Benssousan, P-L Lions, G. Papanicolaou, “Asymptotic Analysis for Periodic
Structures”, North-Holland Publishing Co, 1978.
6. L.C. Evans, “Weak convergence methods for nonlinear PDEs”, NSF-CBMS
regional conference notes, 1988.
7. V. Jikov, S. Kozlov, O. Oleinik, “Homogenization of differential operators and
integral functions”, Springer, 1994.
8. J. Xin, “An Introduction to Fronts in Random Media”, Surveys and Tutorials in
Applied Math Sciences, No. 5, Springer, 2009.
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