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课程大纲

COURSE SYLLABUS

1.

课程名称(中英文)

Course Title(Chinese and English)

数学英文写作

Mathematical Writing in English

2.

课程类别Course Type

必修 Required

3.

授课院系

Originating Department

数学系

Department of Mathematics

4.

可选课学生所属院系

Open to Which Majors

所有院系

5.

课程学时Credit Hours

32

6.

课程学分Credit Value

2

7.

授课语言

Teaching Language

中英文混合板书及教材，中英文混合讲解。

Textbook and lecture notes are written in a combination of Chinese and English. The languages of instruction are Mandarin and English.

8.

授课教师Instructor(s)

汤涛，曹敏

Tao TANG and Min TSAO

9.

先修课程、其它学习要求

Pre-requisites or Other Academic Requirements

三年大学公共英语课程，高等数学，线性代数

Three years of university English courses， Calculus, Linear Algebra

10.

教学目标Course Objectives

本课程旨在提高研究生和科研人员的科技英语尤其是数学英语的写作讲解和使用能力。主要内容包含英文数学文章的结构及常用句型和词语，英文数学文章的投稿和修改。本课程也将讨论怎样用英文给数学讲座，怎样用英文写科研计划和经费申请，并对常用英文数学写作软件LaTeX做一个基本介绍。

This course is designed for mathematics graduate students and researchers who need advanced training in English writing and communication skills. Topics include structures of English mathematical research papers, commonly used sentence structures and phrases, and tips on how to submit and revise papers in English. The course also covers how to give mathematics presentations in English and how to write research or grant proposals in English. It also introduces LaTeX, an open-source software for high-quality typesetting widely used for mathematical papers.

11.

教学方法及授课创新点 Teaching Methods and Innovations

本课程选用汤涛和丁玖攥写的《数学之英文写作》为课本，采用让学生多读多写多交流的教学方式，为学生打下稳固的数学英文写作基础，培养和提高学生的英文科研工作能力。

The textbook of this course is ``Mathematical Writing in English’’ written by professors Tao TANG and Jiu DING. The course offers students a lot of opportunities to practice reading, writing and presenting in scientific English. It will help students develop a solid foundation in mathematical writing in English and acquire valuable skills in using English for research and teaching.

12.

教学内容及学时分配Course Contents and Course Schedule

1. 数学文章的结构 (2 weeks)

2. 数学文章的词句 (3 week)

3. 怎样修改文章 (1 weeks)

4. 英文数学写作软件LaTeX (1 week)

5. 文章投稿 (1 weeks)

6. 数学综合写作 (2 weeks)

7. 其它文体的书写 (2 week)

8. 怎样写科研计划和经费申请 (2 weeks)

9. 怎样给数学讲座 (2 week)

13.

课程考核Course Assessment

Homework 20% + in class group presentation 40% + Final presentation 40%

14.

教材及其它参考资料Textbook and Supplementary Readings

汤涛和丁玖,数学之英文写作，高等教育出版社，2014， ISBN：9787040368932

课程大纲

COURSE SYLLABUS

1.



代数

Algebra

2.


必修 Required

3.

授课院系


数学系

4．



所有院系

5.


48学时

6.


3学分

7.

授课语言

Teaching Language

中文/英文

8.


李勤，李才恒

9.



线性代数，近世代数

10.


了解和掌握代数学的基本对象包括群，环，模，域的基本思想和基础知识，培养学生在今后的科研中应用基本的代数工具的能力。为进一步学习交换代数或同调代数打下基础。

11.


课堂教学+讨论

12.


1. 群论（包括Jordan-Holder定理和Sylow定理）

2. 环理论（包括环的基本性质和理想，Unique factorization domains和Principal ideal domains）

3. 模理论（包括链条件）

4. 域理论及伽罗瓦理论（包括有限域和伽罗瓦理论基本定理）

13.


笔试，作业

14.


1. Algebra (GTM 211)，Serge Lang.

2. Algebra (GTM 73), Thomas W. Hungerford.

课程大纲

COURSE SYLLABUS

1.



测度论与积分

Measure Theory and Integration

2.


必修 Required

3.

授课院系


数学系


4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language

根据学生的情况可以是英文、中文或者两者相结合。

English, or Chinese, or both depending on the need of the students.

8.


苏琳琳，助理教授; 王学锋, 教授

Linlin Su, Assistant Professor；Xuefeng Wang, Professor

9.



本科实、复变函数, 本科生泛函分析

Undergraduate Lebesgue Theory， Complex Analysis， and Functional Analysis

10.


本课是大学实变函数课程的继续。在大学课程里，学生已经掌握了实轴上的Lebesgue测度及积分理论，故而自然地本课将以抽象的测度论开始，再讲抽象可测空间上的积分理论，Lp空间等。这些内容为其它研究生课程如概率论打下基础。

课程的最后部分把学生重新带回Rn空间中，学习调和分析的一些最基本的内容，包括傅氏变换的有界性和傅里叶级数的收敛性等等广泛应用于应用数学和偏微分方程的内容。

11.


将采用传统方式教授此课(版书，课堂讨论，作业，课外答疑，闭卷考试)。

The course will be taught in the standard way (“chalk and board”, in-class discussion, homework, office hours, closed-book exams).

12.


1. General Measure Spaces: Their Properties and Construction

1.1. Measures and Measurable Sets

1.2. Signed Measures: The Hahn and Jordan Decompositions

1.3. The Carath6odory Measure Induced by an Outer Measure

1.4. The Construction of Outer Measures

1.5. The Caratheodory-Hahn Theorem: The Extension of a Premeasure to a Measure

2. Integration Over General Measure Spaces

2.1. Measurable Functions

2.2. Integration of Nonnegative Measurable Functions

2.3. Integration of General Measurable Functions

2.4. The Radon-Nikodym Theorem

3. The Construction of Particular Measures

3.1. Product Measures: The Theorems of Fubini and Tonelli

3.2. Lebesgue Measure on Euclidean Space R"

3.3. Cumulative Distribution Functions on R and Lebesgue-Stieltjes integral

4. General LP Spaces: Completeness, Duality, and Weak Convergence

4.1. The Completeness of LP(X, μ)

4.2. The Riesz Representation Theorem for the Dual of LP(X, μ), 1 ≤ p < ∞

4.3. The Kantorovitch Representation Theorem for the Dual of L∞ (X, μ)

4.4. Weak Sequential Compactness in LP(X, p.), 1 < p <∞

4.5. Weak Sequential Compactness in L1(X, μ): The Dunford-Pettis Theorem

5. Some Basics in Harmonic Analysis

5.1. The Fourier transform on L1 and L2

5.2. Riesz-Thorin interpolation theorem and the Fourier transform on Lp, 1

5.3. The Marcinkiewicz interpolation theorem

5.4. Hardy-Littlewood maximal function and Hardy-Littlewood maximal inequality

5.5. Hardy-Littlewood-Sobolev inequality

5.6. Trigonometric Fourier Series

5.6.1.1. Convergence pointwise: Dirichlet–Dini Theorem, Dirichlet-Jordan Theorem for functions of bounded variation

5.6.1.2. Convergence in Lp, 1

13.


作业 40%+期中考试（闭卷）20%+期末考试(闭卷) 40%

Homework 40%+ Mid-term Exam (closed-book) 20%+Final Exam (closed-book) 40%

14.


1. Real Analysis, fourth edition, by Halsey L. Royden and Patrick M. Fitzpatrick, 2010.

2. Topics in Real and Functional Analysis by Gerald Teschl, University of Vienna, 2015 version.

3. Measure and Integral, R. Wheeden and A. Zygmund, 1997.

课程大纲

COURSE SYLLABUS

1.



泛函分析

Functional Analysis

2.


必修 Required

3.

授课院系


数学系


4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language



8.


苏琳琳，助理教授; 王学锋，教授

Linlin Su, Assistant Professor；Xuefeng Wang, Professor

9.



本科实、复变函数, 本科泛函分析

Undergraduate Lebesgue Theory，Complex Analysis，and Functional Analysis

10.


本课程是本科泛函课程的继续与深入，着重介绍有重要应用价值的经典理论，为学生的其他研究生数学课程和相关的科研工作打下基础。

11.


将采用传统方式教授此课(版书，课堂讨论，作业，课外答疑，闭卷考试)。强调抽象理论和具体应用的结合。

The course will be taught in the standard way (“chalk and board”, in-class discussion, homework, office hours, closed-book exams). The course is a balanced mix of abstract theories and applications.

12.


1. Hahn-Banach Theorem

1.1. The extension theorem

1.2. Hyperplane separation of convex sets

1.3. Applications

1.3.1.1. Extension of positive linear functionals

1.3.1.2. Lagrange multipliers of convex programming problems

2. Weak and weak * topologies

2.1. Weak convergence and weak compactness of unit ball in reflexive Banach spaces

2.2. Weak* convergence and weak* sequential compactness—Helly’s Theorem

2.3. Banach-Alaoglu Theorem

2.4. Applications

2.4.1. Approximation of the delta-function by continuous functions

2.4.2. Approximate quadrature

2.4.3. Existence of PDE via Galerkin’s method

3. General spectral theory

3.1. Spectral radius and Gelfand’s theorem

3.2. Functional calculus, spectral mapping theorem

3.3. Spectral decomposition/separation theorem

3.4. Isolated eigenvalues

3.4.1.1. Algebraic multiplicity

3.4.1.2. Laurent expansion of the resolvent operator near isolated eigenvalue

3.4.1.3. Stability of a finite set of isolated eigenvalues under small operator perturbation

3.5. Spectrum of the adjoint operator

3.6. The case of unbounded but closed operators

4. Compact operators and Fredholm operators

4.1. Riesz-Schauder theory

4.2. Hilbert-Schmidt theorem, min-max characterization of eigenvalues

4.3. Positive compact operators: Krein-Rutman theorem (for the special case of Banach space C(Q), where Q is a compact Hausdorff space)

4.4. Fredholm operators

4.4.1.1. Characterization of Fredholm operators, pseudoinverse

4.4.1.2. Fredholm index: index of product of two operators, constancy of index under small or compact perturbation

4.4.1.3. Essential spectrum of a bounded operator, and its constancy under compact perturbation

4.5. Applications

4.5.1. Second order elliptic operators

4.5.2. Non-local diffusion operators

4.5.3. Toeplitz operators

5. Spectral theory of bounded symmetric, normal and unitary operators

5.1. The spectrum of symmetric operators

5.2. Functional calculus for symmetric operators

5.3. Spectral resolution of symmetric operators

5.4. Absolutely continuous, singular, and point spectra

5.5. The spectral representation of symmetric operators

5.6. Spectral resolution of normal operators

5.7. Spectral resolution of unitary operators

5.8. Examples

6. Unbounded self-adjoint operators

6.1. Spectral resolution via Cayley transform

6.2. The extension of unbounded symmetric operators, deficiency indices

6.3. The Friedrichs extension

6.4. Examples

7. Semigroups of operators

7.1. Strongly continuous one-parameter semigroups

7.2. The generation of semigroups: Hille-Yosida theorem

7.3. Exponential decay of semigroups

7.4. Examples: semigroups defined by parabolic equations, and by nonlocal diffusion equations

13.


作业 40%+期中考试（闭卷）20%+期末考试(闭卷) 40%

Homework 40%+ Mid-term Exam (closed-book) 20%+Final Exam (closed-book) 40%

14.


1. 泛函分析讲义（上、下），张恭庆等编著

2. Functional Analysis, by Peter Lax.

3. Perturbation Theory for Linear Operators, by T. Kato.

课程大纲

COURSE SYLLABUS

1.



拓扑

Topology

2.


必修 Required

3.

授课院系


数学系

4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language

英文教材，中文授课

8.


夏志宏教授

9.



基础拓扑学，近世代数

10.


了解和掌握拓扑学的基础理论，基本方法，重要例子以及主要结果。培养学生在拓扑学领域初步的科研能力

11.


课堂授课+学生的课程project

12.


Chapter 1: The fundamental groups,

Chapter 2: Homology,

Chapter 3: Cohomology,

Chapter 4: Homotopy.

13.


20% Homework+ 30% Course Project +50% Final

14.


Algebraic Topology, by Allen Hatcher

Differential forms in algebraic topology, by Raoul Bott and Loring Tu.

课程大纲

COURSE SYLLABUS

1.



微分流形

Differentiable manifolds

2.


必修 Required

3.

授课院系


数学系

4．



所有院系

5.


48学时

6.


3学分

7.

授课语言

Teaching Language

中文/英文

8.


李勤

9.



数学分析，线性代数，拓扑学，基础微分几何

10.


微分流形是现代几何与拓扑学的基本研究对象。本课程的基本目标是让学生了解和掌握微分流形的基本定义和理论，知晓最基本和重要的例子，并能够将已经学到的分析，代数和拓扑知识应用在微分流形这一几何对象上。并为有志于在几何，拓扑及数学物理等方向从事研究的学生提供最基础的知识和准备。

11.


课堂教学+讨论

12.


1. 微分流形的定义，例子及流形之间的映射。

2. 切丛，余切丛，法向丛。

3. Sard's定理。　

4. 映射度

5. 微分形式及de Rham上同调

6. 李群的基础知识

13.


笔试+口试

14.


1. Foundations of differentiable manifolds and Lie groups (GTM 94). Frank W. Warner

2. Differential forms in algebraic topology (GTM 82) Raoul Bott and Loring W. Tu

课程大纲

COURSE SYLLABUS

1.



偏微分方程数值解

Numerical Solutions to Partial Differential Equations

2.


必修 Required

3.

授课院系


数学Mathematics

4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language

中英文Chinese and English

8.


李景治副教授

9.



微积分Calculus, 线性代数Linear Algebra, 常微分和偏微分方程Ordinary and Partial Differential Equations

10.


教授偏微分方程数值基本解法及其理论

Teach numerical methods for partial differential equations and the underlying theory.

11.


理论与编程并重，并辅以前沿课题应用

Teaching in both theory and programming, including applications to cutting edge problems

12.


1. 偏微分方程基本概念和分类：抛物，椭圆和双曲方程

Concepts and classification of partial differential equations: Parabolic, elliptic, and hyperbolic equations

2. 有限差分方法简介：网格剖分，差分格式的推导，截断误差，差分算子和多项式插值

Finite difference methods: Mesh, Derivation of finite difference, truncation error, finite difference operators, and polynomial interpolation

3. 一维抛物型方程的显示差分格式：收敛性和Fourier稳定性分析

Explicit schemes for one-dimensional parabolic equations: Convergence and Fourier analysis for stability

4. 一维抛物型方程的隐示差分格式：Thomas算法，加权法和最大值原理

Implicit schemes for one-dimensional parabolic equations: Thomas algorithm, weighted method, and maximum principle

5. 一维抛物型方程：多层时间格式，一般性边界条件处理，极坐标处理

One-dimensional parabolic equations: multilevel scheme, general boundary conditions and polar coordinates

6. 相容性，收敛性和稳定性

Consistency, convergence and stability

7. 二维和三维抛物型方程：交错方向(ADI)格式和LOD格式

Two and three dimensional parabolic equations: Alternating Direction Implicit scheme and Locally One-Dimensional (LOD) scheme

8. 一维双曲型方程：特征线法，CFL条件和迎风格式

One dimensional hyperbolic equations: Method of characteristics, CFL condition, and upwind scheme

9. 一维双曲型方程：Lax-Wendroff格式，盒式格式，蛙跳格式，有限体积法及守恒律

One dimensional hyperbolic equations: Lax-Wendroff scheme, box scheme, leapfrog scheme, finite volume method and conservation law

10. 对流扩散方程：中心显示格式，迎风格式，隐式格式，及特征差分格式

Convection diffusion equations: central explicit scheme, upwind scheme, implicit scheme, and characteristic difference scheme

11. 椭圆型方程：五点和九点格式，收敛性和稳定性分析，高斯消去法，和快速Poisson解法器

Elliptic equations: five-point and nine-point scheme, convergence and stability analysis, Gaussian elimination, and fast Poisson solver

12. 线性方程组的迭代解法：Jacobi迭代，Gauss-Seidel迭代，松弛(SOR)迭代

Iterative methods for linear systems of equations: Jacobi, Gauss-Seidel, and SOR iterations

13. 再论相容性，收敛性和稳定性：截断误差与逼近阶，Lax等价定理，von Neumann条件

Consistency, convergence and stability revisit: truncation error and approximation orders, Lax equivalence theorem, von Neumann condition

14. 其他专题I--更多迭代方法：Newton迭代，最速下降法，共轭梯度法

Other topics I—More iteration methods: Newton iteration, steepest descent, conjugate gradient method

15. 其他专题II--稳定性讨论：绝对稳定性和0-稳定性

Other topics II—Stability revisit: Absolute stability and zero-stability

16. 其他专题III—函数空间方法简介：配值和谱方法, 变分和有限元

Other topics III—Function space methods: Collocation and spectral methods, variations and finite element methods，variations and finite element methods

17. 其他专题V—快速Fourier变换和快速Poisson解法器

Other topics IV—FFT and Fast Poisson solvers

18. 其他专题IV—不可压缩Navier-Stokes equations, 投影法，

Other topics V—Incompressible Navier-Stokes equations, Projection methods

13.


作业（30%）+报告（20%）+期末考试编程（50%）

Assignment (30%) + Presentation (20%) + Final Programming Exam (50%)

14.


参考教材Textbook：

偏微分方程数值解Numerical Solution of Partial Differential Equations: An Introduction, by K. W. Morton, D. F. Mayers, 人民邮电出版社, 2006.

偏微分方程的数值方法Numerical Partial Differential Equations: Finite Difference Methods, by J. W. Thomas, 世界图书出版社, 2005.

偏微分方程数值解讲义，李治平编著，北京大学出版社, 2010.

其他参考资料Supplementary Readings：

偏微分方程数值解法，陆金甫，关治编著，清华大学出版社, 2004.

Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, by Randall J. LeVeque, SIAM, 2007.

Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, by L. N. Trefethen, Cornell University, 1996.

R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser-Verlag, 1990.

MATLAB Tutorial, to accompany “Partial Differential Equations: Analytical and Numerical Methods”, 2nd edition by Mark S. Gockenbach, SIAM, 2010.

课程大纲

COURSE SYLLABUS

1.



应用数学方法：扰动理论和变分法

Methods of Applied Math: Perturbation Theory and Variational Approach

2.


必修 Required

3.

授课院系


数学系

4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language

中文讲授，英文板书

Chinese speaking, English writing

8.


张振

9.



常微分方程，偏微分方程，实分析

Ordinary and Partial Differential Equations, Numerical Analysis

10.


掌握计算和应用数学模型建立和分析的基本方法

Master the basic methods of modelling and analysis in computational and applied mathematics

11.


专题性质授课，并辅以前沿课题应用

Teaching in topics, and application to cutting edge problems

12.


Week 1: Introduction to asymptotic approximations

Week 2: Matched asymptotic expansion

Week 3: Interior layers, application of boundary layer theory to PDEs

Week 4: Multiple scale expansion and its applications

Week 5: The method of multiple scale expansion, the WKB method

Week 6: The WKB method, inhomogeneous linear equations, turning points

Week 7: Application of WKB to wave equations; The homogenization method

Week 8: The homogenization method for problems in multiple dimentions; applicaiton to porous medium flow

Week 9: Asymptotic expansion of integrals, Laplace's method

Week 10: Asymptotic expansion of integrals, method of stationary phase, method of steepest descents

Week 11: Variational methods, the first variation

Week 12: Isoperimetric problems, holonomic and nonholonomic constraints

Week 13: Free boundary problems and its applications

Week 14: Direct method, gamma-convergence

13.


平时作业（30%）+报告（20%）+闭卷考试（50%）

Assignment (30%) + Presentation (20%) + Closed-book Written Exam (50%)

14.


1. M. H. Holmes, Introduction to perturbation methods, Springer-Verlag, 1995.

2. C. M. Bender and S. A. Orszag, Advanced mathematical methods for scientists and engineers, Springer, 1999.

3. Bruce van Brunt, The Calculus of Variations, Springer-Verlag, 2004.

4. M. Giaquinta and S. Hildebrandt, Calculus of Variations, Vol. I and II, Springer, 1996.

课程大纲

COURSE SYLLABUS

1.



高等统计学

Advanced satistics

2.


必修 Required

3.

授课院系


数学系Department of Mathematics

4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language

英语为主，辅以中文解释

English with Detailed Explanations in Chinese

8.


田国梁教授

9.



Calculus, Matrix Algebra, Probability Theory, Mathematical Statistics

10.


通过此课程的学习，能够让学生掌握高等统计学的基本概念和基本理论,为进入统计领域的研究打基础。

This course aims to enable students to master the basic concept and theory of Advanced Statistics and to lay a foundation for the research in statistics.

11.


将采用传统方式教授此课(版书，课堂讨论，作业，课外答疑，闭卷考试)。强调抽象理论和具体应用的结合。

The course will be taught in the standard way (“chalk and board”, in-class discussion, homework, office hours, closed-book exams). The course is a balanced mix of abstract theories and applications.

12.


Chapter 1: Background

1.1 Probability Theory Background

1.2 Mathematical Statistics Background

Theory of Point Estimation

Chapter 2 UMVU Estimators

Chapter 3 Average Risk Optimality

3.1 First Examples

3.2 Single-Prior Bayes

3.3 Hierarchical Bayes

3.4 Empirical Bayes

3.5 Risk Comparisons

Chapter 4: Minimaxity and Admissibility

4.1 Minimax Estimation

4.2 Admissibility and Minimaxity in Exponential Families

4.3 Simultaneous Estimation

Chapter 5: Asymptotic Optimality

5.1 Performance Evaluations in Large Samples

5.2 Asymptotic Efficiency

5.3 Efficient Likelihood Estimation

Testing Statistical Hypotheses(Small- Sample Theory)

Chapter 6 Uniformly Most Powerful Tests

6.1 Stating The Problem

6.2 The Neyman–Pearson Fundamental Lemma

6.3 p-values

6.4 Distributions with Monotone Likelihood Ratio

Chapter7: Small- Sample Theory: The Minimax Principle

7.1 Tests with Guaranteed Power

7.2 Examples

Large-Sample Theory

Chapter 8： Convergence in Probability and in Law

8.1 Convergence in Probability

8.2 Applications

8.3 Convergence in Law

8.4 The Central Limit Theorem

8.5 Taylor’s Theorem and the Delta Method

8.6 Uniform Convergence

13.


作业(20%) + 期中考试(30%) + 期末闭卷考试(50%)

Assignment (20%) + Mid-term Test (30%) + Closed-book Written Examination (50%)

14.


[1]Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation (2nd Edition). Springer Texts in Statistics, Springer-Verlag, New York. [Each student will be provided an e-book of this monograph]

[2]Lehmann, E. L. (1999). Elements of Large-Sample Theory. Springer Texts in Statistics, Springer-Verlag, New York. [Each student will be provided an e-book of this monograph]

[3]Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses. Springer Texts in Statistics,

Springer-Verlag, New York.

课程大纲

COURSE SYLLABUS

1.



教学实习

Teaching Practice

2.


必修 Required

3.

授课院系


数学系

4．



数学系

5.


32学时

6.


2

7.

授课语言

Teaching Language

中文/英文

8.


教师

9.



数学系硕士与博士生

10.


通过教学实习，培养研究生教学能力和积累教学经验，为研究生以后在科研和教育领域的职业生涯做好准备。

11.


先由资深教师对研究生进行教学培训，之后让研究生带习题课，期间资深教师通过听课帮助研究生不断改进教学技术。

12.


1. 由资深教师对新入学的研究生进行教学培训；

2. 每学期给研究生分配习题课和改作业的工作；

3. 资深教师到研究生所带的习题课听课，并将其授课优点和需改正之处反馈给研究生，促进其在教学方面不断进步。

13.


资深教师评语+学生评语

14.


无

课程大纲

COURSE SYLLABUS

1.



辛拓扑与Hamilton动力系统

Symplectic Topology and Hamiltonian Dynamical Systems

2.


选修

3.

授课院系


数学系

4．



所有院系

5.


48学时

6.


3

7.

授课语言

Teaching Language

中文/英文

8.


夏志宏教授

9.



动力系统引论，微分拓扑，高等微分几何

10.


了解和掌握当代Hamilton动力系统理论的基本理论和方法

掌握辛拓扑基本理论以及保持辛结构的微分动力系统的特性

培养学生在Hamilton动力系统及相关的几何拓扑领域的科研能力

11.


课堂教学+讨论

12.


1. 分析力学的基本原理， 4学时

2. 流形上的Hamilton及Lagrange动力学，6学时

3. 微分形式与辛形式，8学时

4. 辛流形的拓扑学，9学时

5. 可积Hamilton动力系统， 6学时

6. 近可积系统的稳定性，6学时

7. 正定Langrange系统的Mather理论， 9学时

13.


笔试+口试

14.


1. Mathematical Methods of Classical Mechanics，V. I. Arnold．

2. Introduction to Symplectic Topology D. McDuff and D. Salamon．

3. Symplectic Invariants and Hamiltonian Dynamics，H. Hofer and E. Zehnder.

课程大纲

COURSE SYLLABUS

1.



光滑遍历论

Introduction to smooth ergodic theory

2.


选修

3.

授课院系


数学系

4．



所有院系

5.


48学时

6.


3

7.

授课语言

Teaching Language

中文、英文

8.


夏志宏教授

9.



动力系统引论，实分析,测度论

10.


掌握当代微分动力系统及遍历理论的重要的理论，方法和主要结果

掌握光滑遍历理论，非一致双曲微分动力系统理论的主要思想方法，并对当前学科前沿问题有所了解

培养学生在动力系统领域较深厚的科研能力

11.


以课堂教学为主，并辅以课堂讨论。

12.


1. 遍历理论， 6学时

2. Lyapunov指数及非一致双曲动力系统的Pesin理论，12学时

3. 熵论，6学时

4. 部分双曲系统理论，9学时

5. 光滑动力系统的遍历理论， 12学时

6. 重要例子，3学时

13.


笔试+口试

14.


1. Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, M. Pollicott.

2. Lyapunov Exponent and Smooth Ergodic Theory, L. Barreira and Y. Pesin.

3. An Introduction to Ergodic Theory, P. Walters.

4. Ergodic Theory and Differentiable Dynamics, R. Mane.

课程大纲

COURSE SYLLABUS

1.



动力系统引论

Introduction to Dynamical Systems

2.


选修，微分方程与动力系统方向必修

3.

授课院系


数学系

4．



所有院系

5.


48学时

6.


3

7.

授课语言

Teaching Language

中文、英文

8.


夏志宏教授

9.



数学分析，常微分方程，实变函数，测度论

10.


了解和掌握动力系统理论的基础理论，基本方法，重要模型以及主要结果

培养学生在动力系统及常微分方程领域初步的科研能力

11.


以课堂教学为主。

12.


1. 动力系统的基本概念， 6学时

2. 一维动力系统，6学时

3. 双曲不动点，9学时

4. Smale马蹄与Anosov环面自同构，9学时

5. 双曲不变集、结构稳定性， 9学时

6. 遍历理论基础，9学时

13.


课堂考试+口试

14.


1. 微分动力系统原理，文兰著。

2. Dynamical systems. Stability, symbolic dynamics, and chaos，by C. Robinson.

3. Introduction to the Modern Theory of Dynamical Systems，by A. Katok and B. Hasselblatt.

4. Introduction to Dynamical Systems by M. Brin and G. Stuck.

课程大纲

COURSE SYLLABUS

1.



群表示论　

Representation theory of groups

2.


选修

3.

授课院系


数学系

4．



数学系，计算机系，物理系

5.


48

6.


3

7.

授课语言

Teaching Language


8.


李才恒教授

9.



近世代数，线性代数

10.


了解和掌握群表示的基础理论，基本方法，重要例子以及主要结果

培养学生在将来的学习和研究中使用群表示理论的能力

11.


讲授+讨论

12.


1. Groups, group actions, group representations

2. FG-modules, group algberas, FG-homomorphisms

3. Maschke’s Theorem

4. Schur’s Lemma and applications

5. Irreducible modules and group algebras

6. Character theory

7. Normal subgroups and lifted characters

8. Tensor products

9. Induced modules and characters

10. Representations on real number fields

11. Representations on finite fields

12. Important examples and applications.

13.


笔试，作业

14.


Representations and Characters of Groups, by Gordon James and Martin Liebeck

Lecture Notes (李才恒编写)

Linear Representations of Finite Groups, by Jean-Pierre Serre.

课程大纲

COURSE SYLLABUS

1.



李群与李代数　

Lie groups and Lie algebras

2.


选修

3.

授课院系


数学系

4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language


8.


教师

9.



微分流形，拓扑

10.


了解和掌握李群和李代数的基础理论，基本方法，重要模型以及主要结果

培养学生在李理论领域初步的科研能力

11.


讲授，讨论

12.


Chapter 1, Lie groups and Lie algebras,

Chapter 2, Elementary representation theory,

Chapter 3, Representative functions,

Chapter 4, The Maximal torus of a compact Lie group,

Chapter 5, Root systems,

Chapter 6, Irreducible characters and weights.

13.


笔试，作业

14.


Representations of compact Lie groups, by Theodor Brocker and Tammo tom Dieck.

课程大纲

COURSE SYLLABUS

1.



组合数学

Combinatorics

2.


选修

3.

授课院系


数学系

4．



数学系，物理系，计算机系

5.


48

6.


3

7.

授课语言

Teaching Language


8.


李才恒教授

9.



近世代数，线性代数

10.


组合数学是一门由众多分支组成的学科，包括组合计数和组合机构，比如图论，有限几何，编码和密码学，组合设计。本课程将根据学生和教师的兴趣选择侧重点。其目标是让学生了解和掌握组合数学的基础理论，基本方法，重要例子以及主要结果培养学生在组合数学领域初步的科研能力。

11.


讲授，讨论

12.


Combinatoial Counting and Enumeration will cover fundamental methods, such as, counting by two ways, recurrence relations, and generating functions, and applications to typical objects like distributions and partitions.

Graph Theory will cover most fundamental topics such as Euler cycles, Hamiltonian cycle, graph minors, graph coloring, graph factoriszations, Cayley graphs, symmetrical graphs, self-complementary graphs and Ramsey numbers.

Finite Geometry will cover important objects, such as Linear Spaces, Projective Spaces, Affine Spaces, Polar Spaces, Gneralized Quadrangles, and group actions on geometries.

Coding Theory will cover basic theory for encoding and decoding, linear codes, Hamming codes, cyclic linear codes, and BCH codes.

13.


笔试，作业

14.


Enumerative Combinatorics, by Richard Stanley

Graph Theory, by Richard Diestel

Algebraic Coding Theory, by Elwyn Berlekamp

Combinatorics of Finite Geometries, by Lynn Batten

Combinatorics, by N. Ya. Vilenkin.

Lecture Notes (由授课教师编写)

课程大纲

COURSE SYLLABUS

1.



交换代数　

Commutative algebra

2.


选修

3.

授课院系


数学系

4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language


8.


教师

9.



近世代数

10.


了解和掌握交换代数基础理论，基本方法，培养学生在交换代数领域初步的科研能力，并为代数几何的深入学习和研究打下基础。

11.


讲授，讨论

12.


Chapter 1, Rings, ideals and modules,

Chapter 2, Rings and modules of fractions,

Chapter 3, Primary decompositions,

Chapter 4, Integral dependence and valuations,

Chapter 5, Chain conditions, noetherian and artin rings,

Chapter 6, Discrete valuation rings and Dedkind domains,

Chapter 7, Completions.

13.


考试，作业

14.


Introduction to Commutative Algebra, by Michael Atiyah and I.G. Macdonald.

课程大纲

COURSE SYLLABUS

1.



微分拓扑

Differential topology

2.


选修

3.

授课院系


数学系

4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language


8.


教师

9.



基础拓扑学，微分流形

10.


了解和掌握微分拓扑的基础理论，基本方法，重要例子以及主要结果

培养学生在微分拓扑领域初步的科研能力

11.


课堂授课+学生课程project

12.


Chapter 1: Manifolds and maps,

Chapter 2: Function spaces,

Chapter 3: Transversality,

Chapter 4: Vector bundles and tubular neighborhoods,

Chapter 5: Degrees and Euler characteristics,

Chapter 6: Morse theory,

Chapter 7: Cobordism,

Chapter 8: Isotopy.

13.


20% Homework + 30% Course Project + 50% Final

14.


Differential topology, by Morris W. Hirsch

课程大纲

COURSE SYLLABUS

1.



代数几何　

Algebraic Geometry

2.


选修

3.

授课院系


数学系

4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language


8.


教师

9.



微分几何，拓扑

10.


了解和掌握代数几何的基础理论，基本方法，重要例子以及主要结果

培养学生在代数几何领域初步的科研能力

11.


课堂授课+学生的课程project

12.


Chapter 1, Varieties,

Chapter 2, Schemes,

Chapter 3, Cohomology and its applications,

Chapter 4, Curves,

Chapter 5, Surfaces.

13.


20% Homework+ 30% Course Project +50% Final

14.


Algebraic Geometry, by Robin Hartshorne

Algebraic Geometry a first course, by Joe Harris

课程大纲

COURSE SYLLABUS

1.



复几何

Complex Geometry

2.


选修

3.

授课院系


数学系

4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language


8.


教师

9.



微分几何，基础拓扑学，复分析。

10.


了解和掌握复几何的基础理论，基本方法，重要例子以及主要结果，培养学生在复几何领域初步的科研能力

11.


课堂讲授加学生的课程project

12.


Chapter 1: Holomorphic functions of several variables, differential forms

Chapter 2: Complex manifolds,

Chapter 3: Kahler manifolds,

Chapter 4: Vector bundles, sheaves and its cohomology,

Chapter 5: Applications of cohomology

Chpater 6: Deformation of complex structures.

13.


20% Homework+ 30% course Project+ 50% Final

14.


Principles of algebraic geometry, by Phillip Griffiths and Joe Harris,

Complex geometry, an introduction, by Daniel Huybrechts

课程大纲

COURSE SYLLABUS

1.



辛几何

Symplectic geometry

2.


选修

3.

授课院系


数学系

4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language


8.


教师

9.



基础拓扑学，微分流形

10.


了解和掌握辛几何的基础理论，基本方法，重要例子以及主要结果

培养学生在辛几何领域初步的科研能力

11.


课堂授课+ 学生课程project

12.


Chapter 1: Symplectic manifolds,

Chapter 2: Symplectomorphisms,

Chapter 3: Contact manifolds,

Chapter 4: Compatible almost complex structures,

Chapter 5: Kahler manifolds.

13.


20% Homework + 30% Course Project + 50% Final

14.


Lectures on symplectic geometry, by Ana Cannas da Silva

课程大纲

COURSE SYLLABUS

1.



非线性泛函分析

Nonlinear Functional Analysis

2.


选修 elective

3.

授课院系


数学系


4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language



8.


苏琳琳，助理教授; 王学锋，讲座教授

Linlin Su, Assistant Professor；Xuefeng Wang, Chaired Professor

9.



偏微分方程(上)。 (无需学过偏微分方程(下)，但至少需与此课同时学。)

PDE I. (PDE II should be taken prior or at the same time as this course.)

10.


介绍非线性偏微分方程领域中常用的几种泛函分析框架，并展示如何把抽象理论应用到具体的偏微方程上。

The course is to introduce several standard functional analysis frameworks for nonlinear partial differential equations, and to show how to apply these abstract theories to concrete PDEs.

11.


将采用传统方式教授此课(版书，课堂讨论，作业，课外答疑，闭卷考试)。本课讲授者长期从事偏微方程的科研，知道什么是科研中最需要的基础知识。教师将高效地利用有限的课时讲授最重要的内容, 并在适当的时候以授课和作业的形式介绍科研前沿问题。

The course will be taught in the standard way (“chalk and board”, in-class discussion, homework, office hours, close-book exams). The instructors are experienced researchers in the field of partial differential equations, hence know what are the most needed in real research. The instructors will be highly efficient in using the class hours to teach the most important material, and to expose the students to research problems via lectures and homework.

12.


· Inverse and implicit function theorems in Banach spaces.

· Crandall-Rabinowitz local bifurcation theorem; applications to nonlinear elliptic PDE.

· Fixed-point theorems; applications to nonlinear PDE.

· Brouwer degree and Brouwer fixed-point theorem, and applications such as Borsuk-Ulam theorem.

· Leray-Schauder degree.

· Rabinowitz global bifurcation theorem; application to nonlinear PDE.

· Krein-Rutman theory.

· Variational method: the direct minimization method, with or without constraints; applications to PDE such as ground states of Schrodinger equations.

· Variational method: critical point theory; applications to PDE.

13.


作业 60%+期末考试(闭卷) 40%

Homework 60%+ Final Exam (close-book) 40%

14.


Methods in nonlinear analysis, by Kung-Ching Chang

课程大纲

COURSE SYLLABUS

1.



偏微分方程 (上、下)

Partial Differential Equations I & II

2.


选修 elective

3.

授课院系


数学系


4．



所有院系

5.


96 (两学期 two semesters)

6.


3+3

7.

授课语言

Teaching Language



8.


苏琳琳，助理教授; 王学锋，讲座教授

Linlin Su, Assistant Professor；Xuefeng Wang, Chaired Professor

9.



线性代数, 高等微积分, 实分析, 泛函分析，复分析

Linear Algebra, Advanced Calculus, Real Analysis, Functional Analysis and Complex Analysis

10.


学生通过本课程的学习将掌握偏微分方程中的基本概念、基本理论以及基本方法。此课将为学生们以后阅读参考文献和开展相关的科研工作打好基础。

The students will learn fundamental concepts, theories, and methods in partial differential equation and be able to apply them to solve problems. Students will be well prepared to read more advanced literature and carry out related research in the future.

11.


将采用传统方式教授此课(版书，课堂讨论，作业，课外答疑，闭卷考试)。本课讲授者长期从事偏微方程的科研，知道什么是科研中最需要的基础知识。教师将高效地利用有限的课时讲授最重要的内容, 并在适当的时候以授课和作业的形式介绍科研前沿问题。

The course will be taught in the standard way (“chalk and board”, in-class discussion, homework, office hours, close-book exams). The instructors are experienced researchers in the field of partial differential equations, hence know what are the most needed in real research. The instructors will be highly efficient in using the class hours to teach the most important material, and to expose the students to research problems via lectures and homework.

12.


· Classical weak and strong maximum principles for 2nd order elliptic and parabolic equations, Hopf boundary point lemma, and their applications.

· Sobolev spaces, weak derivatives, approximation, density theorem, Sobolev inequalities, Kondrachov compact imbedding.

· L² theory for second order elliptic equations, existence via Lax-Milgram Theorem, Fredholm alternative, L² estimates, Harnack inequality, eigenvalue problem for symmetric and non-symmetric, second order elliptic operators.

· L² theory for second order parabolic and hyperbolic equations, existence via Galerkin method, uniqueness and regularity via energy method.

· Semigroup theory applied to second order parabolic and hyperbolic equations.

· A brief introduction to elliptic and parabolic regularity theories, the Lp and Schauder estimates.

· Nonlinear elliptic equations, variational method—the direct minimization method, method of upper and lower solutions, fixed point method.

· Nonlinear parabolic equations, global existence, stability of steady states, traveling wave solutions.

· Conservation laws, Rankine-Hugoniot jump condition, uniqueness issue, vanishing viscosity method, entropy condition, Riemann problem for Burger's equation, p-systems.

13.


作业 60%+ 期末考试40%

Homework 60%+ Final Exam 40%

14.


1. Partial Differential Equations, by Lawrence C. Evans, American Mathematical Society.

2. Elliptic and Parabolic Equations, by Wu Zhuoqun,Yin Jinxue and Wang Chunpeng, World Scientific Publishing Co.

3. Elliptic Partial Differential Equations of second Order, by David Gilbarg and Neil S.Trudinger, Springer.

4. Linear and Quasilinear Equations of Parabolic Type, by Q. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, American Mathematical Society.

5. Partial Differential Equations, 2nd edition, by R. McOwen, Prentice-Hall

课程大纲

COURSE SYLLABUS

1.



现代概率论

Advanced Probability

2.


选修

3.

授课院系


数学系


4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language

英语，辅以部分中文解释

English with some explanations in Chinese

8.


Chen Anyue 陈安岳

9.



高等数学（上、下），线性代数（上、下），概率论（二年级秋季），实分析，泛函分析，随机过程基础，测度论与积分，Calculus (I & II)，Linear Algebra (I&II)， Introduction to Probability (2nd year, fall semester)，Real Analysis，Functional Analysis， Introduction of Stochastic Processes， Measure Theory & Integration

10.


After learning this course, students should be able

1.to deeply understand and master the basic concepts and conclusions of modern probability theory, not only to remember these basic concepts and the basic probability laws including conditions and conclusions, but also deeply to understand the basic principles and ideas of modern probability;

2．to fully master the four basic convergence theorems ( Monotone Convergence Theorem, Fatou Lemma, Dominated Convergence Theorem, and Bounded Convergence Theorems ) and be able to apply them in many important topics and different problems;

3．to clearly understand the probability meaning, difference, and relationships of several kind of convergence concepts ( almost everywhere convergence; convergence in measure/probability; Convergence in Lp Norm; Weak Convergence) and be able to apply them in different problems;

4．to fully master the very important concepts of conditional expectations and conditional probabilities and to improve the ability of solving practical problems by applying the basic probability methods of “ conditioning” .

5. to clearly understand and master the basic concepts regarding martingales including the existence, uniqueness, properties and applications of martingales, super- and sub-martingales and be able to apply the important martingale method in the study of modern theory of stochastic processes, stochastic analysis and financial mathematics.

11.


1. Pay attention to the newly and recently obtained conclusions. In the teaching process, I’ll combine the important classic results with the newly obtained results together to ensure the advance of this course.

2. In the teaching, I’ll pay much attention to the important and difficult concepts and conclusions. In particular, the concepts of conditional expectations and martingales are very abstract and thus are quite difficult for students to master. To help students master these important yet difficult concepts, I’ll give extremely detailed explanations and provide many examples so that students could understand the exact meaning and thus fully master the related conclusions and then could apply them in many different problems.

3. Emphasize the applications. In the teaching, emphases will be given to the application aspect. For some difficult theoretic conclusions, the strict proof could be skimmed over for the time being, yet I’ll teach students to understand the meaning of these results and encourage students to apply them first. For some important methods and techniques, fully applications and examples will be provided.

12.


Chapter 1: Measure and Probability (6h): -algebra of sets (definition, basic properties), -algebra generated by a collection of sets, -systems & d-systems, Monotone class theorems (set form), Dynkin lemma, Measure & Probability, Probability space, Monotone class theorems (function form)

Chapter 2. Independence, Expectation and Convergence (12h): The -system Lemma, Borel-Cantelli Lemma, Convergence Theorems, Jensen’s Inequality for Convex Functions, The Schwarz Inequality, Orthogonal Projection, Holder from Jensen; Convergence in Probability, Weak Convergence, Convergence in Distributions, Characteristic Functions.

Chapter 3. Conditional probability and conditional expectation (8h): Definition of conditional expectation and conditional probability, Existence & Uniqueness, Properties of conditional expectation and conditional probability, Tower Property, Some Important Inequalities.

Chapter 4. Martingales (12h): Definition of martingales, Properties of martingales, Super-martingales, Sub-martingales, Examples, Convergence of martingales, Stopping times, Optional Sampling Theorem.

Chapter 5. Super-martingales and Sub-martingales (8h): Definitions and Properties of Super-martingales and Sub-martingales, Examples, Doob Decomposition. Convergence of martingales, Stopping times, Optional Sampling Theorem.

Chapter 6. Martingale Inequality and Martingale Convergence Theorems (8h): Uniform Integrability; UI Martingales; Martingale Convergence Theorems; Backwards Martingale Convergence Theorems; Strong Law of Large Numbers; Martingale Central Limit Theorem.

13.


1. Assignments: 20%

2. Mid-term test: 30%

3. Final Exam (2h): 50%

14.


Textbook: Jean Jacod & Philip Protter，《Probability Essentials》，Springer-Verlag, Berlin Heidelberg.

Supplementary Readings:

1. David Williams, ，《Probability with Martingales》，Cambridge University Press, Cambridge, 1991.

2. 严士健，王隽骧，刘秀芳，《概率论基础》，科学出版社

课程大纲

COURSE SYLLABUS

1.



连续时间马氏链

Continuous Time Markov Chains & Their Applications

2.


Optional 选修

3.

授课院系


数学系


4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language



8.


Chen Anyue 陈安岳

9.



高等数学（上、下），线性代数（上、下），概率论（二年级秋季），实分析，泛函分析，随机过程基础， Calculus (I & II)，Linear Algebra (I&II)， Introduction to Probability (2nd year, fall semester)，Real Analysis，Functional Analysis， Introduction of Stochastic Processes

10.


学习本课程后，学生应能够掌握连续时间马氏链的基本理论，能够对现实生活中的某些随机问题建立相应的马氏链模型。熟练掌握在Q矩阵给定的条件下，解决马氏链的存在性、唯一性以及过程的构造等问题的一般性研究方法。能够使用本课程中的基本方法来分析Q过程的概率性质，包括常返性、遍历性、衰减性以及灭绝概率等。

After learning this course, the students should master the fundamental concepts and conclusions of continuous time Markov chains as well as the basic methods and techniques in analyzing such kind of stochastic processes. Students should be able, after studying and practicing, to apply these conclusions and methods in practical problems. In particular, they should be able to solve the existence, uniqueness and construction problems when the infinitesimal generator, the so-called q-matrix Q is given. They should also be able to analyzing the properties of the related Q-processes including to reveal the extinction, delay, and/or, ergodic properties when certain conditions are satisfied.

11.


教学过程中，会特别突出理论与实践相结合，注重由实际问题推导出相应的随机数学模型，给学生以清晰的背景解释。着重分析几类重要的连续时间马氏链模型，如生灭过程、分枝过程及排队论等，增强学生对随机过程的认识。课程讲授过程中，教会学生能综合使用概率和分析的方法来处理问题，使学生既能深刻理解概率直观含义，又能给出分析上的严格证明。

In the teaching process, the emphases will be put to the application aspect, i.e. the main approach will be application-oriented. The very useful sub-classes of Continuous Time Markov Chains including birth-death processes, branching processes and the queuing theory will be detailed analyzed. For some difficult points, the strict proofs could be postponed for a while but must be given later when the students have got enough experience and knowledge.

Also, we shall combine the two approaches, the analytic and probabilistic, closely so that, on the one hand, students should be able to get the conclusions strictly by using analytic approach and , on the other hand, they should clearly aware of the intuitive probabilistic ideas.

12.


Chapter 1 Introduction (4h): Stochastic Processes; Discrete-Markov Chain, Continuous-time Markov Chains.

Chapter 2 Transition Probability and Rate-matrix (6h): Properties of Transition Probabilities, Chaperman-Kolmogorov Equations, Kolmogorov Backward Equations, Kolmogorov Forward Equations; Resolvents; Rate-matrix, q-processes, Examples.

Chapter 3 Existence and Uniqueness of q-processes (10h): Construction of Feller Minimal Processes, Properties of Feller Minimal Processes, Uniqueness I (Conservative case), Uniqueness II (Non-conservative case), Reuter Lemma, Examples, Applications.

Chapter 4 Classification of States (10h): Communications between states; Recurrence, Positive-Recurrence, Embedded Markov Chains; Jump-Chains, Ergodicity, Equilibrium Distributions, Examples.

Chapter 5 Decay Parameter and Invariant Measures (10h): Communications Sub-invariant Measures & Sub-invariant Functions, Invariant Measures & Invariant Functions, Decay Parameters, Quasi-Limit-Distributions, Applications.

Chapter 6 Birth-Death Processes (10h): Reversibility of Birth-Death Processes, Duality of Birth-Death Processes, Boundaries of Birth-Death Processes, Examples, Applications.

13.


1. Assignments: 20%

2. Mid-term test: 30%

3. Final Exam (2h): 50%

14.


Textbook: W. Anderson: Continuous-Time Markov Chains, Springer, Berlin 1991

课程大纲

COURSE SYLLABUS

1.



矩阵计算

Matrix Computations

2.


应用数学研究生课程 Course for graduated students, Applied Mathematics

3.

授课院系



4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language

中文， Chinese

8.


何炳生 He， Bingsheng

9.



The background required of the students is a good knowledge of linear algebra, numerical analysis and programming experience. Computing assignments are in Matlab so about a year's worth of programming experience in C or C++ is more than enough.

10.


Matrix is the key mathematical tool for describing the problems in science, engineering, economics, and industry. This course is for students interested in understanding or further developing stable and efficient algorithms for systems of linear equations, least squares problems, eigenvalue problems, singular value problems and some of their generalizations and applications. In this course, techniques for dense and sparse, structured problems, parallel techniques and direct and iterative methods will be covered. Students will come to appreciate many state-of –the –art algorithms or matrix methods, and will have the ability to quantify and analyze many practical applications and be far easier to deal with them by applying the matrix methods.

11.


By presenting the motivating ideas for each algorithm, we try to stimulate the students intuition and make the technical details easier to follow.

12.


1. Matrix Multiplication Problems 6 Hours

2. Matrix Analysis 4 Hours

3. General Linear Systems 5 Hours

4. Special Linear Systems 9 Hours

5. Orthogonalization and Least Squares 8 Hours

6. Parallel Matrix Computations 4 Hours

7. The Unsymmetric Eigenvalue Problem 9 Hours

8. The Symmetric Eigenvalue Problem 9 Hours

9. Lanczos Methods 4 Hours

10. Iterative Methods for Linear Systems 6 Hours

11. Functions of Matrices 3 Hours

12. Special Topics 5 Hours

13.


Exercise (20%), and Semester examination (80%)

14.


Textbook:

Gene H. Golub, Charles F. Van Loan, Matrix Computations, 3rd Edition, The John Hopkins University Press, Baltimore and London, 1996.

Supplementary Readings:

1. Gibert Strang, Introduction to Linear Algebra, 4th Edition, Wellesley-Cambridge and SIAM, 2009.

2. Carl D. Meyer ,Matrix Analysis and Applied Linear Algebra, SIAM.

3. David Watkins, Fundamentals of Matrix Computations, Wiley, 1991.

课程大纲

COURSE SYLLABUS

1.



数值优化

Numerical Optimization

2.



3.

授课院系



4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language

中文， Chinese

8.


何炳生 He, Bingsheng

9.



The only background required of the students is a good knowledge of advanced calculus and numerical linear algebra.

10.


This course is for students interested in solving optimization problems. Because of the wide (and growing) use of optimization in science, engineering, economics, and industry, it is essential for students and practitioners alike to develop an understanding of optimization algorithms. Knowledge of the capabilities and limitations of these algorithms leads to a better understanding of their impact on various applications, and points the way to future research on improving and extending optimization algorithms and software. The goal of this course is to give a comprehensive description of the most powerful, state-of-the-art, techniques for solving continuous optimization problems.

11.


By presenting the motivating ideas for each algorithm, we try to stimulate the students intuition and make the technical details easier to follow.

12.


1. Introduction 2 Hours

2. Fundamentals of Unconstrained Optimization 2 Hours

3. Line Search Methods 2 Hours

4. Trust-Region Methods 4 Hours

5. Conjugate Gradient Methods 4 Hours

6. Practical Newton Methods 4 Hours

7. Quasi-Newton Methods 6 Hours

8. Nonlinear Least-Squares Problems 4 Hours

9. Theory of Constrained Optimization 8 Hours

10. Linear Programming: The Simplex Method 8 Hours

11. Linear Programming: Interior-Point Methods 8 Hours

12. Fundamentals of Algorithms for Nonlinear Constrained Optimization 8 Hours

13. Penalty, Barrier, and Augmented Lagrangian Methods 6 Hours

14. Sequential Quadratic Programming 6 Hours

13.


Exercise (20%), and Semester examination (80%)

14.


R. Fletcher, Practical Methods of Optimization, John Wiley & Sons, New York, 1987.

J. Nocedal and Stephen J. Wright, Numerical Optimization, Springer, 1999

课程大纲

COURSE SYLLABUS

1.



凸优化和单调变分不等式的分裂收缩算法

Splitting contraction methods for convex optimization and monotone variational inequalities

2.



3.

授课院系



4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language

中文， Chinese

8.


何炳生 He，Bingsheng

9.



The only background required of the students is a good knowledge of advanced calculus and numerical linear algebra.

10.


Optimization problems arising from big data problem, dimension reduction, machine learning, image processing and etc, can be translated and/or relaxed to a large scale structured convex optimization. This course is devoted for the students interested in solving practical large optimization problems in the different areas of science and technology.

For solving the large scale problems, it is recognized that the first order algorithms is practical and relative effective. The alternating direction method of multipliers (ADMM) is a benchmark for solving a linearly constrained convex minimization model with a two-block separable objective function. In this course, we will introduce the new development of the ADMM-like methods, both in theoretical convergence, and the practical implementations and applications.

最优化理论与方法是运筹学与计算数学的交叉学科. 近 20 年来, 信号处理, 图像恢复, 机器学习等信息技术领域以及统计学、数据科学中涌现了大量的优化问题. 有效地求解这些问题, 是当今一些世界一流应用数学家关切的课题, 也是应用数学的一个新的研究热点.

数据科学中大规模计算问题的很大一部分可以归结为（或松弛成）一个可分离算子的凸优化问题. 由于问题规模大, 传统的优化求解方法往往难以凑效. 根据问题的结构特点, 设计简单易行的一阶分裂算法已渐成学界共识.

变分不等式是运筹学中许多问题的一种统一表述模式. 经济活动中的最优平衡问题、政策性调控问题, 都可以用变分不等式来描述. 最优化和变分不等式有着紧密的联系. 凸优化的一阶最优性条件就是一个单调变分不等式. 在变分不等式的框架下考虑凸优化的求解方法, 就像微积分中用导数求一元二次函数的极值, 常常会带来很大的方便.

本课程中介绍凸规划的分裂收缩算法, 始终追求简单统一的原则，都纳入统一的框架. 统一框架揭示方法之间的内在联系, 简化算法的收敛性证明, 又能对设计新的算法, 提高算法效率提供指导性帮助.

11.


By presenting the motivating ideas for each algorithm, we try to stimulate the student intuition and make the technical details easier to follow.

12.


1. Introduction for convex optimization and monotone variational inequality 4 Hours

2. Projection and contraction methods for monotone variational inequalities 8 Hours

3. Customized Proximal Point Algorithms and Relaxed PPA for linearly constrained optimization. 6 Hours

4. Alternating direction methods of multiplies for structured convex optimization 6 Hours

5. New developments of ADMM 8 Hours

6. Study of the Convergence rate of the splitting methods 8 Hours

7. ADMM-like methods for multi-blocks linearly constrained convex optimization 6 Hours

8. Group-wise ADMM for multi-blocks linearly constrained convex optimization 6 Hours

9. Splitting contraction in a unified framework. 6 Hours

13.


Programming (40%), and Semester examination (60%)

14.


自编教材，主要内容取自 (http://math.nnju.edu.cn/~hebma)上《凸优化和单调变分不等式的收缩算法》

参考书籍： Stephen Boyd and Lieven Vandenberghe， Convex Optimization

课程大纲

COURSE SYLLABUS

1.



随机分析

Stochastic Analysis

2.


选修

3.

授课院系


数学系

4．



所有院系

5.


48

6.


3

7.

授课语言

Teaching Language



8.


陈安岳 Chen Anyue 欧阳顺湘Ouyang Shunxiang

9.



概率论基础，测度论，泛函分析，微分方程，随机过程

10.


在概率论和随机过程论基础上，掌握随机分析的基础理论与方法，为进一步研究金融数学，金融工程等学科提供必要的随机分析基础。

11.


由具体的金融问题，给出传统分析学所面临的困境，进而导出引入随机分析的必要性。注重实例分析，增强学生对随机分析的理解。注重理论与实践相结合，深入分析随机积分的一般理论，通过实例给出随机积分在金融中运用。

12.


第一章：离散时间鞅（6课时）

第二章：布朗运动（8课时）

第三章：Ito 积分（16课时）

第四章：关于Levy过程的随机积分（12课时）

第五章：随机微分方程简介（3课�

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