分析 代数 几何/拓扑 概率/计算 大一暑假 数学分析原理 代数II 曲线和曲面的微分几何 大二暑假 实分析 交换代数 代数拓扑 概率论I

l  数学分析原理

Principles of Mathematical Analysis, Walter Rudin, 机械工业出版社

l  实分析

Real Analysis, Stein and Shakarchi, 世界图书出版公司

n  代数II

[Ar] M. Artin, Algebra (Second Edition), 机械工业出版社.

[Ar] 11 Rings, 12 Factoring, 14 Linear Algebra in a Ring, 9 Linear Groups, 10 Group Representations.

□ 主理想整区上的模论，可替代 [Ar] 14 Linear Algebra in a Ring，参考教材：

[Ja] N. Jacobson, Basic Algebra I, Chapter 3, Modules over a PID.

□ 二次数域，参考教材：

[Co] D.A. Cox, Primes of the Form x2+ny2.

[Mar] D. Marcus: Number Fields, Chapter 1 and 2, Springer-Verlag.

[Se1] J.P. Serre, A course in Arithmetic.

□ 表示论可选专题（依指导老师倾向选择）：

1. Young Table;

2. Classification of Representations of GL2(Fp).

[Feng2] 冯克勤, 章璞, 李尚志, 群与代数表示引论.

[Se2] J.P. Serre, Linear Representations of Finite Groups, Springer.

n  交换代数

[AM] Atiyah and MacDonald, Commutative Algebra.

[AM] 1 Rings and Ideals，第2Modules，第3Ring and Modules of Fractions，第4Primary Decomposition，第5Integral Dependence and Valuations，第6Chain Conditions，第7Noetherian Rings，第8Artin Rings，第9Discrete Valuation Rings and Dedekind Domains，第10Completions，第11 Dimension Theory.

u  曲线和曲面的微分几何

[C1] Differential Geometry of Curves and Surfaces, M. do Carmo, 机械工业出版社

[C1] 1 Curve 2 Regular surfaces 3 The geometry of the Gauss map 4 The intrinsic geometry of surfaces

u  代数拓扑

Ø  概率论I

1课程名称：ZFC set theory and elementary logic

On the set theory side, we will discuss Zermelo-Fraenkel-Choice (ZFC) set theory, answering the questions:

1. What exactly are the ZFC axioms?
2. Why must we be cautious when choosing axioms for set theory?
3. How do the ZFC axioms serve as a foundation for mathematics?

We will also prove Zorn’s lemma and the basic facts about cardinal arithmetic.

The remaining lectures will cover basic results in proof theory and model theory, the two subjects that directly analyze logical reasoning.  There are two complementary goals:

1. Write down a complete axiomatization for the complex numbers.
2. Show that there is no complete axiomatization for the integers (Goedel’s incompleteness theorem).

2课程名称：广义相对论