核 心 課 程

Real Analysis (高等實分析 I, II)
Course Description

 

This course aims to cover extensions of Lebesgue Theory in contemporary analysis and probability. The axiomatic method of exposition is chosen. The course is a graduate level course. It requires regular participation, the completion of homework and the mid-term/final exams.


 

 I. Fall, 2007

 

Chap I (Foundation of Real Analysis), Chap II (Functional Analysis), Chap IV Sect. 6 (Theory of Differentiations). Some material from Chap III (Fourier Analysis) 

 

II. Spring, 2008

 

Chap IV (Foundation of Probability Theory), Chap V (Functional-Stochastic Analysis). Some material from Chap III and Appendix (Spectral Analysis).

Textbook: P. Malliavin, Integration and Probability, GTM 157 Springer

References:

[1] F.C. Liu, Lecture notes

[2] J.L. Doob, Measure Theory, GTM 143 Springer

[3] P. Malliavin, Integration and Probability, GTM 157 Springer

[4] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC.

[5] M. Taylor, Measure Theory and Integration, Graduate Studies in Math, Vol. 76, AMS.

Instructors: 張清輝 (Fall 2007-Spring 2008)

 

Fall 2007

Fri. 15:30 ~ 17:30 Sat. 09:30 ~ 11:30

   

First meeting :

Fri. Sept. 21st, 2007

 

Spring 2008

Mon. 15:30~ 17:30 Wed. 10:10~12:10

   

First meeting :

Wed. Feb. 20th, 2008

Algebra (高等代數 I, II)
Course Description

Part 1:

Set and Group Theory

 

Zorn's lemma, cardinality, basic group theory, group action, Sylow's
theorem, abelian groups, symmetric groups, nilpotent ?and solvable
groups, normal series, simple groups.

Part 2:

Ring and Field Theory

 

Basic commutative ring theory, modules, modules over PID's,
tensor products, field extensions, separable extensions, splitting field,
Galois theory, finite fields, cyclotomic and cyclic extensions.

Part 3:

 Non-commutative Ring Theory

 

Simple, primitive rings and algebras, Schur's lemma,
density theorem, Artin-Wedderburn theorem, Jacobson's radical,
semi-simple rings and algebras, Noether-Skolem theorem, Bi-commutant
theorem.

Part 4:

Group Representation Theory

 

Representations, characters, Group algebras, orthogonal relations,
Frobenius reciprocity, induced representations, Burnside's theorem,
permutation representations, more examples, Brauer's theorem.

Part 5:

Central Simple Algebra

 

General theory and Brauer groups, maximal and splitting subfields,
Cross product, Galois cohomologies, Inflation, cyclic division
algebras, division algebras over local fields, Brauer invariants,
division over number fields.

Part 6:

Commutative Algebra

 

Integral extension, Noetherian rings and modules, primary
decomposition, Nakayama's lemma, Hilbert's Nullstellensatz.

References:

(1) Hungerford: Algebra, GTM 73.
(2) Pierce: Associative Algebra, GTM 88.
(3) Artin: Algebra, Prentice Hall
(4) Atiyah and MacDonald: Introduction to commutative algebra.

Prerequisite: one year undergraduate algebra

Instructor: 余家富

Time:

Friday

9:00 ~ 12:00

 

Saturday

14:00 ~ 16:00

 

First meeting : Sat. Sept. 29 2007 (Fall)

 

First meeting : Sat. Feb. 23 2008 (Spring)

主持單位: 中央研究院數學研究所

贊助單位: 慈澤教育基金會    中央研究院數學研究所

聯絡人: 焦源鳴 (02)2785-1211-341  ymchiao@math.sinica.edu.tw

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