# 核心課程

## Real Analysis (高等實分析 I, II) Unit of Credit: 3 (學分:3)

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, 2008

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, 2009

Chap I (Foundation of Real Analysis), Chap II (Functional Analysis), Chap IV Sect. 6 (Theory of Differentiations). Some material from Chap III (Fourier 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: 劉太平 (Fall 2009-Spring 2009)

Time:

 Friday 13:20 ~ 15:10 Saturday 10:20 ~ 12:10

First meeting:
Friday February 20th 2009 (Spring)
Friday September 19th 2008 (Fall)

Meeting location:
Spring Friday & Saturday Class - 舊數館103室，國立台灣大學

Fall Friday Class - 新生大樓201室，國立台灣大學
Fall Saturday Class - 舊數館103室，國立台灣大學

## Algebra (高等代數 I, II) Unit of Credit: 3 (學分:3)

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.

Par
t 2: Field Theory
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.

Pa
rt 4: Group Representation Theory
Representations, characters, Group algebras, orthogonal relations, Frobenius reciprocity, induced representations, Burnside's theorem, permutation representations, more examples, Brauer's theorem, Representations of symmetric groups, Sprecht modules, introduction to symmetric functions.

Par
t 5: Commutative Algebra
Integral extension, Noetherian rings and modules, primary decomposition, Nakayama's lemma, Hilbert's Nullstellensatz.

References:
[1] Hungerford: Algebra. GTM 73.

[2] Fulton and Harris, Representation Theory. A first course. GTM 129

[3] Sagan: The Symmetric Group. GTM 203.

[4] Atiyah and MacDonald: Introduction to Commutative Algebra.

Instructors: 余家富 (Fall), 程舜仁 (Spring)

Time:

 Friday 10:20 ~ 12:10 Saturday 13:20 ~ 15:10

First meeting:
Friday February 20th 2009 (Spring)
Friday September 19th 2008 (Fall)

Meeting location:
Spring Friday & Saturday Class - 舊數館103室，國立台灣大學

Fall Friday Class - 新生大樓502室，國立台灣大學
Fall Saturday Class - 舊數館103室，國立台灣大學

1. 住宿費補助: 檢附發票 (上限為NT\$1,000元，補助地區為新竹以南)
2. 車票費補助: 檢據(票根)核實列支(金額上限比照火車自強號車種價格，補助地區為新竹以南)
3. 6月及12月攜帶發票及票根辦理請款。(無票根或遺失者不予補助)

1. 發票抬頭: 財團法人慈澤教基金會
2. 收銀機統一發票，則免抬頭，應輸入機關統一編號：14300339
3. 統一發票，僅有貨品代號者，應由經手人加註貨品名稱、數量、單價並簽名。

*課程可選單一學期進行研習；遠道之學生可申請補助住宿及交通費。