# 核心課程

*登記修課請於9/11(三)前將學生資料表email至 rynnj@math.sinica.edu.tw

102學年第一學期(秋季)行事曆

## Unit of Credit: 3 (學分:3)

Instructors: 劉豐哲

Course Description:

This course aims to cover extensions of Lebesgue Theory in contemporary analysis and probability;
emphasis will be placed on functions of real variables and their role in modern analysis.
The course is a graduate level course. It requires regular participation in recitation sessions
.

Part 1: Introduction
and Preliminaries in Abstract Analysis
Summability of systems of real numbers, modeling of independent coin tossings, metric spaces and normed vector spaces,
compactness and its characterizations.

Part 2: Measure theory and Construction of measures
Lebesgue theory of measure and integration, monotone convergence and Lebesgue dominated convergence theorem,
Lp-spaces and Holder inequality, outer measures and construction of puter measures, Caratheodory outer measures
and Lebesgue-Stieltzes measures, measure-theorical approximation of sets in Rn .

Part 3: Differentiation of Measures and Functions of Real Variables
Lusin theorm, Riemann and Lebesgue integral, Representation of general integrals as Lebesgue-Stieltzes integrals,
Covering theorms and differentiation of Radon measures on Rn, Functions of bounded variation and absolute continuity,
Change of variables formula for Lebesgue integrals on Rn, Smoothing of functions.

Part 4: Elements of Functional Analysis
The Baire Category Theorm and its consequences, The open mapping theorm and the closed graph theorm,
Separation principles and Hahn-Banach theorm, Hilbert spaces, Riesz representation and Lebesgue-Nikodym theorem,
Fourier expansion in separable Hilbert spaces, Lp-spaces and their dual spaces.

Part 5: Fourier Integrals
Fourier integral for integrable functions and for rapidly decreasing functions, extension of Fourier integral to L2 functions,
Fourier inversion formula, Soboler space
Hs and application to smoothness of weak solutions to elliptic partial differential equations.

Part 6: Miscellaneous Topics

Topics in probability theory and calculus of variations

Lecture notes:

Chapter1: Introduction and Preliminaries [2013/10/18 revised]

References:
[1] G.B. Folland, Real Analysis
[2] S.L. Royden, Real Analysis
[3] S. Saks, Theory of Integral
[4] E.M. Stein & R. Shakarchi, Real Analysis

*習題課上課時間:每週日晚間7點至9點於天文數學館6樓638教室。(自9月29日開始第一堂實習課。)

## Unit of Credit: 3 (學分:3)

Instructors: 余家富

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, construction of Galois extensions.

Part
3: Non-commutative Ring Theory
Simple rings and algebras, Schur's lemma,
density theorem, Artin-Wedderburn theorem, Jacobson's radical,
semi-simple rings and algebras, Noether-Skolem theorem.

Part 4: Group Representation Theory
Representations, characters, group algebras, orthogonality relations,
induced representations, Frobenius reciprocity, Burnside'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,
central simple algebras over global fields.

Part
6: Commutative Algebra
Noetherian rings and modules, Hilbert basis theorem, Hilbert Nullstellensatz,
Integral extensions, Noether normalization theorem

.

References:
[1] Hungerford: Algebra. GTM 73.
[2] Sagan: The Symmetric Group. GTM 203.
[3] Atiyah and MacDonald: Introduction to Commutative Algebra
[4] R. Pierce, Assocaited Algebras, GTM 88
[5] W. Fulton and J. Harris, Representation Theory, GTM 129

[6] I.Reiner, Maximal orders

Prerequisite: One year undergraduate algebra course with strong mathematics maturity.

Time:

 Real Analysis 劉豐哲 TA Session (Tentative) Mondays 8:10 - 10:00 - Wednesdays 10:20 - 12:10 Room 102

 Algebra 余家富 TA Session (Tentative) Fridays 15:10 - 17:00 19:00 - 21:00 Saturdays 9:10 - 10:00 Room 638

Location

6th Floor, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei

(map)

1.住宿: 統一由中研院數學所代訂,自行安排住宿者無法補助(申請表請於102年9月6日前提出)
2.交通: 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)
(6月及12月攜帶申請表格、發票及票根辦理請款。(無票根或遺失者不予補助)

*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。

*The disccusion during the courses will mainly be held in Mandarin.