# 核心課程

## Unit of Credit: 3 (學分:3)

Instructor: 劉豐哲

Time: Mon.8:10-10:00；Wed. 10:20-11:10

Venue: Room 102 (Astro.-Math. Building)

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:

Chapter 1-7 [ 2014/9/11 revised ]

*習題課上課時間:每周三上午 8:10 於6樓638教室 [10/15(三)開始]

## Undergraduate Independent Study in PDE

Instructor:劉太平

Course Description: Topics in partial differential equations, materials are flexible with study plan to be designed for the individual students.

Time: There will be one meeting per week at a time agreed upon by student and instructor.

Further information: Highly motivated undergraduate students in any university in Taiwan with strong mathematical maturity who intend to continue post-graduate studies in mathematics are encouraged to apply. Interested students should send a statement of purpose, all available college transcripts, and a list of 2-3 possible references (with e-mail addresses and/or phone numbers) familiar with the student by e-mail to rynnj@math.sinica.edu.tw. In the statement of purpose the applicant should indicate clearly why he/she intends to study PDE.

## Undergraduate Independent Study in Representation Theory

Instructor:程舜仁

Course Description:Topics in representation theory, materials are flexible with study plan to be designed for the individual students.

Time: There will be one meeting per week at a time agreed upon by student and instructor.

Further information:Highly motivated undergraduate students in any university in Taiwan with strong mathematical maturity who intend to continue post-graduate studies in mathematics are encouraged to apply. Interested students should send a statement of purpose, all available college transcripts, and a list of 2-3 possible references (with e-mail addresses and/or phone numbers) familiar with the student by e-mail to rynnj@math.sinica.edu.tw. In the statement of purpose the applicant should indicate clearly why he/she intends to study representation theory.

Location

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

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。
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)。
(6月及12月攜帶申請表格、發票\票根辦理請款。(無票根或遺失者不予補助)

*住宿及交通補助地區為新竹以南