擬登記修課學生，請填妥基本學生資料表後，依各班登記修課規定準備申請資料。

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,

L^{p}-spaces and Holder inequality, outer measures and construction of puter measures, Caratheodory outer measures

and Lebesgue-Stieltzes measures, measure-theorical approximation of sets in R^{n }.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 R^{n}, Functions of bounded variation and absolute continuity,

Change of variables formula for Lebesgue integrals on R^{n}, 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, L^{p}-spaces and their dual spaces.

Part 5: Fourier Integrals

Fourier integral for integrable functions and for rapidly decreasing functions, extension of Fourier integral to L^{2}functions,

Fourier inversion formula, Soboler space H^{s}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 ]

Prerequisite: One year undergraduate advanced calculus.

*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]

選課:國立臺灣大學學生請透過學校選課系統即可，不需另行登記；非國立臺灣大學學生請email基本學生資料表至rynnj@math.sinica.edu.tw，將再通知後續流程。

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.

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

台北市羅斯福路4段1號(台大校區)天文數學館 (map)

補助資訊 Subsidy Information : (accepted applicant)

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**

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:林思潔 TEL:(02)2368-5999#341 FAX:(02)2368-9771 Email: rynnj@math.sinica.edu.tw