2011 Core Curriculum

Instructors: 劉豐哲 (Fall 2011-Spring 2012)

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.

Part 1: Introduction
Summability of system of real numbers, Convolution of sequences and random walks
Modeling of independent coin tossing, A quick introdution to measure and integration

Part 2: Outer measures
Measures arising from outer measures, Regularity of outer measures,
1-dimensional Lebesgue measure, Caratheodory outer measures, Hausdorff measures-length measure and surface measure,
n-dimensional Lebesgue measure and transformation of integrals

Part 3: L²-spaces
Completeness of L²-spaces, Orthonormal systems-completeness and Fourier expansion,
Best approximation, Orthonormalization-Legendre, Laguerre and Hermite polynomials,
Conditional expectation, Fourier-transform-Poisson summation formula and central limit theorem

Part 4: Elements of Functional Analysis
Metric spaces-completeness and compactness, Normed vector spaces,
Linear functionals, The Baire category theorem and its consequences,
Strong and weak convergences, Hilbert spaces

Part 5: Decomposition and Differentiation of Measures
Signed measures, The Lebesgue-Randon-Nikodym theorem, Conditional expectation,
Differentiation on Rn, Functions of bounded variation, Convex functions

Part 6: L^p-spaces and Sobolev spaces
Dual of L^p, Distribution functions and interpolation of L^p-spaces,
Distributions and Sobolev spaces, Variational methods in mathematical physics,

Part 7: Some Miscellaneous Topics
Topics in Fourier analysis, probability theory and calculus of variations,

Lecture notes:
CH1-2

References:
[1] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC.
[2] G.B. Folland, Real Analysis: Modern Techniques and Their Applications, Measure Theory, Wiley-Interscience
[3] F.C. Liu, Lecture notes
[4] P. Malliavin, Integration and Probability, GTM 157 Springer

高等實分析習題分配

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

Instructors: 林正洪 (Fall 2011 - Spring 2012)

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.

Part 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.
representations of symmetric groups, Young diagrams, Specht modules.

Part 5: Central Simple Algebra
Noetherian rings and modules, Hilbert basis theorem, Hilbert Nullstellensatz,
Integral extensions, Noether normalization theorem.

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

Prerequisite: one year undergraduate algebra

Time:

	Real Analysis		Algebra
	劉豐哲	TA Session (Tentative)	林正洪	TA Session (Tentative)
Fridays	20:00 - 21:00	-	16:00 - 18:00	-
Saturdays	10:00 - 12:00	13:30 - 15:30	9:00 - 10:00	15:30 - 17:30
Room	638		638
Location	6th Floor, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 台北市羅斯福路4段1號天文數學館6樓 (map)

補助資訊 Subsidy Information :
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。( 請於100年9月23日前提出申請)
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月攜帶申請表格、發票及票根辦理請款。(無票根或遺失者不予補助)

開立發票注意事項 :
1. 發票抬頭: 財團法人慈澤文教基金會
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。

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:林宛嫻 TEL:(02)2368-5999-341 FAX:(02)2368-9771 Email:felicia@math.sinica.edu.tw

*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 Mandrian.

核心課程

*注意: 12月16日、12月17日上課教室改為617研討室

100學年第一學期(秋季)行事曆-請按此下載

學生資料表-請按此下載

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

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