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
 G.B. Folland, Real Analysis
 S.L. Royden, Real Analysis
 S. Saks, Theory of Integral
 E.M. Stein & R. Shakarchi, Real Analysis
Prerequisite: One year undergraduate advanced calculus.
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.
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
 Hungerford: Algebra. GTM 73.
 Sagan: The Symmetric Group. GTM 203.
 Atiyah and MacDonald: Introduction to Commutative Algebra
 R. Pierce, Assocaited Algebras, GTM 88
 W. Fulton and J. Harris, Representation Theory, GTM 129
 I.Reiner, Maximal orders
Prerequisite: One year undergraduate algebra course with strong mathematics maturity.
8:10 - 10:00
10:20 - 12:10
15:10 - 17:00
19:00 - 21:00
9:10 - 10:00
6th Floor, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei
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:firstname.lastname@example.org
*The disccusion during the courses will mainly be held in Mandarin.