INTRODUCTION TO ADVANCED ANALYSIS
Term 1
Level: 4/5-MSc/PhD
Type:
Pure Mathematics.
Abstract:
This course provides
an essential basis for a broad spectrum of Advanced Analysis and
Applications (including linear and nonlinear PDEs, Spectral Theory,
..., etc.)
Aims : To teach rigorous techniques of
modern analysis.
Prerequisites:
Complex
Analysis, Lebesgue Measure and Integration Theory, Functional
Analysis.
Syllabus:
Banach Spaces:
(L_p and Orlicz Spaces);
(Review of Theorems: Banach-Alaoglou,
Banach Contraction Principle, Hahn-Banach, Riesz, ...);
Schwartz
Function Space & Fourier transform;
Minkowski, Hoelder,
Young, Hausdorf-Young Inequalities;
Interpolation Theorems
Riesz–Thorin theorem;
Poincare and Sobolev Inequalities;
Sobolev Spaces; Rellich-Kondrachov Theorem;
Coercive
Inequalities on Manifolds and Weighted Spaces;
[Applications to
Isoperimetry and Transportation theory];
Markov Semigroups;
Duhamel formula;
Basic spectral Theory;
[Basic
Hypoellipticity];
Smoothness and Ergodicity Estimates
Some Books
Ellias M. Stein & Rami Shakarchi, Functional Analysis-Introduction to Further Topics in Analysis.
Elliott H. Lieb and Michael Loss, Analysis.
William P. Ziemer, Weakly Differentiable Functions.
Examples & Problem 2015 autumn
P.1 ; P.2 (16 Nov); P.3 ; P.4 (30 Nov); P.5 (17 Dec).
Hints, Remarks & Some Solutions
S.1 ; S.2 ; S.3 ; S.4 ; S.5 ; S.6.