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.