Spring 2019 Analysis on Manifolds
Preliminary: multivarible calculus，real analysis, complex analysis, functional analysis.
Plan：
Chapter I Differential and pseudodifferential operators on manifolds, including the sobolev space, Sobolev embedding theorem, Rellich theorem, elliptic estimates and elliptic regularity;
Chapter II Spectral theory, including symmetric and self-adjoint, Friedich extension, Kato-Rellich theorem, functional calculus, etc;
Chapter III Heat equation and heat kernels on manifolds.
Pdf version of the note (not completed 2019.06.03)
Reference:
Partial Differential Equation II, by Taylor
The analysis of Linear Partial Differential Operators III, by Hormander
Spin Geometry, by Lawson and Michelsohn
Holomorphic Morse Inequalities and Bergman Kernels, by Ma and Marinescu
广义函数论, by 施瓦兹
Analysis of Real and Complex Manifolds, by Narashimhan
拟微分算子和Nash-Moser定理, by 阿里纳克 and 热拉尔
Pseudodifferential Operators and Spectral Theory, by Shubin
Topology and Analysis, by Booss and Blacker
The Neumann Problem for the Cauchy-Riemann complex, by Folland and Kohn
泛函分析讲义下册, by 张恭庆
Methods of Modern Mathematical Physics I, by Reed and Simon
Heat Kernels and Dirac Operators, by Berline, Gezler and Vergne
A Course in Functional Analysis, GTM 96, by Conway
Classes of Linear Operators I, by Gohberg, Goldberg and Kaashoek
Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, by Gilkey