“钱塘江上潮信来,今日方知我是我。” –《水浒传·第一百一十九回》

Nonlinear PDEs (2024)

Reference: Differential Equations IIB, Alex Kaltenbach

Ⅲ. Application of Monotone Operator Theory: P-Laplace Equations and P-Stokes Equations

Manifolds and Riemannian Geometry (2024)

Reference:[1]. Differential Geometry II, J.M.Sullivan [2].Introduction to Smooth Manifolds, John M. Lee [3]. Riemannian Manifolds: An Introduction to Curvature, John M. Lee [4]. Riemannian geometry, Zuoqin Wang

Ⅰ. 1.Introduction and Basic Def 2.Tangent Vector and Submanifolds 3.Tangent space II 4.Vector Field and Flow 5.Lie Derivative 6. Vector Bundle and Dual Space

Ⅱ. 1. Bilinear Forms 2.Partition of Unity 3. Riemannian Metric

Ⅲ. 1. Tensor product & Exterior Algebra 2. Differential Form & Exterior Derivative 3. Integration on Manifold 4. Stokes THM 5. Lie derivative for K-form

Ⅳ. 1. Connections 2. Parallel Transport & Geodesic

Ⅴ. 1. Riemannian Curvature 2. Flat Metric 3. Ricci Curvature

Ⅵ. Variation of Energy

Ⅶ. Lie Groups

Harmonic Analysis (2022)

References: [1] Fourier analysis, Javier Duoandikoetxea (primary) [2] Functional Analysis: Introduction to Further Topics in Analysis, Elias M. Stein, Rami Shakarchi [3] Fourier analysis: an introduction, Elias M. Stein, Rami Shakarchi [4] Classical Fourier analysis, Loukas Grafakos [5] Modern Fourier analysis, Loukas Grafakos

Ⅰ.Fourier Series and Integral

Ⅱ.Hardy Littlewood Maximal Functions

Ⅲ.Hilbert Transform

Ⅳ.Distributions

Ⅴ.Singular Integrals and C-Z operators

Ⅵ.Hardy Space and BMO

Ⅶ.Weight Inequality and Ap condition

Ⅷ.Littlewood-Paley Theory and Application on Multipliers and Singular Integrals

Ⅸ.The T1 Theorem

Multidimensional Complex Analysis (2023)

References: [1] Lecture notes on multidimensional complex analysis, Harold P. Boas [2] 多元复分析, 涂振汉

Ⅰ. Multivariate Holomorphics: Properties, Complex Submanifolds, Injective Holomorphic, Biholomorphics on Domains

Ⅱ. Bergman Kernels and Bergman Metric

Ⅲ. Dbar Equations and Extensions: Dolbeault Solvable THM, Hatogs Extensions, Bochner-Martinelli Formula and Bochner-Severi Extension

Ⅳ. Holomorphic Sets, Weierstrass Polynomials and Locallization of Principle Holomorphic Sets

Ⅴ. Holomorphic Convex and Domains of Holomorphy

Ⅵ. Pluriharmonicity

Partial Differential Equations (2022)

References:[1] Partial Differential Equations, Lawrence C. Evans [2] Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis

Ⅰ.Preliminary

Ⅱ.Four Kinds of PDEs

Ⅲ.Envelope and Charateristic Method

Ⅳ.Solution Representation

Ⅴ.Soblev Space

Ⅵ.Second-Order Elliptic Equations

Ⅶ.Linear Evolution Equations

(Linear) Functional Analysis (2022)

References:[1] Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis [2] Principles of Functional Analysis, Martin Schechter [3] Functional analysis, Peter D. Lax

Ⅰ.Linear Space: Hahn-Banach THM, Dimension, Duality, Orthogonal, Conjugate convex Functions

Ⅱ.Linear Operators

Ⅲ.Weak Topology, Reflexive, Separable and Uniformly Convex

Ⅳ.L^p Space, Regularization and Its Strong Compactness

Ⅴ.Hilbert Spaces and Normal Operators

Ⅵ.Compact Operators and Spectral Theory

Ⅶ.Banach Algebra and Riesz Calculus

Ⅷ.Semigroups

Ⅸ.Sesquilinear Form

Complement: Existence of Projection

Complement: Finite Dimensional Operators

Complement: Quotient Space