“il cessa de calculer et de vivre.” ― Marquis de Condorcet
Preface
In July 2020, I walked into the realm of advanced MATH. However, it wasn’t until I read Topology by Munkres and Algebra by Hungerford during the Winter of 2021 that I realized it’s overwhelming. The concepts and theorems in these books proved to be challenging, prompting me to take inspiration from Kunihiko Kodaira and develop my own approach to learning MATH.
Conformal Loop Ensembles
References: [1]. Conformal Loop Ensembles: The Markovian characterization and the loop-soup construction, Scott Sheffield, Wendelin Werner [2]. Autour des relations entre SLE, CLE, champ libre Gaussien, et leur cons´equences, Hao Wu
IN PROGRESS
Random Interlacements
Reference:[1].An Introduction to Random Interlacements, A. Drewitz , B. Ráth , A. Sapozhnikov [2]. Vacacnt Set of Random Interlacements and Percolation, A. Sznitman [3].Lectures on the Poisson Process, Gunter Last, Mathew Penrose
Ⅰ. General Theory of Poisson Process
Ⅱ. Discrete Time Random Interlacement: Existence, Properties and Stochastic Domination
Ⅲ. Discrete Time Random Interlacement Point Process: Construction and Properties
Ⅳ. Percolation of Vacant Sets on Discrete Time Random Interlacements
Loops and Occupation Time
Reference:[1]. Topics in Occupation Times and Gaussian Free Fields, A. Sznitman [2].Lecture notes on the Gaussian Free Field, W. Werner, E. Powell
Ⅲ. Continuous Time RW Loops: Rooted Loops, Pointed Loops and Unrooted Loops
Discrete Gaussian Free Fields
References:[1].Statistical Mechanics of Lattice Systems:a Concrete Mathematical Introduction, Chap.8, Friedli, S. and Velenik, Y.[2].Lecture notes on the Gaussian Free Field, W. Werner, E. Powell
Ⅱ. Random Spanning Trees and GFFs: Wilson’s Algorithm and Occupation Fields
Schramm-Loewner Evolutions
References:[1].Schramm–Loewner Evolution, Antti Kemppainen [2].Schramm–Loewner Evolution, Jason Miller [3].Schramm-Loewner Evolution, G. Lawler [4].Lectures on Schramm–Loewner Evolution, N. Berestycki & J.R. Norris
Ⅰ. Preliminary: Conformal Mapping, BM, Harmonic Measure and Green Function
Ⅱ. Mapping-out Functions: Compact Hulls and Capacity
Ⅲ. Chordal Loewner Theory: Loewner Transform and Differential Equation
Ⅳ. SLE: Properties, Bessel Flows, Hitting probability, Phase Transition and Conformal Transform
Ⅵ. SLE(8/3): Brownian Excursions and Restriction
Ⅶ. SLE(4) and Continumm Gaussian Free Fields
Percolation Theory: Conformal Invariance of 2D Lattice Models
References:[1]Random Planar Curves and Schramm-Loewner Evolutions, Wendelin Werner[2] Lectures on two-dimensional critical percolation, Wendelin Werner [3] Conformal invariance of lattice models, Hugo Duminil-Copin, Stanislav Smirnov [4]Percolation Theory, Kim Christensen
Mostly base on the lectures given by Hao WU, YMSC
Ⅰ.Bernoulli Bond Percolation: Phase Transition
Ⅱ.Bernoulli Site Percolation: Critical Values, Cardy’s Formula
Ⅲ.Random Cluster Model: Phase Transition and Self-Dual Points
Ⅵ.FK-Ising Model: Discrete Complex Anaysis, Conformal Invariance
Ⅴ.Ising Model: Critical Value, Conformal Invaraince
Stochastic Differential Equations
References:[1] Stochastic Differential Equations, Øksendal B. [2]Stochastic Partial Differential Equation, Étienne Pardoux [3] An Introduction to Stochastic PDEs, Martin Hairer
Ⅰ.Ito Integral and Ito Process
Complement: Stationary Distribution of Diffusions
Harmonic Analysis
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
Ⅱ.Hardy Littlewood Maximal Functions
Ⅴ.Singular Integrals and C-Z operators
Ⅶ.Weight Inequality and Ap condition
Ⅷ.Littlewood-Paley Theory and Application on Multipliers and Singular Integrals
Stochastic Calculus
References:[1]Brownian Motion, Martingales, and Stochastic Calculus, Le Gall, Jean-François(primary) [2] Continuous martingales and Brownian motion, Daniel Revuz, Marc Yor [3]Brownian Motion and Stochastic Calculus, Ioannis Karatzas, Steven E. Shreve
Ⅲ.Stochastic Integration: Construction, Ito Formula, Representation of Marts, and Girsanov Thm
Ⅳ.Local Time and Generalized Ito Formula
Ⅴ.General Theory of Markov Process
Advanced Stochastic Process
References:[1]Probability Theory: Theory and Examples, Richard Durrett [2]Lecture Notes on Measure-theoretic Probability Theory, Sebastien Roch
Partial Differential Equations
References:[1]Partial Differential Equations, Lawrence C. Evans [2]Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis
Ⅲ.Envelope and Charateristic Method
Ⅵ.Second-Order Elliptic Equations
(Linear) Functional Analysis
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
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
Complement: Existence of Projection
Complement: Finite Dimensional Operators
Complement: Vector Space over Complex Domain
Advanced Probability Thoery
References:[1] A course in probability theory, Kai Lai Chung[2] Probability Theory: Theory and Examples, Richard Durrett [3] Advanced Probability Theory, Bingyi Jing
Complement: Infinite divisiblility
Linear Model
References:[1]Plane answers to complex questions: the theory of linear models, Ronald Christensen [2]Lecture Note from Aijiao Deng, Wuhan University
Ⅰ.Projection and Generalized Inverse
Applied Stochastic Process
References:[1]Stochastic Processes, J. Chang [2]Lecture Notes on Stochastic Modeling, Karl Sigman [3]Stochastic Processes, Sheldon M. Ross
Ⅳ.Continuous Time Markov Chains
Ⅵ.Markov Random Fields and Hidden Markov Models
Ⅶ.Brownian Motion:Levy’s construction, Conditional Dist, and Shifted Hitting Time
Complement: Stochastic Approximation
Complement: Gamble’s Ruin Problem
Complement: Poisson Clumping Heuristic
Multivariate Statistics
References:[1] Applied Multivariate Statistical Analysis, Richard A. Johnson, Dean W. Wichern [2]应用多元统计分析, 高惠璇 [3] Applied Multivariate Statistical Analysis, Hardle, Simar.
Ⅶ.Principle Component Analysis
Ⅸ.Canonical Correlation Analysis
Complement: Multidimensional scaling
Real Analysis
References:[1]Real analysis: modern techniques and their applications, Gerald B. Folland [2]Real analysis:measure theory, integration and Hilbert spaces, Elias M. Stein, Rami Shakarchi [3]Real and Complex Analysis, Rudin [4]Measure Theory, Cohn
Ⅰ.Lebesgue Measure: Properties, Integration Theory, and Differentiation Theory
Ⅱ.Abstract Measure: Properties, Integration Theory, and Differentiation Theory
Ⅲ.Radon Measure: Representation, Reguarity, Duality, and Product measure
Ⅳ.Hausdorff Measure: Properties and Self-similarity
Ⅴ.Haar Measure on LCH topological groups
Complex Analysis
References:[1]Complex analysis, Elias M. Stein, Rami Shakarchi [2]Invitation to Complex Analysis, Ralph P. Boas [3]Complex analysis, Ahlfors L.[4]Real and Complex Analysis,Rudin
Ⅶ.Linear Fractional Transformations
Ⅸ.Fourier Transformations in Complex Domain
Differential Manifolds
In fact, most manifolds in my notes are SMOOTH.
Reference:[1]Manifolds, Ed Segal [2]Introduction to Manifolds, Loring W. Tu
Mathematical Statistics
References:[1]Statistical Inference, George Casella, Roger L. Berger [2]Mathematical Statistics and Data Analysis, Rice, John A.
Ⅱ.Principles of Data Reduction
Ⅶ.Decision Theory and Bayesian Inference
Complement: Fisher Information
Complement: Examine Statistical model
Ordinary Differential Equations
Reference:[1]常微分方程教程,丁同仁; 李承治(primary)[2]Ordinary Differential Equations and Dynamical Systems, Gerald Teschl
Ⅴ.Parameters and Initial Value in High Order Equations
General Topology
I will “refresh” the note which’s totally a DISASTER because it’s my first time to make a note.
References:[1]Topology, James R. Munkres(primary) [2]Lectures on Basic Topology, Chengye You
Set Theory and General Topology
Advanced Algebra
I will “refresh” the note which’s totally a DISASTER because it’s my first time to make a note.
References:[1]Algebra, Thomas W. Hungerford(primary) [2]Abstract algebra, David S. Dummit, Richard M. Foote