Notes

Muhammad Haris Rao

Here, I have collected various writings I have collected notes and summaries on most standard mathematical content taught in a usual undergraduate degree in pure mathematics. Most of these are adapted from writings I have myself written over the course of my undergraduate degree at the University of Melbourne, and there may be minors errors. If you find any, feel free to to contact me about them, and I will correct them.

Logic and Foundations

Set Theory

• Ultrafilters

First-Order Logic

• Syntax of First-Order Logic
• Semantics of First-Order Logic
• Natural Deduction

Model Theory

• Gödel's Completeness and Compactness Theorems
• Löwenheim-Skolem Theorem
• Ultraproducts and Łoś Theorem

Algebra

• Matrices over Commutative Rings

Analysis

Single-Variable Real Analysis

• Taylor Series

Fourier Analysis

• Poisson Summation Formula

Transcendental Functions

• Axiomatic Construction of Logarithm, Exponential and Trigonometric Functions
• The Gamma Function
• Particular Values of Gamma Function

Measure Theory

• Measure Spaces and Measurable Functions
• Approximation by Simple Functions
• Lebesgue Integration
• Integral Convergence Theorems
• Dynkin $\pi$-$\lambda$ Theorem
• Caratheodery Extension Theorem
• Lebesgue-Stielties Measure
• Pushforward-Measures
• Densities and Change of Measure

Functional Analysis

• Weierstrass Approximation Theorem
• $L^p$ Spaces

Topology and Geometry

Point-Set Topology

• Separation Axioms
• Countability Axioms
• Compactness in Product Spaces
• Heine-Borel Theorem

Probablity and Statistics

• Probability Spaces and Random Variables
• Borel-Cantelli Lemmas
• Independence and Product Spaces
• Distributions on $\mathbb{R}$ and Distribution Functions
• Expectation
• Computational Formulas for Expectation
• Convergence of Random Variables
• Markov and Chebychev Inequalities and the Weak Law of Large Numbers
• Strong Law of Large Numbers
• Characteristic Functions

Probability Distributions

• Bernoulli and Binomial Distributions
• Geometric and Negative Binomial Distrbutions
• Poisson Distribution
• Exponential and Gamma Distrbutions
• Normal Distribution
• $\chi^2$-distribution
• $t$-distribution
• $F$-distribution

Stochastic Processes

• Poisson Process

Number Theory

Analytic Number Theory

• Dirichlet Series and Euler Products
• Chebychev Bounds for Prime Counting Function
• Convolution of Arithmetic Functions and Möbius Inversion
• Bernoulli Numbers and Sum of Powers
• Perron's Formula
• Equivalent Forms of Prime Number Theorem