Notes

Combinatorics

06/26/2023: Fedor Petrov's Counting Proof of Oddtown 

10/17/2022: Proof of Dirac's Theorem: every graph on n>2 vertices with min. degree at least n/2 has a hamiltonian cycle

10/11/2022: EKR via Katona Circle (From Liana Yepremyan's Combinatorics Class)

09/20/2022: Littlewood-Offord (when the x_i are in R) Theorem of Erdös (From Liana's Combinatorics class)

09/15/2022: When is LYM Tight? (From Liana's Combinatorics class)

09/12/2022: Proof of Lemma 2.1 from this paper by Cosmin Pohoata and Fedor Petrov

09/08/2022: Problem 2 from Chapter 3 of Bollobás' book Combinatorics

09/07/2022: Proof of Local LYM Inequality (from Liana's Combinatorics class)

09/06/2022: One proof of the LYM Inequality (from Liana's Combinatorics class)

09/02/2022: Proof of Sperner's Theorem via symmetric chain decomposition (from Liana's Combinatorics class)

08/24/2022: Proof of a theorem we went over today in Liana Yepremyan's Combinatorics class 

08/22/2022: Proof of Kövári, Sós, Turán Theorem -- standard double counting + convexity argument

08/14/2022: Frieze and Karonski's Random Graphs -- Exercise 1.4.3

Mod 6 Oddtowns

Croot-Lev-Pach Polynomial Method

Complex Analysis 

In preparation for the complex analysis portion of the analysis qual I made the below set of notes. They are based off of Stein and Shakarchi's Complex Analysis and David Borthwick's Lecture notes. 

The Argument Principle

Cauchy Integral Formula

Cauchy Riemann Equations

Chapter 6 - Exercise 1 - Stein and Shakarchi

Chapter 8 - Exercise 10 - Stein and Shakarchi

Liouville's Theorem

Morera's Theorem 

Open Mapping and Max Mod Principle

Power Series pt 1

Power Series pt 2

Cauchy's Theorem (basic version in a convex domain) 

Residue Theorem

Rouché's Theorem 

Schwarz Lemma

Singularities and Poles

Isolation of zeros of holomorphic functions

Measure Theory/Real Analysis

In preparation for the measure theory portion of the analysis qual I made the below set of notes. They are based off of Stein and Shakarchi's Real Analysis and David Borthwick's Lecture notes. 

Borel-Cantelli Lemma (two proofs)

Caratheodory

Convergence Theorems (bounded convergence, Fatou, monotone convergence, dominated convergence)

Density of Various Families of Functions in L^1

Extension of Fourier Transform from Schwartz Functions to L^2 functions

Construction of the Lebesgue Integral: 

• Integrating simple functions

• Integrating bounded functions with finite support

• Integrating nonnegative functions 

Littlewood Principles (Egorov, Lusin, and approximating measurable sets by rectangles)

Properties of Measurable Functions

Nested Sequences Lemma, Borel Approximation

Orthonormal Basis Theorem

Properties of Lebesgue Outer Measure

Properties of Lebesuge-Measurable Sets

Riemann-Lebesgue Lemma

Riesz-Fischer Theorem