Notes
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All mistakes are my own and all merits go to the lecturers.
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
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.
Chapter 6 - Exercise 1 - Stein and Shakarchi
Chapter 8 - Exercise 10 - Stein and Shakarchi
Open Mapping and Max Mod Principle
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)
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:
Littlewood Principles (Egorov, Lusin, and approximating measurable sets by rectangles)
Properties of Measurable Functions
Nested Sequences Lemma, Borel Approximation
Properties of Lebesgue Outer Measure
(in progress)
Fall 2025 -- my notes based off of David Borthwick's functional analysis lectures
Lecture 1 - Introduction to course and some basic preliminary notions
Lecture 2 - bounded Operators, operator topologies
Lecture 3 - strong convergence, weak convergence, uniform boundedness principle
Lecture 4
Lecture 5 - Orthogonal Decomposition and the Riesz Lemma
Lecture 6 - Orthonormal sequences, Weak Convergence, Weak Sequential Compactness
(in progress)
Many of the below proofs are heavily inspired by those that appear in Barany's Book Combinatorial Convexity.
Caratheodory
Caratheodory via Perron's Theorem (from a paper of Naszódi and Polyanskii)
Separating Hyperplane Theorems
1st Separation theorem (two disjoint convex, compact sets)
Farkas Lemma (One of many different versions)
Radon's Theorem
Standard proof (from Barany's book)