18.217 Graph Theory and Additive Combinatorics (Fall 2019, MIT OCW) | Mathematics

18.217 Graph Theory and Additive Combinatorics

18.217 Graph Theory and Additive Combinatorics (Fall 2019, MIT OCW). Instructor: Professor Yufei Zhao. This course examines classical and modern developments in graph theory and additive combinatorics, with a focus on topics and themes that connect the two subjects. The course also introduces students to current research topics and open problems. (from ocw.mit.edu)

A Bridge between Graph Theory and Additive Combinatorics

Lecture 01 - A Bridge between Graph Theory and Additive Combinatorics

Lecture 02 - Forbidding a Subgraph I: Mantel's Theorem and Turan's Theorem

Lecture 03 - Forbidding a Subgraph II: Complete Bipartite Subgraph

Lecture 04 - Forbidding a Subgraph III: Algebraic Constructions

Lecture 05 - Forbidding a Subgraph IV: Dependent Random Choice

Lecture 06 - Szemeredi's Graph Regularity Lemma I: Statement and Proof

Lecture 07 - Szemeredi's Graph Regularity Lemma II: Triangle Removal Lemma

Lecture 08 - Szemeredi's Graph Regularity Lemma III: Further Applications

Lecture 09 - Szemeredi's Graph Regularity Lemma IV: Induced Removal Lemma

Lecture 10 - Szemeredi's Graph Regularity Lemma V: Hypergraph Removal and Spectral Proof

Lecture 11 - Pseudo-random Graph I: Quasirandomness

Lecture 12 - Pseudo-random Graph II: Second Eigenvalue

Lecture 13 - Sparse Regularity and the Green-Tao Theorem

Lecture 14 - Graph Limits I: Introduction

Lecture 15 - Graph Limits II: Regularity and Counting

Lecture 16 - Graph Limits III: Compactness and Applications

Lecture 17 - Graph Limits IV: Inequalities between Subgraph Densities

Lecture 18 - Roth's Theorem I: Fourier Analytic Proof over Finite Field

Lecture 19 - Roth's Theorem II: Fourier Analytic Proof in the Integers

Lecture 20 - Roth's Theorem III: Polynomial Method and Arithmetic Regularity

Lecture 21 - Structure of Set Addition I: Introduction to Freiman's Theorem

Lecture 22 - Structure of Set Addition II: Groups of Bounded Exponent and Modeling Lemma

Lecture 23 - Structure of Set Addition III: Bogolyubov's Lemma and the Geometry of Numbers

Lecture 24 - Structure of Set Addition IV: Proof of Freiman's Theorem

Lecture 25 - Structure of Set Addition V: Additive Energy and Balog-Szemeredi-Gowers Theorem

Lecture 26 - Sum Product Problem and Incidence Geometry

References

18.217 Graph Theory and Additive Combinatorics (Fall 2019)
Instructor: Professor Yufei Zhao. Lecture Notes. Assignments: Problem Sets (No Solutions). This course examines classical and modern developments in graph theory and additive combinatorics.