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.
What Will You Learn?
- Master fundamental concepts in Graph Theory
- Understand Eulerian and Hamiltonian paths and circuits. And many related topics to Paths.
- Get to know a wide range of different Graphs, and their properties.
- Be able to preform Elementary, Advanced Operations on Graphs to produce a new Graph
- Understand Graph Coloring.
- Know how to turn a Graph into a Matrix and vice versa.
- Obtain a solid foundation in Trees, Tree Traversals, and Expression Trees.
Forbidding a Subgraph I: Mantel’s Theorem and Turán’s Theorem00:00
Forbidding a Subgraph II: Complete Bipartite Subgraph00:00
Forbidding a Subgraph III: Algebraic Constructions00:00
Forbidding a Subgraph IV: Dependent Random Choice00:00
Szemerédi’s Graph Regularity Lemma I: Statement and Proof00:00
Szemerédi’s Graph Regularity Lemma II: Triangle Removal Lemma00:00
Szemerédi’s Graph Regularity Lemma III: Further Applications00:00
Szemerédi’s Graph Regularity Lemma IV: Induced Removal Lemma00:00
Szemerédi’s Graph Regularity Lemma V: Hypergraph Removal and Spectral Proof00:00
Pseudorandom Graphs I: Quasirandomness00:00
Pseudorandom Graphs II: Second Eigenvalue00:00
Sparse Regularity and the Green-Tao Theorem00:00
Graph Limits I: Introduction00:00
Graph Limits II: Regularity and Counting00:00
Graph Limits III: Compactness and Applications00:00
Graph Limits IV: Inequalities between Subgraph Densities00:00
Roth’s Theorem I: Fourier Analytic Proof over Finite Field00:00
Roth’s Theorem II: Fourier Analytic Proof in the Integers00:00
Roth’s Theorem III: Polynomial Method and Arithmetic Regularity00:00
Structure of Set Addition I: Introduction to Freiman’s Theorem00:00
Structure of Set Addition II: Groups of Bounded Exponent and Modeling Lemma00:00
Structure of Set Addition III: Bogolyubov’s Lemma and the Geometry of Numbers00:00
Structure of Set Addition IV: Proof of Freiman’s Theorem00:00
Structure of Set Addition V: Additive Energy and Balog-Szemerédi-Gowers Theorem00:00
Sum-Product Problem and Incidence Geometry00:00