| 1 | Introduction (PDF) |
| Fundamental Examples of the Polynomial Method |
| 2 | The Berlekamp-Welch Algorithm (PDF) |
| 3 | The Finite Field Nikodym and Kakeya Theorems (PDF) |
| 4 | The Joints Problem (PDF) |
| 5 | Why Polynomials? (PDF) |
| Background in Incidence Geometry |
| 6 | Introduction to Incidence Geometry (PDF) |
| 7 | Crossing Numbers and the Szemeredi-Trotter Theorem (PDF) |
| 8 | Crossing Numbers and Distance Problems (PDF) |
| 9 | Crossing Numbers and Distinct Distances (PDF) |
| 10 | Reguli; The Zarankiewicz Problem (PDF) |
| 11 | The Elekes-Sharir Approach to the Distinct Distance Problem (PDF) |
| Algebraic Structure |
| 12 | Degree Reduction (PDF) |
| 13 | Bezout Theorem (PDF) |
| 14 | Special Points and Lines of Algebraic Surfaces (PDF) |
| 15 | An Application to Incidence Geometry (PDF) |
| 16 | Taking Stock (PDF) |
| Cell Decompositions |
| 17 | Introduction to the Cellular Method (PDF) |
| 18 | Polynomial Cell Decompositions (PDF) |
| 19 | Using Cell Decompositions (PDF) |
| 20 | Incidence Bounds in Three Dimensions (PDF) |
| 21 | What's Special About Polynomials? (A Geometric Perspective) (PDF) |
| Ruled Surfaces and Projection Theory |
| 22 | Detection Lemmas and Projection Theory (PDF) |
| 23 | Local to Global Arguments (PDF) |
| 24 | The Regulus Detection Lemma (PDF) |
| The Polynomial Method in Number Theory |
| 25 | Introduction to Thue's Theorem on Diophantine Approximation (PDF) |
| 26 | Thue's Proof (Part I) (PDF) |
| 27 | Thue's Proof (Part II): Polynomials of Two Variables (PDF) |
| 28 | Thue's Proof (Part III) (PDF) |
| Introduction to the Kakeya Problem |
| 29 | Background on Connections Between Analysis and Combinatorics (Loomis-Whitney) (PDF) |
| 30 | Hardy-Littlewood-Sobolev Inequality (PDF) |
| 31 | Oscillating Integrals and Besicovitch's Arrangement of Tubes (PDF) |
| 32 | Besictovitch's Construction (PDF) |
| 33 | The Kakeya Problem (PDF) |
| 34 | A Version of the Joints Theorem for Long Thin Tubes (PDF) |
