18.S997 | Fall 2012 | Graduate

The Polynomial Method

Lecture Notes

Lecture notes are courtesy of MIT students and are used with permission.

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)

Course Info

As Taught In
Fall 2012
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Lecture Notes