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Browse materialsMIT 18.S997 · Fall 2012

The Polynomial Method

Art: Anonymous painter,15th century, Westerner and Arab practicing geometry 15th century manuscript, 15th… · Public domain

Instructor: Prof. Lawrence D Guth
Term: Fall 2012
Level: Graduate
What's inside: 4 problem sets · 1 lecture note · 35 other files · 2 web pages
Materials open: MIT OCW

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About this course

Source: MIT OpenCourseWare 18.S997 The Polynomial Method, Fall 2012, Prof. Lawrence D Guth — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

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Lecture Notes

34 files

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MIT OpenCourseWare, Prof. Lawrence D Guth, 18.S997 The Polynomial Method, Fall 2012.

License: CC BY-NC-SA 4.0 · Edition: Fall 2012 · Open the original course on OCW ↗

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The Polynomial Method

Before you start

What's here, and what isn't

Not included here

Materials by section

Lecture Notes

Assignments

Lecture 1: Introduction (PDF)

Lecture 2: The Berlekamp-Welch Algorithm (PDF)

Lecture 3: The Finite Field Nikodym and Kakeya Theorems (PDF)

Lecture 4: The Joints Problem (PDF)

Lecture 5: Why Polynomials? (PDF)

Lecture 6: Introduction to Incidence Geometry (PDF)

Lecture 7: Crossing Numbers and the Szemeredi-Trotter Theorem (PDF)

Lecture 8: Crossing Numbers and Distance Problems (PDF)

Lecture 9: Crossing Numbers and Distinct Distances (PDF)

Lecture 10: Reguli; The Zarankiewicz Problem (PDF)

Lecture 11: The Elekes-Sharir Approach to the Distinct Distance Problem (PDF)

Lecture 12: Degree Reduction (PDF)

Lecture 13: Bezout Theorem (PDF)

Lecture 14: Special Points and Lines of Algebraic Surfaces (PDF)

Lecture 15: An Application to Incidence Geometry (PDF)

Lecture 16: Taking Stock (PDF)

Lecture 17: Introduction to the Cellular Method (PDF)

Lecture 18: Polynomial Cell Decompositions (PDF)

Lecture 19: Using Cell Decompositions (PDF)

Lecture 20: Incidence Bounds in Three Dimensions (PDF)

Lecture 21: What’s Special About Polynomials? (A Geometric Perspective) (PDF)

Lecture 22: Detection Lemmas and Projection Theory (PDF)

Lecture 23: Local to Global Arguments (PDF)

Lecture 24: The Regulus Detection Lemma (PDF)

Lecture 25: Introduction to Thue’s Theorem on Diophantine Approximation (PDF)

Lecture 26: Thue’s Proof (Part I) (PDF)

Lecture 27: Thue’s Proof (Part II): Polynomials of Two Variables (PDF)

Lecture 28: Thue’s Proof (Part III) (PDF)

Lecture 29: Background on Connections Between Analysis and Combinatorics (Loomis-Whitney) (PDF)

Lecture 30: Hardy-Littlewood-Sobolev Inequality (PDF)

Lecture 31: Oscillating Integrals and Besicovitch’s Arrangement of Tubes (PDF)

Lecture 32: Besictovitch’s Construction (PDF)

Lecture 33: The Kakeya Problem (PDF)

Lecture 34: A Version of the Joints Theorem for Long Thin Tubes (PDF)