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18.156

Projection Theory

Prof. Lawrence D Guth | Spring 2025

Science & Math Mathematics Differential Equations Mathematical Analysis

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CC BY-NC-SA 4.0

Course introduction

The class studies projection theory, starting from the first questions and building up to recent developments. Projection theory studies how a set X behaves under different orthogonal projections. Questions of this type aren’t usually emphasized in the graduate analysis curriculum, but they come up in many areas of math, including harmonic analysis, analytic number theory, additive combinatorics, and homogeneous dynamics.

We will survey several applications of projection theory. For each topic, we will introduce and motivate the topic and see how it connects with projection theory. We will prove something about each topic but not necessarily the strongest results.

Course Information

Source	MIT 開放式課程
Department	Mathematics
Language	English
Number of videos	24

Course videos (24)

Lecture 01: Introduction to Projection Theory

Lecture 01: Introduction to Projection Theory

Lecture 02: Fundamental Methods of Projection Theory

Lecture 02: Fundamental Methods of Projection Theory

Lecture 03: Projection Theory in Euclidean Space

Lecture 03: Projection Theory in Euclidean Space

Lecture 04: The Fourier Method in Euclidean Space

Lecture 04: The Fourier Method in Euclidean Space

Lecture 05: The Large Sieve

Lecture 05: The Large Sieve

Lecture 06: Projections and Smoothing

Lecture 06: Projections and Smoothing

Lecture 07: Applications of the Large Sieve to Number Theory

Lecture 07: Applications of the Large Sieve to Number Theory

Lecture 08: The Szemeredi-Trotter Theorem

Lecture 08: The Szemeredi-Trotter Theorem

Lecture 09: Reflections on the Szemeredi-Trotter Theorem

Lecture 09: Reflections on the Szemeredi-Trotter Theorem

Lecture 10: Sum-Product Theory

Lecture 10: Sum-Product Theory

Lecture 11: Contagious Structure in Projection Theory

Lecture 11: Contagious Structure in Projection Theory

Lecture 12: The Bourgain-Katz-Tao Projection Theorem

Lecture 12: The Bourgain-Katz-Tao Projection Theorem

Lecture 13: The Balog-Szemeredi-Gowers Theorem

Lecture 13: The Balog-Szemeredi-Gowers Theorem

Lecture 14: The Bourgain Projection Theorem Part 1 (over the Real Numbers)

Lecture 14: The Bourgain Projection Theorem Part 1 (over the Real Numbers)

Lecture 15: The Bourgain Projection Theorem, Part 2

Lecture 15: The Bourgain Projection Theorem, Part 2

Lecture 16: The Bourgain Projection Theorem, Part 3

Lecture 16: The Bourgain Projection Theorem, Part 3

Lecture 17: Random Walks on Finite Groups, Part 1

Lecture 17: Random Walks on Finite Groups, Part 1

Lecture 18: Random Walks on Finite Groups, Part 2

Lecture 18: Random Walks on Finite Groups, Part 2

Lecture 19: Random Walks on Finite Groups, Part 3

Lecture 19: Random Walks on Finite Groups, Part 3

Lecture 20: Homogeneous Dynamics, Part 1

Lecture 20: Homogeneous Dynamics, Part 1

Lecture 21: Homogeneous Dynamics, Part 2

Lecture 21: Homogeneous Dynamics, Part 2

Lecture 22: Sharp Projection Theorems, Part 1: Introduction and Beck's Theorem.

Lecture 22: Sharp Projection Theorems, Part 1: Introduction and Beck's Theorem.

Lecture 23: Sharp Projection Theorems, Part 2: AD Regular Case

Lecture 23: Sharp Projection Theorems, Part 2: AD Regular Case

Lecture 24: Sharp Projection Theorems, Part 3: Combining Different Scales

Lecture 24: Sharp Projection Theorems, Part 3: Combining Different Scales