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Projection Theory

Prof. Lawrence D Guth · Massachusetts Institute of Technology

Start with lecture 1 At MIT OpenCourseWare

About

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.

Lectures

Lecture Videos

  1. Lecture 01: Introduction to Projection Theory Video
  2. Lecture 02: Fundamental Methods of Projection Theory Video
  3. Lecture 03: Projection Theory in Euclidean Space Video
  4. Lecture 04: The Fourier Method in Euclidean Space Video
  5. Lecture 05: The Large Sieve Video
  6. Lecture 06: Projections and Smoothing Video
  7. Lecture 07: Applications of the Large Sieve to Number Theory Video
  8. Lecture 08: The Szemeredi-Trotter Theorem Video
  9. Lecture 09: Reflections on the Szemeredi-Trotter Theorem Video
  10. Lecture 10: Sum-Product Theory Video
  11. Lecture 11: Contagious Structure in Projection Theory Video
  12. Lecture 12: The Bourgain-Katz-Tao Projection Theorem Video
  13. Lecture 13: The Balog-Szemeredi-Gowers Theorem Video
  14. Lecture 14: The Bourgain Projection Theorem Part 1 (over the Real Numbers) Video
  15. Lecture 15: The Bourgain Projection Theorem, Part 2 Video
  16. Lecture 16: The Bourgain Projection Theorem, Part 3 Video
  17. Lecture 17: Random Walks on Finite Groups, Part 1 Video
  18. Lecture 18: Random Walks on Finite Groups, Part 2 Video
  19. Lecture 19: Random Walks on Finite Groups, Part 3 Video
  20. Lecture 20: Homogeneous Dynamics, Part 1 Video
  21. Lecture 21: Homogeneous Dynamics, Part 2 Video
  22. Lecture 22: Sharp Projection Theorems, Part 1: Introduction and Beck's Theorem. Video
  23. Lecture 23: Sharp Projection Theorems, Part 2: AD Regular Case Video
  24. Lecture 24: Sharp Projection Theorems, Part 3: Combining Different Scales Video

Syllabus and materials

At MIT OpenCourseWare, as the course publishes them.

Credits and licence

Institution
Massachusetts Institute of Technology
Department
Mathematics
Teacher
Prof. Lawrence D Guth
Term
Spring 2025
Level
Graduate
Language
English
Subjects
Mathematics
Source
MIT OpenCourseWare
Licence
CC BY-NC-SA 4.0, as MIT OpenCourseWare states it. Non-commercial use only. MIT OpenCourseWare's terms
Attribution
Prof. Lawrence D Guth. Projection Theory. Spring 2025. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu/. License: Creative Commons BY-NC-SA.
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