Talk202527 h 39 min640×360Subtitles

MIT 18.156 Projection Theory, Spring 2025

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About

Instructor: Lawrence D Guth

View the complete course: https://ocw.mit.edu/courses/18-156-projection-theory-spring-2025

YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62K9_7Pg26ytrAOXwXFpN43

Given a set or measure in Euclidean space, we consider its projection

onto many different subspaces. Informally, we study an object by looking at its shadows in many different directions. How do the features of the object compare with the features of the shadows? If many shadows are highly concentrated, is the original object highly concentrated too? This type of question connects geometry and harmonic analysis and combinatorics.

And understanding this type of question has applications in number theory, homogeneous dynamics and harmonic analysis. Some of these connections were just found recently. Seeing all the connections between projection theory and other areas was one motivation for me to teach this class.

One of the fundamental problems in the subject is called the exceptional set problem. It is a quantitative version of the question from the first paragraph: If many shadows are highly concentrated, is the original object highly concentrated too? This question was introduced in the 1960s. It was fully answered in 2024 by Orponen-Shmerkin-Ren-Wang. This breakthrough was a second motivation for me to teach this class.

The full proof of Orponen-Shmerkin-Ren-Wang is quite complex, and we won't study every detail, but we will introduce the background and describe the main new ideas.

License: Creative Commons BY-NC-SA

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