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18.156 | Spring 2025 | Graduate

# Projection Theory

### Course Meeting Times

Lectures: 2 sessions/week, 1.5 hours/session

### Prerequisites

18.155 Differential Analysis

### Description

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. It is an especially good time to study projection theory, because there have been some striking recent applications, and because one of the central problems of the field was very recently solved. At the same time, there are many interesting open problems which I am excited to discuss and reflect on.

The goals of the course are as follows:

- Learn the classical techniques and results of projection theory (with full details).
- Learn about applications in several areas.
- Learn about open questions.
- Learn some of the main ideas in the recent work in the field. Level of detail will depend on everyone’s interest.

### Requirements

The work for the class will be as follows. For the first half or so of the course, we will have short weekly homework assignments to digest the fundamental ideas. Each student will also scribe a couple of lectures. Each student will do a final project on a topic related to the class. This could involve doing a harder problem or exploring a research direction or studying a topic in the literature that we didn’t cover in lecture and writing some exposition of it.

### Grading

The course grade will be based 50% on the problem sets and 50% on the final project.

## Course Info

##### Instructor

- Prof. Lawrence D Guth

##### Departments

- Mathematics

##### As Taught In

Spring 2025

##### Level

Graduate

##### Topics

- Mathematics
  - Differential Equations
  - Mathematical Analysis

##### Learning Resource Types

theaters Lecture Videos
notes Lecture Notes
assignment Problem Sets
group_work Projects
Editable Files

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