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\begin{document}

\title{Projection theory: class titles and descriptions}

% Revised class titles and descriptions

\maketitle

Title of the class as a whole: Projection theory

Description: 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.

Class 1: Introduction to projection theory

Description:  We formulate the main questions and goals of the class,  especially the exceptional set problem.  

Keywords: Projection theory,  Marstrand,  Kaufman,  Furstenberg,  

Class 2: Fundamental methods of projection theory

Description: We introduce two fundamental methods to study projections, developed by Kaufman and Falconer in the 1960s and 70s.   The first is based on double counting.   The second is based on Fourier analysis.   

Keywords: Projection theory,  Fourier analysis,  Kaufman,  Falconer,  Vinh,  finite field

Class 3: Projection theory in Euclidean space

Description: We adapt the fundamental methods of projection theory to Euclidean space,  taking account of multiple scales.  

Keywords: Projection theory,  Marstrand,  Kaufman,  multiscale analysis,  Frostman,  Hausdorff

Class 4: The Fourier method in Euclidean space

Description: We adapt the Fourier method to Euclidean space,  taking into account many different scales.

Keyword: Projection theory,  Fourier,  Kaufman,  Falconer

Class 5: The large sieve

Description: We introduce a classical topic from number theory which is closely related to projection theory.

Keywords: Large sieve,  sieve theory,  Linnik,  projection theory

Class 6: Projections and smoothing

Description: The projection of a rough function at a typical angle is usually smoother than the original function.   

Keywords: Projection theory,  smoothing,  Falconer

Class 7: Applications of the large sieve to number theory

Description: We discuss one of the main applications of the large sieve in number theory: the Bombieri-Vinogradov theorem about the distribution of prime numbers modulo $q$ for different $q$.

Keywords: Large sieve, projection theory,  Bombieri-Vinogradov, 

Class 8: The Szemeredi-Trotter theorem

Description: The Szemeredi-Trotter theorem gives the sharp answer to a natural discrete projection problem in the plane.   The proof of the theorem is based on topology,  and it is completely different from the proofs we have explored earlier.

Keywords: Incidence geometry,  projection theory,  Szemeredi-Trotter,  

Class 9: Reflections on the Szemeredi-Trotter theorem

Description: We discuss the analogy between the Szemeredi-Trotter theorem and the exceptional set problem.   We explain why it is hard to adapt the proof of Szemeredi-Trotter to the exceptional set problem.  

Keywords: Szemeredi-Trotter,  Furstenberg,  Marstrand,  projection theory,  complex numbers

Class 10: Sum-product theory

Description: We introduce sum-product theory and tools from additive number theory.

Keywrods: sum-product,  Plunnecke, Ruzsa and Edgar-Miller.  Bourgain-Katz-Tao.

Class 11: Contagious structure in projection theory

Description: We study algebraic structure in the set of exceptional directions.

Keywords: Bourgain-Katz-Tao,  sum-product,  projection theory,  Plunnecke,  Ruzsa,  Edgar-Miller

Class 12: The Bourgain-Katz-Tao projection theorem

Description: The Bourgain-Katz-Tao projection theorem, from the early 2000s,  is the first result in projection theory that really goes beyond the two fundamental techniques we introduced in Lecture 2.  

Keywords: Bourgain-Katz-Tao, projection theory,  finite field

Class 13: The Balog-Szemeredi-Gowers theorem

Description: We develop the last tool in our toolkit from combinatorial number theory. 

Keywords: Balog-Szemeredi-Gowers,  approximate subgroup,  additive energy,  additive number theory. 

Class 14: The Bourgain projection theorem (over the real numbers)

Description: We introduce Bourgain's projection theorem,  a central result in the field.   It is the analogue of the Bourgain-Katz-Tao projection theorem over the real numbers.

Keywords: Bourgain projection theorem,  projection theory,  multiscale,  Furstenberg

Class 15: The Bourgain projection theorem 2

Description: We introduce the key idea of uniform sets and branching functions.

Keywords: Bourgain projection theorem,  projection theory,  multiscale,  Furstenberg

Class 16: The Bourgain projection theorem 3

Description: We discuss the role of robust estimates in multiscale arguments.

Keywords: Bourgain projection theorem,  projection theory,  multiscale,  Furstenberg

Class 17: Random walks on finite groups 1

Description: We give a friendly introduction to the field of random walks on finite groups.   We discuss theorems of Selberg and Bourgain-Gamburd.  

Keywords: Random walk,  mixing,  finite group,  Selberg,  Sarnak,  Bourgain-Gamburd,  sum product

Class 18: Random walks on finite groups 2

Description: We sketch the proof of Selberg's random walk estimate,  following the approach of Sarnak-Xue.

Keywords: Random walk,  mixing,  finite group,  Selberg,  Sarnak,  Bourgain-Gamburd,  sum product

Class 19: Random walks on finite groups 3

Description: We discuss the results of Bourgain-Gamburd and some of the proof ideas.   

Keywords: Random walk,  mixing,  finite group,  Selberg,  Sarnak,  Bourgain-Gamburd,  sum product

Class 20: Homogeneous dynamics part 1

Description: We give a friendly introduction to the field of homogeneous dynamics and how projection theory can help understand it.

Keywords: Mixing,  unipotent orbit,  Margulis,  Ratner,  Lindenstrauss,  Oppenheim,  projection theory

Class 21: Homogeneous dynamics part 2

Description: We discuss the work of Lindenstrauss-Mohammadi on quantitative Ratner theorems,  and how projection theory plays a role in the proof. 

Keywords: Mixing,  unipotent orbit,  Margulis,  Ratner,  Lindenstrauss,  Oppenheim,  projection theory

Class 22: Sharp projection theorems part 1: Introduction and Beck's theorem.

Description: We begin studying the recent breakthrough in projection theory by  Orponen-Shmerkin and Ren-Wang.    We discuss how to use a non-sharp bound many times to get a sharp bound.

Keywords: Furstenberg,  projection,  Wolff,  Orponen-Shmerkin,  Ren-Wang,  

Class 23: Sharp projection theorems part 2: AD regular case

Description:  Orponen and Shmerkin proved a sharp projection theorem for sets with self-similar spacing.  

Keywords: Furstenberg,  projection,  Wolff,  Orponen-Shmerkin,  Ren-Wang,  

Class 24: Sharp projection theorems part 3: Combining different scales

Description: Ren and Wang proved the full Furstenberg conjecture by applying the AD regular case at many different scales.

Keywords: Furstenberg,  projection,  Wolff,  Orponen-Shmerkin,  Ren-Wang,  














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