\contentsline {section}{\tocsection {}{1}{Introduction and overview}}{3}
\contentsline {subsection}{\tocsubsection {}{1.1}{What is projection theory?}}{3}
\contentsline {subsection}{\tocsubsection {}{1.2}{Applications of projection theory}}{5}
\contentsline {section}{\tocsection {}{2}{Fundamental methods of projection theory}}{8}
\contentsline {subsection}{\tocsubsection {}{2.1}{Double Counting}}{10}
\contentsline {subsection}{\tocsubsection {}{2.2}{Orthogonality / Fourier method}}{11}
\contentsline {subsection}{\tocsubsection {}{2.3}{Projection theory for balls in Euclidean space}}{15}
\contentsline {section}{\tocsection {}{3}{Projection theory for balls in Euclidean space}}{18}
\contentsline {subsection}{\tocsubsection {}{3.1}{Fourier Method}}{22}
\contentsline {section}{\tocsection {}{4}{The Fourier method in Euclidean space}}{25}
\contentsline {subsection}{\tocsubsection {}{4.1}{Sieve Theory}}{31}
\contentsline {section}{\tocsection {}{5}{The large sieve}}{33}
\contentsline {subsection}{\tocsubsection {}{5.1}{The Large Sieve}}{33}
\contentsline {subsection}{\tocsubsection {}{5.2}{Proof of Linnik's Large Sieve inequality}}{36}
\contentsline {section}{\tocsection {}{6}{Projections and smoothing}}{44}
\contentsline {section}{\tocsection {}{7}{Applications of the Large Sieve to Number Theory}}{46}
\contentsline {subsection}{\tocsubsection {}{7.1}{Distribution of primes mod $q$}}{46}
\contentsline {subsection}{\tocsubsection {}{7.2}{Multiplicative Convolution and Primes}}{47}
\contentsline {subsection}{\tocsubsection {}{7.3}{Multiplicative Convolution and Projections}}{47}
\contentsline {subsection}{\tocsubsection {}{7.4}{Large Sieve and Multiplicative Convolution}}{49}
\contentsline {section}{\tocsection {}{8}{The Szemeredi-Trotter theorem}}{51}
\contentsline {subsection}{\tocsubsection {}{8.1}{The Szemeredi-Trotter projection theorem}}{51}
\contentsline {subsection}{\tocsubsection {}{8.2}{Question: Are there other sharp examples for the Szemerédi–Trotter theorem?}}{53}
\contentsline {subsection}{\tocsubsection {}{8.3}{Proof of the Szemeredi-Trotter theorem}}{53}
\contentsline {section}{\tocsection {}{9}{Reflections on the Szemeredi-Trotter theorem}}{59}
\contentsline {subsection}{\tocsubsection {}{}{Structure of Sharp Examples}}{59}
\contentsline {subsection}{\tocsubsection {}{}{Projection Theory over Finite Fields}}{61}
\contentsline {subsection}{\tocsubsection {}{}{Projection Theory of Unit Balls}}{61}
\contentsline {section}{\tocsection {}{10}{Sum-product theory}}{62}
\contentsline {subsection}{\tocsubsection {}{10.1}{Sum-product theory}}{62}
\contentsline {subsection}{\tocsubsection {}{10.2}{Freiman-Ruzsa theorem}}{63}
\contentsline {subsection}{\tocsubsection {}{10.3}{Ruzsa triangle inequality}}{64}
\contentsline {subsection}{\tocsubsection {}{10.4}{Plunnecke inequality}}{64}
\contentsline {section}{\tocsection {}{11}{Contagious structure in projection theory}}{67}
\contentsline {subsection}{\tocsubsection {}{11.1}{Contagious Structure Lemma}}{67}
\contentsline {section}{\tocsection {}{12}{Proof of Bourgain-Katz-Tao projection theorem}}{70}
\contentsline {subsection}{\tocsubsection {}{12.1}{Proof of BKT}}{70}
\contentsline {subsection}{\tocsubsection {}{12.2}{Additive energy and robust estimates}}{72}
\contentsline {section}{\tocsection {}{13}{Proof of the Balog-Szemeredi-Gowers Theorem}}{74}
\contentsline {subsection}{\tocsubsection {}{13.1}{Simple Bounds About $P_K(a, b)$}}{75}
\contentsline {section}{\tocsection {}{14}{Bourgain's projection theorem over $\mathbb {R}$, part 1}}{78}
\contentsline {subsection}{\tocsubsection {}{14.1}{Finite field case}}{78}
\contentsline {subsection}{\tocsubsection {}{14.2}{Real case}}{78}
\contentsline {section}{\tocsection {}{15}{Bourgain's Projection Theorem II}}{81}
\contentsline {section}{\tocsection {}{16}{Bourgain's projection theorem III}}{87}
\contentsline {subsection}{\tocsubsection {}{16.1}{Why robust estimates are useful}}{87}
\contentsline {subsection}{\tocsubsection {}{16.2}{How to prove robust estimates}}{89}
\contentsline {section}{\tocsection {}{17}{Random walks on finite groups I}}{91}
\contentsline {section}{\tocsection {}{18}{Random walks on finite groups II}}{99}
\contentsline {section}{\tocsection {}{19}{Random walks on finite groups III}}{107}
\contentsline {section}{\tocsection {}{20}{Homogenenous Dynamics I}}{108}
\contentsline {section}{\tocsection {}{21}{Homogeneous Dynamics II}}{112}
\contentsline {section}{\tocsection {}{22}{Sharp Projection Theorems I: Introduction and Beck's Theorem}}{118}
\contentsline {subsection}{\tocsubsection {}{22.1}{History of the Furstenberg conjecture}}{119}
\contentsline {subsection}{\tocsubsection {}{22.2}{Outline of OSRW proof of Furstenberg conjecture}}{123}
\contentsline {section}{\tocsection {}{23}{Sharp Projection Theorems II: AD Regular Case}}{126}
\contentsline {subsection}{\tocsubsection {}{23.1}{Brute force proof}}{127}
\contentsline {subsection}{\tocsubsection {}{23.2}{General AD vs Projective AD}}{128}
\contentsline {subsection}{\tocsubsection {}{23.3}{Sketch of the proof for the AD regular case}}{129}
\contentsline {section}{\tocsection {}{24}{Sharp Projection Theorems III: Combining different scales}}{133}
\contentsline {subsection}{\tocsubsection {}{24.1}{Well spaced case}}{133}
\contentsline {subsection}{\tocsubsection {}{24.2}{Combining scales}}{136}
