PDF files

• • An older draft of my book “Incidence Theory” (with a focus on the polynomial method). For the published version, see here.
  • Polynomial method lecture notes (with a focus on incidences). These are mostly subsumed by the above book. The main exception is a proof of the complete Guth-Katz distinct distances theorem that involves ruled surfaces and flat points.
    • Chapter 1. Introduction to geometric incidences, related problems in discrete geometry, and first applications.
    • Chapter 2. Some basic real algebraic geometry.
    • Chapter 3. Polynomial partitioning and how to use it to obtain incidence bounds.
    • Chapter 4. Constant-sized polynomial partitioning.
    • Chapter 5. The joints problem.
    • Chapter 6. The Elekes-Sharir-Guth-Katz
      Framework.
    • Chapter 7. Lines in .
    • Chapter 8. Distinct intersection points (finishing the Guth-Katz distinct distances problem).
    • Chapter 9. More distinct distances problems.
  • Additive combinatorics lecture notes:
    • Chapter 1. Small doubling.
    • Chapter 2. The sum product problem.
    • Chapter 3. The Balog-Szemerédi-Gowers theorem.
    • Chapter 4. Arithmetic progressions in dense sets.
    • Chapter 5. The quasi-polynomial Freiman-Ruzsa.
    • Chapter 6. Third moment energy (incomplete).
    • (Chapter 0. Preliminaries.)
  • The Konyagin-Shkredov sum-product bound – The proof of the sum-product bound of Konyagin and Shkredov, explained in detail.

• • Distinct Distances: Open Problems and Current Bounds – This is my ongoing attempt to survey the many open variants of the distinct distances problem, and the best known bounds for them. Occasionally I also include proofs, if they are short and elegant.

• The second-partitioning-polynomial technique.

Comments, questions, and complaints are welcome.