Polynomial Method Lecture Notes #2

Following my previous post, this is an update about my lecture notes for the “polynomial method” class that I am currently teaching. In the last post I briefly described the first three chapters of the lecture notes. Since then, four more chapters are already online and another will be uploaded in the next couple of days. The new chapters are:

  • Chapter 4 describes the constant-degree polynomial partitioning technique, which was introduced by Solymosi and Tao. This technique is useful for deriving incidence bounds in higher dimensions. We prove the complex Szemerédi–Trotter theorem using this technique. (I plan to discuss more recent techniques for handling incidences in higher dimensions in a later chapter.)
  • Chapter 5 consists of a short proof for the joints theorem.
  • Chapter 6 describes the Elekes-Sharir-Guth-Katz framework, which reduces the distinct distances problem to a problem involving line intersections in {\mathbb R}^3  .
  • Chapter 7 revolves around incidences with lines in {\mathbb R}^3  . This is a main step in the proof of Guth and Katz’s distinct distances theorem. The proof of this theorem is completed in Chapter 8.
I am happy to receive corrections and suggestions for improvement. Once again, many thanks to Frank de Zeeuw, for providing a lot of great feedback!


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s