Additive Combinatorics and Number Theory


Math 8440 - Spring 2020


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          NOTE:   Classes will continue online starting March 31st, 11:15-12:15. We will use ZOOM, please register at:

https://zoom.us/meeting/register/vpQsdOCtrD8vYf8XQMqV-zE0ufYP3ptS2Q
Textbook:      There is no official textbook for the course but I will use the Lecture notes and papers listed below.
                      

Description:  Description:  The aim is to give an introduction to some fundamental results of additive combinatorics and number theory.
More specifically I plan to discuss the following topics:




I. Freiman's theorem and Four term arithmetic progressions


We introduce the tools from additive number theory needed for the Fourier analytic proof  (due to Gowers) of Szemeredi's theorem for 4-term arithmetic progressions (AP's).
Notes: K. Soundararjan: Additive Combinatorics

Lecture 1      Lecture 2      Lecture 3      Lecture 4     Lecture 5       Lecture 6        Lecture 7       Lecture 8    Lecture 9    Lecture 10    Lecture 11      Lecture 12     Lecture 13



II. Hypergraph Regularity and Arithmetic Progressions in the Primes

We introduce the (weak) hypergraph regularity lemma and use it to prove the Green-Tao theorem on long arithmetic progressions in the primes. The missing facts from analytic number theory i.e. the so-called linear forms conditions will be discussed afterwards.
Notes: Tao: A variant of the Hypergraph Removal Lemma
            Conlon-Fox-Zhao: A relative Szemeredi Theorem

       Lecture 14    Lecture 15        Lecture 16     Lecture 17      Lecture 18      Lecture 19      Lecture 20


 
III. Sieve theory I. - The linear forms condition

We introduce a variant of the sieve methods due to Goldston-Yildirim to finish the proof the Green-Tao theorem.
Notes: Green-Tao: The primes contain arbitrary long arithmetic progressions

   







                                                                                                         IV. Sieve theory II. - Bounded gaps between the primes

       
We discuss the Maynard-Tao theorem on bounded gaps between the primes.
         Notes: A. Granville: Primes in intervals of bounded length

         Lecture 21      Lecture 22      Lecture 23      Lecture 24     Lecture 25       Lecture 26