Math 8440 - Spring 2020

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:

More specifically I plan to discuss the following topics:

- Gowers uniformity norms and arithmetic progressions in sets of positive density
- Freimann's theorem: structure of sets with small sumsets

- Randomness of almost primes: the linear forms condition of Green-Tao
- The Green-Tao theorem: arithmetic progressions in the primes
- Bounded gaps between the primes

I. Freiman's theorem and Four term 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

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 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8

II. Hypergraph Regularity and
Arithmetic Progressions in the Primes

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

Lecture 17 Lecture 18 Lecture 19 Lecture 20

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 Szemredi Theorem

Lecture 9 Lecture 10 Lecture 11 Lecture 12 Lecture 13 Lecture 14 Lecture 15 Lecture 16

Conlon-Fox-Zhao: A relative Szemredi Theorem

Lecture 9 Lecture 10 Lecture 11 Lecture 12 Lecture 13 Lecture 14 Lecture 15 Lecture 16

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.

Lecture 17 Lecture 18 Lecture 19 Lecture 20

We discuss the Maynard-Tao theorem on bounded gaps between the primes.

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