Back to TGINF

# Dense Model Theorem

Speaker: Nic
Title: Dense Model Theorem
Date: 19 Mar 2018 5:30pm-7:00pm
Location: Pierce 320
Food: Indian + Chinese food
Audience background:
Talk style:

Abstract: One of the most celebrated results of 21st century mathematics is arguably the Green-Tao Theorem, which states that the prime numbers contain arbitrarily long arithmetic progressions. At a high level, their proof can be broken down into two steps:

1. A dense model theorem, which roughly states that if R is a “pseudorandom” set in a universe X which contains D as a “dense” subset, then one can find a “model” subset M which is dense in X such that D and M are “indistinguishable” (M is a “dense model” for R). For the Green-Tao theorem, the notion of indistinguishability guarantees that R and M contain roughly the same proportion of arithmetic progressions, and Szemeredi’s theorem guarantees that M contains enough arithmetic progressions.

2. The construction of a pseudorandom set of integers containing of the primes as a dense subset.

I plan to highlight step 1 of this argument, and explain how it relates to certain notions of computational indistinguishability. There is even a nontrivial chance I’ll be able to present most of the proof!

I will mostly be following this paper although I’ve also referred to these two. The original paper by Green and Tao is a bit impenetrable…