In the previous set of notes we saw that functions that were holomorphic on an open set enjoyed a large number of useful properties, particularly if the domain was simply connected. 11,858 more words

## Tags » Math.CA

#### 246A, Notes 1: complex differentiation

At the core of almost any undergraduate real analysis course are the concepts of differentiation and integration, with these two basic operations being tied together by the… 6,311 more words

#### The two-dimensional case of the Bourgain-Demeter-Guth proof of the Vinogradov main conjecture

In this blog post, I would like to specialise the arguments of Bourgain, Demeter, and Guth from the previous post to the two-dimensional case of the Vinogradov main conjecture, namely… 1,352 more words

#### Decoupling and the Bourgain-Demeter-Guth proof of the Vinogradov main conjecture

Given any finite collection of elements in some Banach space , the triangle inequality tells us that

However, when the all “oscillate in different ways”, one expects to improve substantially upon the triangle inequality. 3,427 more words