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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 … 1,336 more words

Expository

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

Expository

Thanksgiving Latte: Matcha Apple Nut Crisp

Thanksgiving is right around the corner and what better way to celebrate than teatime with family & friends. We have a lovely Matcha latte recipe to share with you, featuring one of our new teas,  182 more words

Tea

275A, Notes 4: The central limit theorem

Let be iid copies of an absolutely integrable real scalar random variable , and form the partial sums . As we saw in the last set of notes… 7,774 more words

Math.CA

275A, Notes 2: Product measures and independence

In the previous set of notes, we constructed the measure-theoretic notion of the Lebesgue integral, and used this to set up the probabilistic notion of expectation on a rigorous footing. 5,661 more words

Math.CA

275A, Notes 0: Foundations of probability theory

Starting this week, I will be teaching an introductory graduate course (Math 275A) on probability theory here at UCLA. While I find myself using probabilistic methods routinely nowadays in my research (for instance, the probabilistic concept of… 7,799 more words

Math.CA