### Description

This site is currently hosting\n\nupdates on my mathematical research;\nexpository articles (such as my articles for the Princeton Companion to Mathematics, or for the tricks wiki);\ndiscussion of open problems;\ntalks that I have given or attended (such as the Distinguished Lectures Series at UCLA);\nmy advice on mathematical careers and mathematical writing;\ninformation about my books;\nmy lecture notes on ergodic theory, on the Poincar conjecture, on random matrices, on graduate real analysis (245B and 245C), and on higher order Fourier analysis;\na campaign to support mathematics, statistics, and computing at the University of Southern Queensland;\nand various other topics, usually related to mathematics.\nWhile most of the posts are aimed at those with a graduate maths background, I will also occasionally have a number of non-technical posts aimed at a lay mathematical audience. My selection of topics is guided by my own personal taste; I do not take requests for specific topics to post about on this blog.

### What's new (with Terry Tao)'s Latest Posts

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The Euler equations for three-dimensional incompressible inviscid fluid flow are where is the velocity field, and is the pressure field. For the purposes of this post, we will ignore all issues of decay or regularity of the fields in question, assuming that they are as smooth and rapidly decreasing as needed to justify all the […]

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An extremely large portion of mathematics is concerned with locating solutions to equations such as or for in some suitable domain space (either finite-dimensional or infinite-dimensional), and various maps or . To solve the fixed point iteration equation (1), the simplest general method available is the fixed point iteration method: one starts with an initial […]

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The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space , and is a vector in that Hilbert space, then one has in the strong topology, where is the -invariant subspace of , and is the orthogonal projection to . […]

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Hoi Nguyen, Van Vu, and myself have just uploaded to the arXiv our paper “Random matrices: tail bounds for gaps between eigenvalues“. This is a followup paper to my recent paper with Van in which we showed that random matrices of Wigner type (such as the adjacency graph of an Erd\H{o}s-Renyi graph) almost surely had […]

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We have seen in previous notes that the operation of forming a Dirichlet series or twisted Dirichlet series is an incredibly useful tool for questions in multiplicative number theory. Such series can be viewed as a multiplicative Fourier transform, since the functions and are multiplicative characters. Similarly, it turns out that the operation of forming […]

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