### 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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I’ve just uploaded to the arXiv my paper “The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture“. As the title suggests, this paper links together the Elliott-Halberstam conjecture from sieve theory with the conjecture of Vinogradov concerning the least quadratic nonresidue of a prime . Vinogradov established the bound and conjectured that for any […]

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The prime number theorem can be expressed as the assertion as , where is the von Mangoldt function. It is a basic result in analytic number theory, but requires a bit of effort to prove. One “elementary” proof of this theorem proceeds through the Selberg symmetry formula where the second von Mangoldt function is defined […]

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In graph theory, the recently developed theory of graph limits has proven to be a useful tool for analysing large dense graphs, being a convenient reformulation of the Szemerédi regularity lemma. Roughly speaking, the theory asserts that given any sequence of finite graphs, one can extract a subsequence which converges (in a specific sense) to […]

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One of the first basic theorems in group theory is Cayley’s theorem, which links abstract finite groups with concrete finite groups (otherwise known as permutation groups). Theorem 1 (Cayley’s theorem) Let be a group of some finite order . Then is isomorphic to a subgroup of the symmetric group on elements . Furthermore, this subgroup […]

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The (presumably) final article arising from the Polymath8 project has now been uploaded to the arXiv as “The “bounded gaps between primes” Polymath project – a retrospective“. This article, submitted to the Newsletter of the European Mathematical Society, consists of personal contributions from ten different participants (at varying levels of stage of career, and intensity […]

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