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Blog Profile / Terence Tao's Blog

Filed Under:Academics / Mathematics
Posts on Regator:474
Posts / Week:1.5
Archived Since:December 12, 2008

Blog Post Archive

A Banach algebra proof of the prime number theorem

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 […]

Additive limits

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 […]

A trivial generalisation of Cayley’s theorem

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 […]

The “bounded gaps between primes” Polymath project – a retrospective

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...Show More Summary

Derived multiplicative functions

Analytic number theory is often concerned with the asymptotic behaviour of various arithmetic functions: functions or from the natural numbers to the real numbers or complex numbers. In this post, we will focus on the purely algebraic properties of these functions, and for reasons that will become clear later, it will be convenient to […]

Narrow progressions in the primes

Tamar Ziegler and I have just uploaded to the arXiv our paper “Narrow progressions in the primes“, submitted to the special issue “Analytic Number Theory” in honor of the 60th birthday of Helmut Maier. The results here are vaguely reminiscent of the recent progress on bounded gaps in the primes, but uses different methods. About […]

Large gaps between consecutive prime numbers

Kevin Ford, Ben Green, Sergei Konyagin, and myself have just posted to the arXiv our preprint “Large gaps between consecutive prime numbers“. This paper concerns the “opposite” problem to that considered by the recently concluded Polymath8 project, which was concerned with very small values of the prime gap. Here, we wish to consider the […]

(Matilde Lalin) Attending conferences with small children

[This guest post is authored by Matilde Lalin, an Associate Professor in the Département de mathématiques et de statistique at the Université de Montréal.  I have lightly edited the text, mostly by adding some HTML formatting. -T.] Mathematicians...Show More Summary

Khot, Osher, Griffiths

In addition to the Fields medallists mentioned in the previous post, the IMU also awarded the Nevanlinna prize to Subhash Khot, the Gauss prize to Stan Osher (my colleague here at UCLA!), and the Chern medal to Phillip Griffiths. Like I did in 2010, I’ll try to briefly discuss one result of each of the […]

Avila, Bhargava, Hairer, Mirzakhani

The 2014 Fields medallists have just been announced as (in alphabetical order of surname) Artur Avila, Manjul Bhargava, Martin Hairer, and Maryam Mirzakhani (see also these nice video profiles for the winners, which is a new initiative of the IMU and the Simons foundation). This time last year, I wrote a blog post discussing one […]

Variants of the Selberg sieve, and bounded intervals containing many primes

I’ve just uploaded to the arXiv the D.H.J. Polymath paper “Variants of the Selberg sieve, and bounded intervals containing many primes“, which is the second paper to be produced from the Polymath8 project (the first one being discussed here). We’ll refer to this latter paper here as the Polymath8b paper, and the former as the […]

Real analysis relative to a finite measure space

In the traditional foundations of probability theory, one selects a probability space, and makes a distinction between deterministic mathematical objects, which do not depend on the sampled state, and stochastic (or random) mathematical objects, which do depend (but in a measurable fashion) on the sampled state. For instance, a deterministic real number […]

The parity problem obstruction for the binary Goldbach problem with bounded error

Two of the most famous open problems in additive prime number theory are the twin prime conjecture and the binary Goldbach conjecture. They have quite similar forms: Twin prime conjecture The equation has infinitely many solutions with prime. Binary Goldbach conjecture The equation has at least one solution with prime for any given even. […]

Hilbert’s fifth problem and approximate groups

Due to some requests, I’m uploading to my blog the slides for my recent talk in Segovia (for the birthday conference of Michael Cowling) on “Hilbert’s fifth problem and approximate groups“.  The slides cover essentially the same range of topics in this series of lecture notes, or in this text of mine, though of course […]

The subspace theorem approach to Siegel’s theorem on integral points on curves via nonstandard analysis

Let be the algebraic closure of, that is to say the field of algebraic numbers. We fix an embedding of into, giving rise to a complex absolute value for algebraic numbers. Let be of degree, so that is irrational. A classical theorem of Liouville gives the quantitative bound for the irrationality […]

Algebraic probability spaces

As laid out in the foundational work of Kolmogorov, a classical probability space (or probability space for short) is a triplet, where is a set, is a -algebra of subsets of, and is a countably additive probability measure on. Given such a space, one can form a number of interesting function spaces, […]

Lebesgue measure as the invariant factor of Loeb measure

There are a number of ways to construct the real numbers, for instance as the metric completion of (thus, is defined as the set of Cauchy sequences of rationals, modulo Cauchy equivalence); as the space of Dedekind cuts on the rationals ; as the space of quasimorphisms on the integers, quotiented by bounded functions. […]

An abstract ergodic theorem, and the Mackey-Zimmer theorem

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. […]

Polymath8: wrapping up

This should be the final thread (for now, at least) for the Polymath8 project (encompassing the original Polymath8a paper, the nearly finished Polymath8b paper, and the retrospective paper), superseding the previous Polymath8b thread (which was quite full) and the Polymath8a/retrospective thread (which was more or less inactive). Show More Summary

When is correlation transitive?

Given two unit vectors in a real inner product space, one can define the correlation between these vectors to be their inner product, or in more geometric terms, the cosine of the angle subtended by and. By the Cauchy-Schwarz inequality, this is a quantity between and, with the extreme positive correlation occurring […]

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