Blog Profile / Terence Tao's Blog

Filed Under:Academics / Mathematics
Posts on Regator:431
Posts / Week:1.1
Archived Since:December 12, 2008

Blog Post Archive

A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound

A capset in is a subset of that does not contain any lines. A basic problem in additive combinatorics (discussed in one of the very first posts on this blog) is to obtain good upper and lower bounds for the maximal size of a capset in. Trivially, one has. Using Fourier methods (and […]

Equivalence of the logarithmically averaged Chowla and Sarnak conjectures

I’ve just uploaded to the arXiv my paper “Equivalence of the logarithmically averaged Chowla and Sarnak conjectures, submitted to the Festschrift “Number Theory – Diophantine problems, uniform distribution and applications” in honour of Robert F. Tichy. This paper is a spinoff of my previous paper establishing a logarithmically averaged version of the Chowla (and Elliott) […]

Visualising random variables

When teaching mathematics, the traditional method of lecturing in front of a blackboard is still hard to improve upon, despite all the advances in modern technology.  However, there are some nice things one can do in an electronic medium, such as this blog.  Here, I would like to experiment with the ability to animate images, […]

Notes on the Nash embedding theorem

Throughout this post we shall always work in the smooth category, thus all manifolds, maps, coordinate charts, and functions are assumed to be smooth unless explicitly stated otherwise. A (real) manifold can be defined in at least two ways. On one hand, one can define the manifold extrinsically, as a subset of some standard space […]

L-functions and modular forms database now out of beta

Over the last few years, a large group of mathematicians have been developing an online database to systematically collect the known facts, numerical data, and algorithms concerning some of the most central types of objects in modern number theory, namely the L-functions associated to various number fields, curves, and modular forms, as well as further data about […]

IMU Graduate Breakout Fellowships – Call for Nominations

The International Mathematical Union (with the assistance of the Friends of the International Mathematical Union and The World Academy of Sciences, and supported by Ian Agol, Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Richard Taylor,...Show More Summary

A quick application of the closed graph theorem

In functional analysis, it is common to endow various (infinite-dimensional) vector spaces with a variety of topologies. For instance, a normed vector space can be given the strong topology as well as the weak topology; if the vector space has a predual, it also has a weak- topology. Similarly, spaces of operators have a number […]

Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors; polynomial patterns in primes

Tamar Ziegler and I have just uploaded to the arXiv two related papers: “Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors” and “polynomial patterns in primes“, with the former developing a “quantitative Bessel inequality” for local Gowers norms that is crucial in the latter. Show More Summary

Biases between consecutive primes

There is a very nice recent paper by Lemke Oliver and Soundararajan (complete with a popular science article about it by the consistently excellent Erica Klarreich for Quanta) about a surprising (but now satisfactorily explained) bias in the distribution of pairs of consecutive primes when reduced to a small modulus. This phenomenon is superficially […]

Sumfree sets in groups

Van Vu and I just posted to the arXiv our paper “sum-free sets in groups” (submitted to Discrete Analysis), as well as a companion survey article (submitted to J. Comb.). Given a subset of an additive group, define the quantity to be the cardinality of the largest subset of which is sum-free in in […]

Finite time blowup for high dimensional nonlinear wave systems with bounded smooth nonlinearity

I’ve just uploaded to the arXiv my paper Finite time blowup for high dimensional nonlinear wave systems with bounded smooth nonlinearity, submitted to Comm. PDE. This paper is in the same spirit as (though not directly related to) my previous paper on finite time blowup of supercritical NLW systems, and was inspired by a question […]

Nominations for 2017 Breakthrough and New Horizons Prizes now open

Nominations for the 2017 Breakthrough Prize in mathematics and the New Horizons Prizes in mathematics are now open.  In 2016, the Breakthrough Prize was awarded to Ian Agol.  The New Horizons prizes are for breakthroughs given by junior mathematicians, usually restricted to within 10 years of PhD; the 2016 prizes were awarded to Andre Neves, […]

Discrete Analysis site now live

Just a quick post to note that the arXiv overlay journal Discrete Analysis, managed by Timothy Gowers, has now gone live with its permanent (and quite modern looking) web site, which is run using the Scholastica platform, as well as the first half-dozen or so accepted papers (including one of my own).  See Tim’s announcement […]

Finite time blowup for a supercritical defocusing nonlinear wave system

I’ve just uploaded to the arXiv my paper Finite time blowup for a supercritical defocusing nonlinear wave system, submitted to Analysis and PDE. This paper was inspired by a question asked of me by Sergiu Klainerman recently, regarding whether there were any analogues of my blowup example for Navier-Stokes type equations in the setting of […]

Finite time blowup for an Euler-type equation in vorticity stream form

I’ve been meaning to return to fluids for some time now, in order to build upon my construction two years ago of a solution to an averaged Navier-Stokes equation that exhibited finite time blowup. (I recently spoke on this work in the recent conference in Princeton in honour of Sergiu Klainerman; my slides for that […]

IPAM begins search for new director

The Institute for Pure and Applied Mathematics (IPAM) here at UCLA is seeking applications for its new director in 2017 or 2018, to replace Russ Caflisch, who is nearing the end of his five-year term as IPAM director.  The previous directors of IPAM (Tony Chan, Mark Green, and Russ Caflisch) were also from the mathematics […]

A new polymath proposal: explaining identities for irreducible polynomials

Over on the polymath blog, I’ve posted (on behalf of Dinesh Thakur) a new polymath proposal, which is to explain some numerically observed identities involving the irreducible polynomials in the polynomial ring over the finite field of characteristic two, the simplest of which is (expanded in terms of Taylor series in ).  Comments on the […]

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 Theorem 1 (Two-dimensional Vinogradov main conjecture) One has as. This particular case of the main conjecture has a classical proof using some elementary number […]

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. For instance, if is a Hilbert space and the are mutually orthogonal, we have the Pythagorean theorem For sake of comparison, […]

A conjectural local Fourier-uniformity of the Liouville function

Let denote the Liouville function. The prime number theorem is equivalent to the estimate as, that is to say that exhibits cancellation on large intervals such as. This result can be improved to give cancellation on shorter intervals. For instance, using the known zero density estimates for the Riemann zeta function, one can […]

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