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Blog Profile / Terence Tao's Blog


URL :http://terrytao.wordpress.com/
Filed Under:Academics / Mathematics
Posts on Regator:449
Posts / Week:1.6
Archived Since:December 12, 2008

Blog Post Archive

Polymath8b, X: writing the paper, and chasing down loose ends

This is the tenth thread for the Polymath8b project to obtain new bounds for the quantity ; the previous thread may be found here. Numerical progress on these bounds have slowed in recent months, although we have very recently lowered the unconditional bound on from 252 to 246 (see the wiki page for more detailed results).  While there may […]

The Cayley-Salmon theorem via classical differential geometry

Let be an irreducible polynomial in three variables. As is not algebraically closed, the zero set can split into various components of dimension between and. For instance, if, the zero set is a line; more interestingly, if, then is the union of a line and a surface (or the product of an […]

Metric entropy analogues of sum set theory

A core foundation of the subject now known as arithmetic combinatorics (and particularly the subfield of additive combinatorics) are the elementary sum set estimates (sometimes known as “Ruzsa calculus”) that relate the cardinality of various sum sets and difference sets as well as iterated sumsets such as,, and so forth. Here, are finite […]

Conserved quantities for the surface quasi-geostrophic equation

As in the previous post, all computations here are at the formal level only. In the previous blog post, the Euler equations for inviscid incompressible fluid flow were interpreted in a Lagrangian fashion, and then Noether’s theorem invoked to derive the known conservation laws for this equation. In a bit more detail: starting with Lagrangian […]

Noether’s theorem, and the conservation laws for the Euler equations

Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. (Rigorous justification of the conservation laws and other identities arising from the formal manipulations below can usually be established in an a posteriori fashion once the identities […]

Conserved quantities for the Euler equations

The Euler equations for incompressible inviscid fluids may be written as where is the velocity field, and is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that is bounded. We will take the dimension to be at least two, with the three-dimensional case being of course especially interesting. […]

Polymath8b, IX: Large quadratic programs

This is the ninth thread for the Polymath8b project to obtain new bounds for the quantity either for small values of (in particular ) or asymptotically as. The previous thread may be found here. The currently best known bounds on can be found at the wiki page. The focus is now on bounding unconditionally […]

Polymath8b, VIII: Time to start writing up the results?

This is the eighth thread for the Polymath8b project to obtain new bounds for the quantity either for small values of (in particular ) or asymptotically as. The previous thread may be found here. The currently best known bounds on can be found at the wiki page. The big news since the last thread […]

“New equidistribution estimates of Zhang type, and bounded gaps between primes” – and a retrospective

There are multiple purposes to this blog post. The first purpose is to announce the uploading of the paper “New equidistribution estimates of Zhang type, and bounded gaps between primes” by D.H.J. Polymath, which is the main output of the Polymath8a project on bounded gaps between primes, to the arXiv, and to describe the main […]

Finite time blowup for an averaged three-dimensional Navier-Stokes equation

I’ve just uploaded to the arXiv the paper “Finite time blowup for an averaged three-dimensional Navier-Stokes equation“, submitted to J. Amer. Math. Soc.. The main purpose of this paper is to formalise the “supercriticality barrier” for the global regularity problem for the Navier-Stokes equation, which roughly speaking asserts that it is not possible to establish […]

Polymath8b, VII: Using the generalised Elliott-Halberstam hypothesis to enlarge the sieve support yet further

This is the seventh thread for the Polymath8b project to obtain new bounds for the quantity either for small values of (in particular ) or asymptotically as. The previous thread may be found here. The currently best known bounds on can be found at the wiki page. The current focus is on improving the … … Continue reading ?

Polymath8b, VI: A low dimensional variational problem

This is the sixth thread for the Polymath8b project to obtain new bounds for the quantity either for small values of (in particular ) or asymptotically as. The previous thread may be found here. The currently best known bounds on can be found at the wiki page (which has recently returned to full functionality, […]

Polymath8b, V: Stretching the sieve support further

This is the fifth thread for the Polymath8b project to obtain new bounds for the quantity either for small values of (in particular ) or asymptotically as. The previous thread may be found here. The currently best known bounds on can be found at the wiki page (which has recently returned to full functionality, […]

A sporadic image resizing bug in WordPress LaTeX

I’m encountering a sporadic bug over the past few months with the way WordPress renders or displays its LaTeX images on this blog (and occasionally on other WordPress blogs).  On most computers, it seems to work fine, but on some computers, the sizes of images are occasionally way off, leading to extremely distorted and fairly […]

Polymath8b, IV: Enlarging the sieve support, more efficient numerics, and explicit asymptotics

This is the fourth thread for the Polymath8b project to obtain new bounds for the quantity either for small values of (in particular ) or asymptotically as. The previous thread may be found here. The currently best known bounds on are: (Maynard) Assuming the Elliott-Halberstam conjecture,. (Polymath8b, tentative). Assuming Elliott-Halberstam,. (Polymath8b, […]

Mertens’ theorems

Mertens’ theorems are a set of classical estimates concerning the asymptotic distribution of the prime numbers: Theorem 1 (Mertens’ theorems) In the asymptotic limit, we have and where is the Euler-Mascheroni constant, defined by requiring that in the limit. The third theorem (3) is usually stated in exponentiated form but in the logarithmic […]

Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves

This is the third thread for the Polymath8b project to obtain new bounds for the quantity either for small values of (in particular ) or asymptotically as. The previous thread may be found here. The currently best known bounds on are: (Maynard) Assuming the Elliott-Halberstam conjecture,. (Polymath8b, tentative). Assuming Elliott-Halberstam,. (Polymath8b, […]

Ultraproducts as a Bridge Between Discrete and Continuous Analysis

(This is an extended blog post version of my talk “Ultraproducts as a Bridge Between Discrete and Continuous Analysis” that I gave at the Simons institute for the theory of computing at the workshop “Neo-Classical methods in discrete analysis“. Some of the material here is drawn from previous blog posts, notably “Ultraproducts as a bridge […]

Polymath8b, II: Optimising the variational problem and the sieve

This is the second thread for the Polymath8b project to obtain new bounds for the quantity either for small values of (in particular ) or asymptotically as. The previous thread may be found here. The currently best known bounds on are: (Maynard). (Polymath8b, tentative). (Polymath8b, tentative) for sufficiently large. (Maynard) Assuming […]

Polymath8b: Bounded intervals with many primes, after Maynard

For each natural number, let denote the quantity where denotes the prime. In other words, is the least quantity such that there are infinitely many intervals of length that contain or more primes. Thus, for instance, the twin prime conjecture is equivalent to the assertion that, and the prime tuples conjecture would imply […]

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