URL : | http://terrytao.wordpress.com/ | |
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Filed Under: | Academics / Mathematics | |

Posts on Regator: | 463 | |

Posts / Week: | 1.1 | |

Archived Since: | December 12, 2008 |

Suppose is a continuous (but nonlinear) map from one normed vector space to another. The continuity means, roughly speaking, that if are such that is small, then is also small (though the precise notion of “smallness” may depend on or, particularly if is not known to be uniformly continuous). If is known to […]

Suppose one has a bounded sequence of real numbers. What kinds of limits can one form from this sequence? Of course, we have the usual notion of limit, which in this post I will refer to as the classical limit to distinguish from the other limits discussed in this point. The classical limit, if […]

Ben Green and I have (finally!) uploaded to the arXiv our paper “New bounds for Szemerédi’s theorem, III: A polylogarithmic bound for “, submitted to Mathematika. This is the sequel to two previous papers (and an erratum to the former paper), concerning quantitative versions of Szemerédi’s theorem in the case of length four progressions. This […]

A sequence of complex numbers is said to be quasiperiodic if it is of the form for some real numbers and continuous function. For instance, linear phases such as (where ) are examples of quasiperiodic sequences; the top order coefficient (modulo ) can be viewed as a “frequency” of the integers, and an element […]

How many groups of order four are there? Technically, there are an enormous number, so much so, in fact, that the class of groups of order four is not even a set, but merely a proper class. This is because any four objects can be turned into a group by designating one of the four […]

Just a short post to note that Norwegian Academy of Science and Letters has just announced that the 2017 Abel prize has been awarded to Yves Meyer, “for his pivotal role in the development of the mathematical theory of wavelets”. The actual prize ceremony will be at Oslo in May. I am actually in Oslo […]

Given a function on the natural numbers taking values in, one can invoke the Furstenberg correspondence principle to locate a measure preserving system – a probability space together with a measure-preserving shift (or equivalently, a measure-preserving -action on ) – together with a measurable function (or “observable”) that has essentially the same statistics as […]

Given a random variable that takes on only finitely many values, we can define its Shannon entropy by the formula with the convention that. (In some texts, one uses the logarithm to base rather than the natural logarithm, but the choice of base will not be relevant for this discussion.) This is clearly a […]

Daniel Kane and I have just uploaded to the arXiv our paper “A bound on partitioning clusters“, submitted to the Electronic Journal of Combinatorics. In this short and elementary paper, we consider a question that arose from biomathematical applications: given a finite family of sets (or “clusters”), how many ways can there be of partitioning […]

The self-chosen remit of my blog is “Updates on my research and expository papers, discussion of open problems, and other maths-related topics”. Of the 774 posts on this blog, I estimate that about 99% of the posts indeed relate to mathematics, mathematicians, or the administration of this mathematical blog, and only about 1% are not […]

I’ve just uploaded to the arXiv my paper “Some remarks on the lonely runner conjecture“, submitted to Contributions to discrete mathematics. I had blogged about the lonely runner conjecture in this previous blog post, and I returned to the problem recently to see if I could obtain anything further. The results obtained were more modest […]

I just learned (from Emmanuel Kowalski’s blog) that the AMS has just started a repository of open-access mathematics lecture notes. There are only a few such sets of notes there at present, but hopefully it will grow in the future; I just submitted some old lecture notes of mine from an undergraduate linear algebra course […]

I’ve just uploaded to the arXiv my paper Finite time blowup for a supercritical defocusing nonlinear Schrödinger system, submitted to Analysis and PDE. This paper is an analogue of a recent paper of mine in which I constructed a supercritical defocusing nonlinear wave (NLW) system which exhibited smooth solutions that developed singularities in finite time. […]

[This guest post is authored by Caroline Series.] The Chern Medal is a relatively new prize, awarded once every four years jointly by the IMU and the Chern Medal Foundation (CMF) to an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics. Funded by the CMF, the […]

I’ve just uploaded to the arXiv my paper “An integration approach to the Toeplitz square peg problem“, submitted to Forum of Mathematics, Sigma. This paper resulted from my attempts recently to solve the Toeplitz square peg problem (also known as the inscribed square problem): Conjecture 1 (Toeplitz square peg problem) Let be a simple closed […]

By an odd coincidence, I stumbled upon a second question in as many weeks about power series, and once again the only way I know how to prove the result is by complex methods; once again, I am leaving it here as a challenge to any interested readers, and I would be particularly interested in […]

In the previous set of notes we introduced the notion of a complex diffeomorphism between two open subsets of the complex plane (or more generally, two Riemann surfaces): an invertible holomorphic map whose inverse was also holomorphic. (Actually, the last part is automatic, thanks to Exercise 40 of Notes 4.) Such maps are also known […]

My colleague Tom Liggett recently posed to me the following problem about power series in one real variable. Observe that the power series has very rapidly decaying coefficients (of order ), leading to an infinite radius of convergence; also, as the series converges to, the series decays very rapidly as approaches. The […]

In the previous set of notes we saw that functions that were holomorphic on an open set enjoyed a large number of useful properties, particularly if the domain was simply connected. In many situations, though, we need to consider functions that are only holomorphic (or even well-defined) on most of a domain, thus they […]

We now come to perhaps the most central theorem in complex analysis (save possibly for the fundamental theorem of calculus), namely Cauchy’s theorem, which allows one to compute a large number of contour integrals even without knowing any explicit antiderivative of. There are many forms and variants of Cauchy’s theorem. To give one such […]

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