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

Posts on Regator: | 448 | |

Posts / Week: | 1.1 | |

Archived Since: | December 12, 2008 |

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 […]

Having discussed differentiation of complex mappings in the preceding notes, we now turn to the integration of complex maps. We first briefly review the situation of integration of (suitably regular) real functions of one variable. Actually there are three closely related concepts of integration that arise in this setting: (i) The signed definite integral, […]

At the core of almost any undergraduate real analysis course are the concepts of differentiation and integration, with these two basic operations being tied together by the fundamental theorem of calculus (and its higher dimensional generalisations, such as Stokes’ theorem). Similarly, the notion of the complex derivative and the complex line integral (that is to […]

Kronecker famously wrote, “God created the natural numbers; all else is the work of man”. The truth of this statement (literal or otherwise) is debatable; but one can certainly view the other standard number systems as (iterated) completions of the natural numbers in various senses. For instance: The integers are the additive completion of the […]

Next week, I will be teaching Math 246A, the first course in the three-quarter graduate complex analysis sequence. This first course covers much of the same ground as an honours undergraduate complex analysis course, in particular focusing on the basic properties of holomorphic functions such as the Cauchy and residue theorems, the classification of singularities, […]

This is a postscript to the previous blog post which was concerned with obtaining heuristic asymptotic predictions for the correlation for the divisor function, in particular recovering the calculation of Ingham that obtained the asymptotic when was fixed and non-zero and went to infinity. It is natural to consider the more general […]

Let be the divisor function. A classical application of the Dirichlet hyperbola method gives the asymptotic where denotes the estimate as. Much better error estimates are possible here, but we will not focus on the lower order terms in this discussion. For somewhat idiosyncratic reasons I will interpret this estimate (and the other analytic […]

Fifteen years ago, I wrote a paper entitled Global regularity of wave maps. II. Small energy in two dimensions, in which I established global regularity of wave maps from two spatial dimensions to the unit sphere, assuming that the initial data had small energy. Recently, Hao Jia (personal communication) discovered a small gap in the […]

[This blog post was written jointly by Terry Tao and Will Sawin.] In the previous blog post, one of us (Terry) implicitly introduced a notion of rank for tensors which is a little different from the usual notion of tensor rank, and which (following BCCGNSU) we will call “slice rank”. This notion of rank could […]

The twin prime conjecture, still unsolved, asserts that there are infinitely many primes such that is also prime. A more precise form of this conjecture is (a special case) of the Hardy-Littlewood prime tuples conjecture, which asserts that as, where is the von Mangoldt function and is the twin prime constant Because is almost […]

I’ve just posted to the arXiv my paper “Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation“. This paper is loosely in the spirit of other recent papers of mine in which I explore how close one can get to supercritical PDE of physical interest (such as the Euler and Navier-Stokes equations), while […]

In logic, there is a subtle but important distinction between the concept of mutual knowledge – information that everyone (or almost everyone) knows – and common knowledge, which is not only knowledge that (almost) everyone knows, but something that (almost) everyone knows that everyone else knows (and that everyone knows that everyone else knows that […]

Note: the following is a record of some whimsical mathematical thoughts and computations I had after doing some grading. It is likely that the sort of problems discussed here are in fact well studied in the appropriate literature; I would appreciate knowing of any links to such. Suppose one assigns true-false questions on an examination, […]

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 […]

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) […]

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, […]

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