Blog Profile / Terence Tao's Blog

Filed Under:Academics / Mathematics
Posts on Regator:417
Posts / Week:1.1
Archived Since:December 12, 2008

Blog Post Archive

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 […]

A cheap version of Halasz’s inequality

A basic estimate in multiplicative number theory (particularly if one is using the Granville-Soundararajan “pretentious” approach to this subject) is the following inequality of Halasz (formulated here in a quantitative form introduced by Montgomery and Tenenbaum. Theorem 1 (Halasz inequality) Let be a multiplicative function bounded in magnitude by, and suppose that,, […]

275A, Notes 5: Variants of the central limit theorem

In the previous set of notes we established the central limit theorem, which we formulate here as follows: Theorem 1 (Central limit theorem) Let be iid copies of a real random variable of mean and variance, and write. Then, for any fixed, we have as. This is however not the end […]

Chains of large gaps between primes

Kevin Ford, James Maynard, and I have uploaded to the arXiv our preprint “Chains of large gaps between primes“. This paper was announced in our previous paper with Konyagin and Green, which was concerned with the largest gap between consecutive primes up to, in which we improved the Rankin bound of to for large […]

Klaus Roth

Klaus Roth, who made fundamental contributions to analytic number theory, died this Tuesday, aged 90. I never met or communicated with Roth personally, but was certainly influenced by his work; he wrote relatively few papers, but they tended to have outsized impact. For instance, he was one of the key people (together with Bombieri) to […]

Analytic number theory program at MSRI: Jan-May 2017 (second announcement)

Chantal David, Andrew Granville, Emmanuel Kowalski, Phillipe Michel, Kannan Soundararajan, and I are running a program at MSRI in the Spring of 2017 (more precisely, from Jan 17, 2017 to May 26, 2017) in the area of analytic number theory, with the intention to bringing together many of the leading experts in all aspects of the […]

275A, Notes 4: The central limit theorem

Let be iid copies of an absolutely integrable real scalar random variable, and form the partial sums. As we saw in the last set of notes, the law of large numbers ensures that the empirical averages converge (both in probability and almost surely) to a deterministic limit, namely the mean of the reference […]

A small remark on the Elliott conjecture

The Chowla conjecture asserts, among other things, that one has the asymptotic as for any distinct integers, where is the Liouville function. (The usual formulation of the conjecture also allows one to consider more general linear forms than the shifts, but for sake of discussion let us focus on the shift case.) This […]

275A, Notes 3: The weak and strong law of large numbers

One of the major activities in probability theory is studying the various statistics that can be produced from a complex system with many components. One of the simplest possible systems one can consider is a finite sequence or an infinite sequence of jointly independent scalar random variables, with the case when the are also identically […]

275A, Notes 2: Product measures and independence

In the previous set of notes, we constructed the measure-theoretic notion of the Lebesgue integral, and used this to set up the probabilistic notion of expectation on a rigorous footing. In this set of notes, we will similarly construct the measure-theoretic concept of a product measure (restricting to the case of probability measures to avoid […]

Sweeping a matrix rotates its graph

I recently learned about a curious operation on square matrices known as sweeping, which is used in numerical linear algebra (particularly in applications to statistics), as a useful and more robust variant of the usual Gaussian elimination operations seen in undergraduate linear algebra courses. Given an matrix (with, say, complex entries) and an index, […]

275A, Notes 1: Integration and expectation

In Notes 0, we introduced the notion of a measure space, which includes as a special case the notion of a probability space. By selecting one such probability space as a sample space, one obtains a model for random events and random variables, with random events being modeled by measurable sets in, and […]

275A, Notes 0: Foundations of probability theory

Starting next week, I will be teaching an introductory graduate course (Math 275A) on probability theory here at UCLA. While I find myself using probabilistic methods routinely nowadays in my research (for instance, the probabilistic concept of Shannon entropy played a crucial role in my recent paper on the Chowla and Elliott conjectures, and random […]

Entropy and rare events

Let and be two random variables taking values in the same (discrete) range, and let be some subset of, which we think of as the set of “bad” outcomes for either or. If and have the same probability distribution, then clearly In particular, if it is rare for to lie in, […]

The logarithmically averaged Chowla and Elliott conjectures for two-point correlations; the Erdos discrepancy problem

I’ve just uploaded two related papers to the arXiv: The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, submitted to Forum of Mathematics, Pi; and The Erdos discrepancy problem, submitted to the new arXiv overlay journal, Discrete Analysis (see this recent announcement on Tim Gowers’ blog). Show More Summary

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