Blog Profile / Terence Tao's Blog

Filed Under:Academics / Mathematics
Posts on Regator:492
Posts / Week:1
Archived Since:December 12, 2008

Blog Post Archive

The De Bruijn-Newman constant is non-negative

Brad Rodgers and I have uploaded to the arXiv its paper “The De Bruijn-Newman constant is non-negative“. This paper affirms a conjecture of Newman regarding to the extent to which the Riemann hypothesis, if true, is only “barely so”. To describe the conjecture, let us begin with the Riemann xi function where is the Gamma […]

Homogeneous length functions on groups

The Polymath14 online collaboration has uploaded to the arXiv its paper “Homogeneous length functions on groups“, submitted to Algebra & Number Theory. The paper completely classifies homogeneous length functions on an arbitrary group, that is to say non-negative functions that obey the symmetry condition, the non-degeneracy condition, the triangle inequality, and […]

Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges

Kaisa Matomaki, Maksym Radziwill, and I have uploaded to the arXiv our paper “Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges“. This is a sequel of sorts to our previous paper on divisor correlations, though the proof techniques in this paper are rather different. As with the previous […]

Metrics of linear growth – the solution

In the tradition of “Polymath projects”, the problem posed in the previous two blog posts has now been solved, thanks to the cumulative effect of many small contributions by many participants (including, but not limited to, Sean Eberhard, Tobias Fritz, Siddharta Gadgil, Tobias Hartnick, Chris Jerdonek, Apoorva Khare, Antonio Machiavelo, Pace Nielsen, Andy Putman, Will […]

Bi-invariant metrics of linear growth on the free group, II

This post is a continuation of the previous post, which has attracted a large number of comments. I’m recording here some calculations that arose from those comments (particularly those of Pace Nielsen, Lior Silberman, Tobias Fritz, and Apoorva Khare). Please feel free to either continue these calculations or to discuss other approaches to the problem, […]

Bi-invariant metrics of linear growth on the free group

Here is a curious question posed to me by Apoorva Khare that I do not know the answer to. Let be the free group on two generators. Does there exist a metric on this group which is bi-invariant, thus for all ; and linear growth in the sense that for all and all natural […]

Embedding the Boussinesq equations in the incompressible Euler equations on a manifold

The Boussinesq equations for inviscid, incompressible two-dimensional fluid flow in the presence of gravity are given by where is the velocity field, is the pressure field, and is the density field (or, in some physical interpretations, the temperature field). In this post we shall restrict ourselves to formal manipulations, assuming implicitly that all fields are […]

An inverse theorem for Kemperman’s inequality

I have just uploaded to the arXiv the paper “An inverse theorem for Kemperman’s inequality“, submitted to a special issue of the Proceedings of the Steklov Institute of Mathematics in honour of Sergei Konyagin. It concerns an inequality of Kemperman discussed previously in this blog, namely that whenever are compact non-empty subsets of a compact […]

Continuous approximations to arithmetic functions

A basic object of study in multiplicative number theory are the arithmetic functions: functions from the natural numbers to the complex numbers. Some fundamental examples of such functions include The constant function ; The Kronecker delta function ; The natural logarithm function ; The divisor function ; The von Mangoldt function, with defined to […]

IPAM program in quantitative linear algebra, Mar 19-Jun 15 2018

Alice Guionnet, Assaf Naor, Gilles Pisier, Sorin Popa, Dimitri Shylakhtenko, and I are organising a three month program here at the Institute for Pure and Applied Mathematics (IPAM) on the topic of Quantitative Linear Algebra.  The purpose of this program is to bring together mathematicians and computer scientists (both junior and senior) working in various […]

UCLA Math Undergraduate Merit Scholarship for 2018

In 2010, the UCLA mathematics department launched a scholarship opportunity for entering freshman students with exceptional background and promise in mathematics. We are able to offer one scholarship each year.  The UCLA Math Undergraduate...Show More Summary

The logarithmically averaged and non-logarithmically averaged Chowla conjectures

Let be the Liouville function, thus is defined to equal when is the product of an even number of primes, and when is the product of an odd number of primes. The Chowla conjecture asserts that has the statistics of a random sign pattern, in the sense that for all and all distinct natural numbers […]

Heat flow and zeroes of polynomials

Let be a monic polynomial of degree with complex coefficients. Then by the fundamental theorem of arithmetic, we can factor as for some complex zeroes (possibly with repetition). Now suppose we evolve with respect to time by heat flow, creating a function of two variables for which On the space of polynomials of degree at […]

Odd order cases of of the logarithmically averaged Chowla conjecture

Joni Teräväinen and I have just uploaded to the arXiv our paper “Odd order cases of of the logarithmically averaged Chowla conjecture“, submitted to J. Numb. Thy. Bordeaux. This paper gives an alternate route to one of the main results of our previous paper, and more specifically reproves the asymptotic for all odd and all […]

An update to “On the sign patterns of entrywise positivity preservers in fixed dimension”

Apoorva Khare and I have updated our paper “On the sign patterns of entrywise positivity preservers in fixed dimension“, announced at this post from last month. The quantitative results are now sharpened using a new monotonicity property of ratios of Schur polynomials, namely that such ratios are monotone non-decreasing in each coordinate of if is […]

Inverting the Schur complement, and large-dimensional Gelfand-Tsetlin patterns

Suppose we have an matrix that is expressed in block-matrix form as where is an matrix, is an matrix, is an matrix, and is a matrix for some. If is invertible, we can use the technique of Schur complementation to express the inverse of (if it exists) in terms of the inverse of, […]

Szemeredi’s proof of Szemeredi’s theorem

Szemerédi’s theorem asserts that all subsets of the natural numbers of positive density contain arbitrarily long arithmetic progressions.  Roth’s theorem is the special case when one considers arithmetic progressions of length three....Show More Summary

Continuous analogues of the Schur and skew Schur polynomials

Fix a non-negative integer. Define an integer partition of length to be a tuple of non-increasing non-negative integers. To each such partition, one can associate Young diagram consisting of left-justified rows of boxes, with the row containing boxes. A semi-standard Young tableau (or Young tableau for short) of shape is a filling […]

Dodgson condensation from Schur complementation

The determinant of an matrix (with coefficients in an arbitrary field) obey many useful identities, starting of course with the fundamental multiplicativity for matrices. This multiplicativity can in turn be used to establish many further identities; in particular, as shown in this previous post, it implies the Schur determinant identity whenever is an invertible […]

An addendum to “arbitrage, amplification, and the tensor power trick”

In one of the earliest posts on this blog, I talked about the ability to “arbitrage” a disparity of symmetry in an inequality, and in particular to “amplify” such an inequality into a stronger one. (The principle can apply to other mathematical statements than inequalities, with the “hypothesis” and “conclusion” of that statement generally playing […]

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