Blog Profile / Terence Tao's Blog


URL :http://terrytao.wordpress.com/
Filed Under:Academics / Mathematics
Posts on Regator:485
Posts / Week:1
Archived Since:December 12, 2008

Blog Post Archive

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 […]

On the sign patterns of entrywise positivity preservers in fixed dimension

Apoorva Khare and I have just uploaded to the arXiv our paper “On the sign patterns of entrywise positivity preservers in fixed dimension“. This paper explores the relationship between positive definiteness of Hermitian matrices, and entrywise operations on these matrices. The starting point for this theory is the Schur product theorem, which asserts that if […]

The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures

Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures“, submitted to Duke Mathematical Journal. Show More Summary

Schur convexity and positive definiteness of the even degree complete homogeneous symmetric polynomials

The complete homogeneous symmetric polynomial of variables and degree can be defined as thus for instance and One can also define all the complete homogeneous symmetric polynomials of variables simultaneously by means of the generating function We will think of the variables as taking values in the real numbers. When one does so, one might […]

On the universality of the incompressible Euler equation on compact manifolds

I’ve just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds“, submitted to Discrete and Continuous Dynamical Systems. This is a variant of my recent paper on the universality of potential well dynamics, but instead of trying to embed dynamical systems into a potential well, here […]

Maryam Mirzakhani

I am totally stunned to learn that Maryam Mirzakhani died today, aged 40, after a severe recurrence of the cancer she has been fighting for several years.  I had planned to email her some wishes for a speedy recovery after learning about the relapse yesterday; I still can’t fully believe that she didn’t make it. […]

On the universality of potential well dynamics

I’ve just uploaded to the arXiv my paper “On the universality of potential well dynamics“, submitted to Dynamics of PDE. This is a spinoff from my previous paper on blowup of nonlinear wave equations, inspired by some conversations with Sungjin Oh. Here we focus mainly on the zero-dimensional case of such equations, namely the potential […]

What are some useful, but little-known, features of the tools used in professional mathematics?

A few days ago, I was talking with Ed Dunne, who is currently the Executive Editor of Mathematical Reviews (and in particular with its online incarnation at MathSciNet).  At the time, I was mentioning how laborious it was for me to create a BibTeX file for dozens of references by using MathSciNet to locate each […]

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