Blog Profile / Terence Tao's Blog


URL :http://terrytao.wordpress.com/
Filed Under:Academics / Mathematics
Posts on Regator:477
Posts / Week:1
Archived Since:December 12, 2008

Blog Post Archive

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 […]

Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges

Kaisa Matomaki, Maksym Radziwill, and I have uploaded to the arXiv our paper “Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges“, submitted to Proceedings of the London Mathematical Society. This paper is concerned with the estimation of correlations such as for medium-sized and large, where is the von Mangoldt […]

PCMI lecture notes on random matrix theory

In July I will be spending a week at Park City, being one of the mini-course lecturers in the Graduate Summer School component of the Park City Summer Session on random matrices.  I have chosen to give some lectures on least singular values of random matrices, the circular law, and the Lindeberg exchange method in […]

Quantitative continuity estimates

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 […]

Generalisations of the limit functional

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 […]

New bounds for Szemerédi’s theorem, III: A polylogarithmic bound for r_4(N)

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 […]

Notes on nilcharacters and their symbols

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 […]

Counting objects up to isomorphism: groupoid cardinality

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 […]

Yves Meyer wins the 2017 Abel Prize

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 […]

Copyright © 2015 Regator, LLC