Blog Profile / Terence Tao's Blog
| URL : | http://terrytao.wordpress.com/ | |
|---|---|---|
| Filed Under: | Academics / Mathematics | |
| Posts on Regator: | 396 | |
| Posts / Week: | 1.7 | |
| Archived Since: | December 12, 2008 | |
Blog Post Archive
(Ingrid Daubechies) Planning for the World Digital Mathematical Library
[This guest post is authored by Ingrid Daubechies, who is the current president of the International Mathematical Union, and (as she describes below) is heavily involved in planning for a next-generation digital mathematical libraryShow More Summary
Planning for the World Digital Mathematical Library
[This guest post is authored by Ingrid Daubechies, who is the current president of the International Mathematical Union, and (as she describes below) is heavily involved in planning for a next-generation digital mathematical libraryShow More Summary
Quasirandom groups and a cheap version of the Brauer-Fowler theorem
Suppose that is a finite group of even order, thus is a multiple of two. By Cauchy’s theorem, this implies that contains an involution: an element in of order two. (Indeed, if no such involution existed, then would be partitioned into doubletons together with the identity, so that would be odd, a contradiction.) Of course, [...]
Notes on the classification of complex Lie algebras
An abstract finite-dimensional complex Lie algebra, or Lie algebra for short, is a finite-dimensional complex vector space together with an anti-symmetric bilinear form that obeys the Jacobi identity for all ; by anti-symmetry one can also rewrite the Jacobi identity as We will usually omit the subscript from the Lie bracket when this will not [...]
The theorems of Frobenius and Suzuki on finite groups
The classification of finite simple groups (CFSG), first announced in 1983 but only fully completed in 2004, is one of the monumental achievements of twentieth century mathematics. Spanning hundreds of papers and tens of thousands of pages, it has been called the “enormous theorem”. A “second generation” proof of the theorem is nearly completed which [...]
An informal version of the Furstenberg correspondence principle
One of the basic objects of study in combinatorics are finite strings or infinite strings of symbols from some given alphabet, which could be either finite or infinite (but which we shall usually take to be compact). For instance, a set of natural numbers can be identified with the infinite string of s and [...]
Rectification and the Lefschetz principle
The rectification principle in arithmetic combinatorics asserts, roughly speaking, that very small subsets (or, alternatively, small structured subsets) of an additive group or a field of large characteristic can be modeled (for the purposes of arithmetic combinatorics) by subsets of a group or field of zero characteristic, such as the integers or the complex numbers [...]
A Fourier-free proof of the Furstenberg-Sarkozy theorem
The following result is due independently to Furstenberg and to Sarkozy: Theorem 1 (Furstenberg-Sarkozy theorem) Let, and suppose that is sufficiently large depending on. Then every subset of of density at least contains a pair for some natural numbers with. This theorem is of course similar in spirit to results such as [...]
Supercommutative gaussian integration, and the gaussian unitary ensemble
The fundamental notions of calculus, namely differentiation and integration, are often viewed as being the quintessential concepts in mathematical analysis, as their standard definitions involve the concept of a limit. However, it is possible to capture most of the essence of these notions by purely algebraic means (almost completely avoiding the use of limits, Riemann [...]
The pseudoconformal and conformal transformations
Consider the free Schrödinger equation in spatial dimensions, which I will normalise as where is the unknown field and is the spatial Laplacian. To avoid irrelevant technical issues I will restrict attention to smooth (classical) solutions to this equation, and will work locally in spacetime avoiding issues of decay at infinity (or at other singularities); [...]
The Harish-Chandra-Itzykson-Zuber integral formula
Let be Hermitian matrices, with eigenvalues and. The Harish-Chandra-Itzykson-Zuber integral formula exactly computes the integral where is integrated over the Haar probability measure of and is a non-zero complex parameter, as the expression when the eigenvalues of are simple, where denotes the Vandermonde determinant and is the constant There are at least two standard [...]
The Harish-Chandra-Itzykson-Zuber integral formula
Let be Hermitian matrices, with eigenvalues and. The Harish-Chandra-Itzykson-Zuber integral formula} exactly computes the integral where is integrated over the Haar probability measure of and is a non-zero complex parameter, as the expression when the eigenvalues of are simple, where denotes the Vandermonde determinant and is the constant There are at least two standard [...]
Some notes on Bakry-Emery theory
[These are notes intended mostly for myself, as these topics are useful in random matrix theory, but may be of interest to some readers also. -T.] One of the most fundamental partial differential equations in mathematics is the heat equation where is a scalar function of both time and space, and is the Laplacian. [...]
Small doubling in groups
Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our survey “Small doubling in groups“, for the proceedings of the upcoming Erdos Centennial. This is a short survey of the known results on classifying finite subsets of an (abelian) additive group or a (not necessarily abelian) multiplicative group that have small doubling [...]
Matrix identities as derivatives of determinant identities
The determinant of a square matrix obeys a large number of important identities, the most basic of which is the multiplicativity property whenever are square matrices of the same size. This identity then generates many other important identities. For instance, if is an matrix and is an matrix, then by applying the previous identity to [...]
A mathematical formalisation of dimensional analysis
Mathematicians study a variety of different mathematical structures, but perhaps the structures that are most commonly associated with mathematics are the number systems, such as the integers or the real numbers. Indeed, the use of number systems is so closely identified with the practice of mathematics that one sometimes forgets that it is possible [...]
An introduction to special relativity for a high school math circle
Things are pretty quiet here during the holiday season, but one small thing I have been working on recently are some notes on special relativity that I will be working through in a few weeks with some bright high school students here at our local math circle. I have only two hours to spend with [...]
Mixing for progressions in non-abelian groups
I’ve just uploaded to the arXiv my paper “Mixing for progressions in non-abelian groups“, submitted to Forum of Mathematics, Sigma (which, along with sister publication Forum of Mathematics, Pi, has just opened up its online submission system). This paper is loosely related in subject topic to my two previous papers on polynomial expansion and on [...]
The spectral proof of the Szemeredi regularity lemma
Perhaps the most important structural result about general large dense graphs is the Szemerédi regularity lemma. Here is a standard formulation of that lemma: Lemma 1 (Szemerédi regularity lemma) Let be a graph on vertices, and let. Then there exists a partition for some with the property that for all but at most of [...]
Lars Hormander
Lars Hörmander, who made fundamental contributions to all areas of partial differential equations, but particularly in developing the analysis of variable-coefficient linear PDE, died last Sunday, aged 81. I unfortunately never met Hörmander personally, but of course I encountered his work all the time while working in PDE. One of his major contributions to the [...]