Blog Profile / Terence Tao's Blog

Filed Under:Academics / Mathematics
Posts on Regator:395
Posts / Week:1.1
Archived Since:December 12, 2008

Blog Post Archive

Heath-Brown’s theorem on prime twins and Siegel zeroes

The twin prime conjecture is one of the oldest unsolved problems in analytic number theory. There are several reasons why this conjecture remains out of reach of current techniques, but the most important obstacle is the parity problem which prevents purely sieve-theoretic methods (or many other popular methods in analytic number theory, such as the […]

A wave equation approach to automorphic forms in analytic number theory

The Poincaré upper half-plane (with a boundary consisting of the real line together with the point at infinity ) carries an action of the projective special linear group via fractional linear transformations: Here and in the rest of the post we will abuse notation by identifying elements of the special linear group with their equivalence […]

Equidistribution for multidimensional polynomial phases

The equidistribution theorem asserts that if is an irrational phase, then the sequence is equidistributed on the unit circle, or equivalently that for any continuous (or equivalently, for any smooth) function. By approximating uniformly by a Fourier series, this claim is equivalent to that of showing that for any non-zero integer (where ), which […]

Analytic Number Theory program at MSRI: Jan-May 2017

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 […]

Deducing the inverse theorem for the multidimensional Gowers norms from the one-dimensional version

This week I have been at a Banff workshop “Combinatorics meets Ergodic theory“, focused on the combinatorics surrounding Szemerédi’s theorem and the Gowers uniformity norms on one hand, and the ergodic theory surrounding Furstenberg’s multiple recurrence theorem and the Host-Kra structure theory on the other. This was quite a fruitful workshop, and directly inspired the […]

Deducing a weak ergodic inverse theorem from a combinatorial inverse theorem.

Note: this post is of a particularly technical nature, in particular presuming familiarity with nilsequences, nilsystems, characteristic factors, etc., and is primarily intended for experts. As mentioned in the previous post, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms: Theorem 1 (Inverse theorem for Gowers norms) Let and […]

An inverse theorem for the continuous Gowers uniformity norm

A few years ago, Ben Green, Tamar Ziegler, and myself proved the following (rather technical-looking) inverse theorem for the Gowers norms: Theorem 1 (Discrete inverse theorem for Gowers norms) Let and be integers, and let. Suppose that is a function supported on such that Then there exists a filtered nilmanifold of degree and complexity […]

A nonstandard analysis proof of Szemeredi’s theorem

Szemerédi’s theorem asserts that any subset of the integers of positive upper density contains arbitrarily large arithmetic progressions. Here is an equivalent quantitative form of this theorem: Theorem 1 (Szemerédi’s theorem) Let be a positive integer, and let be a function with for some, where we use the averaging notation. Then for we […]

Cycles of a random permutation, and irreducible factors of a random polynomial

In analytic number theory, there is a well known analogy between the prime factorisation of a large integer, and the cycle decomposition of a large permutation; this analogy is central to the topic of “anatomy of the integers”, as discussed for instance in this survey article of Granville. Consider for instance the following two parallel […]

Inverse theorems for sets and measures of polynomial growth

I’ve just uploaded to the arXiv my paper “Inverse theorems for sets and measures of polynomial growth“. This paper was motivated by two related questions. The first question was to obtain a qualitatively precise description of the sets of polynomial growth that arise in Gromov’s theorem, in much the same way that Freiman’s theorem (and […]

Gaps between primes

Just a short post here to note that the cover story of this month’s Notices of the AMS, by John Friedlander, is about the recent work on bounded gaps between primes by Zhang, Maynard, our own Polymath project, and others. I may as well take this opportunity to upload some slides of my own talks […]

Nested approximate subgroups

Suppose that are two subgroups of some ambient group, with the index of in being finite. Then is the union of left cosets of, thus for some set of cardinality. The elements of are not entirely arbitrary with regards to. For instance, if is a normal subgroup of, then for […]

A differentiation identity

Here’s a cute identity I discovered by accident recently. Observe that and so one can conjecture that one has when is even, and when is odd. This is obvious in the even case since is a polynomial of degree, but I struggled for a while with the odd case before finding a slick three-line […]

Cancellation for the multilinear Hilbert transform

I’ve just uploaded to the arXiv my paper “Cancellation for the multilinear Hilbert transform“, submitted to Collectanea Mathematica. This paper uses methods from additive combinatorics (and more specifically, the arithmetic regularity and counting lemmas from this paper of Ben Green and myself) to obtain a slight amount of progress towards the open problem of obtaining […]

Failure of the L^1 pointwise and maximal ergodic theorems for the free group

I’ve just uploaded to the arXiv my paper “Failure of the pointwise and maximal ergodic theorems for the free group“, submitted to Forum of Mathematics, Sigma. This paper concerns a variant of the pointwise ergodic theorem of Birkhoff, which asserts that if one has a measure-preserving shift map on a probability space, then for […]

A remark on the lonely runner conjecture

The lonely runner conjecture is the following open problem: Conjecture 1 Suppose one has runners on the unit circle, all starting at the origin and moving at different speeds. Then for each runner, there is at least one time for which that runner is “lonely” in the sense that it is separated by a […]

The standard branch of the matrix logarithm

Because of Euler’s identity, the complex exponential is not injective: for any complex and integer. As such, the complex logarithm is not well-defined as a single-valued function from to. However, after making a branch cut, one can create a branch of the logarithm which is single-valued. For instance, after removing the negative […]

Embedding the SQG equation in a modified Euler equation

The Euler equations for three-dimensional incompressible inviscid fluid flow are where is the velocity field, and is the pressure field. For the purposes of this post, we will ignore all issues of decay or regularity of the fields in question, assuming that they are as smooth and rapidly decreasing as needed to justify all the […]

Newton iteration and the Siegel linearisation theorem

An extremely large portion of mathematics is concerned with locating solutions to equations such as or for in some suitable domain space (either finite-dimensional or infinite-dimensional), and various maps or. To solve the fixed point iteration equation (1), the simplest general method available is the fixed point iteration method: one starts with an initial […]

The ergodic theorem and Gowers-Host-Kra seminorms without separability or amenability

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space, and is a vector in that Hilbert space, then one has in the strong topology, where is the -invariant subspace of, and is the orthogonal projection to. […]

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