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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...

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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...

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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...

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A Fourier-analytic proof of Frobenius’ theorem

A finite group is said to be a Frobenius group if there is a non-trivial subgroup of (known as the Frobenius complement of ) such that the conjugates of are “disjoint as possible” in the sense that...

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Notes on simple groups of Lie type

In this previous post I recorded some (very standard) material on the structural theory of finite-dimensional complex Lie algebras (or Lie algebras for short), with a particular focus on those Lie...

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Expansion in finite simple groups of Lie type

Emmanuel Breuillard, Ben Green, Bob Guralnick, and I have just uploaded to the arXiv our joint paper “Expansion in finite simple groups of Lie type“. This long-delayed paper (announced way back in...

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Qualitative probability theory, types, and the group chunk and group...

The classical foundations of probability theory (discussed for instance in this previous blog post) is founded on the notion of a probability space – a space (the sample space) equipped with a...

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Hilbert’s fifth problem and approximate groups

Due to some requests, I’m uploading to my blog the slides for my recent talk in Segovia (for the birthday conference of Michael Cowling) on “Hilbert’s fifth problem and approximate groups“.  The slides...

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A trivial generalisation of Cayley’s theorem

One of the first basic theorems in group theory is Cayley’s theorem, which links abstract finite groups with concrete finite groups (otherwise known as permutation groups). Theorem 1 (Cayley’s...

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Additive limits

In graph theory, the recently developed theory of graph limits has proven to be a useful tool for analysing large dense graphs, being a convenient reformulation of the Szemerédi regularity lemma....

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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,...

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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...

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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...

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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...

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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...

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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...

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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...

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Heat flow and zeroes of polynomials II: zeroes on a circle

This is a sequel to this previous blog post, in which we discussed the effect of the heat flow evolution on the zeroes of a time-dependent family of polynomials , with a particular focus on the case...

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1% quasimorphisms and group cohomology

Let , be additive groups (i.e., groups with an abelian addition group law). A map is a homomorphism if one has for all . A map is an affine homomorphism if one has for all additive quadruples in , by...

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The alternative hypothesis for unitary matrices

In a recent post I discussed how the Riemann zeta function can be locally approximated by a polynomial, in the sense that for randomly chosen one has an approximation where grows slowly with , and is...

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