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Bill Thurston

Bill Thurston, who made fundamental contributions to our understanding of low-dimensional manifolds and related structures, died on Tuesday, aged 65.

Perhaps Thurston’s best known achievement is the proof of the… 458 more words


The Birkhoff-Kakutani theorem

A topological space is said to be metrisable if one can find a metric on it whose open balls generate the topology.

There are some obvious necessary conditions on the space in order for it to be metrisable. 1,396 more words


The Szemeredi-Trotter theorem via the polynomial ham sandwich theorem

The ham sandwich theorem asserts that, given bounded open sets in , there exists a hyperplane that bisects each of these sets , in the sense that each of the two half-spaces on either side of the hyperplane captures exactly half of the volume of . 1,180 more words


At the AustMS conference

This week I was in my home town of Adelaide, Australia, for the 2009 annual meeting of the Australian Mathematical Society. This was a fairly large meeting (almost 500 participants). 2,037 more words


Concentration compactness and the profile decomposition

One of the most important topological concepts in analysis is that of compactness (as discussed for instance in my Companion article on this topic).  There are various flavours of this concept, but let us focus on… 3,204 more words


The van der Corput trick, and equidistribution on nilmanifolds

This week I was in Columbus, Ohio, attending a conference on equidistribution on manifolds. I talked about my recent paper with Ben Green on the quantitative behaviour of polynomial sequences in nilmanifolds, which I have… 2,023 more words