In this final set of course notes, we discuss how (a generalisation of) the expansion results obtained in the preceding notes can be used for some nnumber-theoretic applications, and in particular to locate almost primes inside orbits of thin groups, following the work of Bourgain, Gamburd, and Sarnak. We will not attempt here to obtain [...]

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Academics / Mathematics : Terence Tao's Blog

In the last three notes, we discussed the Bourgain-Gamburd expansion machine and two of its three ingredients, namely quasirandomness and product theorems, leaving only the non-concentration ingredient to discuss. We can summarise t...

Academics / Mathematics : Terence Tao's Blog

In the Winter quarter (starting on January 9), I will be teaching a graduate course on expansion in groups of Lie type. This course will focus on constructions of expanding Cayley graphs on finite groups of Lie type (such as the sp...

Academics / Mathematics : Terence Tao's Blog

In the previous set of notes we saw how a representation-theoretic property of groups, namely Kazhdan’s property (T), could be used to demonstrate expansion in Cayley graphs. In this set of notes we discuss a different representatio...

Academics / Mathematics : Terence Tao's Blog

This is the seventh thread for the Polymath8b project to obtain new bounds for the quantity either for small values of (in particular ) or asymptotically as. The previous thread may be found here. The currently best known bounds on ...

Academics / Mathematics : Terence Tao's Blog

We have now seen two ways to construct expander Cayley graphs. The first, discussed in Notes 2, is to use Cayley graphs that are projections of an infinite Cayley graph on a group with Kazhdan’s property (T). The second, discussed i...

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