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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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... Read Post
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... Read Post
In this, the final lecture notes of this course, we give discuss one of the motivating applications of the theory developed thus far, namely to count solutions to linear equations in primes (or in dense subsets of primes ). Unfortun... Read Post