DescriptionThe goal of this project is to understand the Bourgain-Gamburd construction of bounded degree uniformly expanding Cayley graphs on SL2(𝔽p). This was the culmination of a series of beautiful results in combinatorics, including the Sum-Product Theorem of Bourgain, Katz and Tao and Helfgott's Theorem on the growth of sets in SL2(𝔽p). The Bourgain-Gamburd argument itself is a gem, involving multiple steps that bring together ingredients from combinatorics, the theory of free groups, and representation theory, and including at its core an ingenious iteration.The project will start with a general study of expander graphs. Students will begin by reading the excellent survey article [3] of Hoory, Linial and Wigderson to get a feel for what expander graphs are, what they are used for and what are the basic methods available. The most relevant sections of that article are 1,2,3 and 11 but sections 4, 7 and 8 also provide some useful context. Students should present to one another on this material. We'll then start to read the paper of Bourgain and Gamburd [1] in detail. The background on additive combinatorics for this paper can be found in Chapters 2 and 6 of Tao and Vu's book [4]. During the course of reading this paper, students will decide to concentrate on some particular part of the theory. Examples of things that students might wish to focus their project on are the Sum-Product phenomenon, the Balog-Szemerédi-Gowers lemma, Helfgott's Theorem [2], extensions of Bourgain-Gamburd's argument, or the applications of uniform expansion of Cayley graphs to number theory. I can also suggest other directions. PrerequisitesAlgebra II.The following would be useful, but not required: Codes and Cryptography III, Analysis IV, Representation Theory IV. Resources
email: Michael Magee |