This is a survey of several exciting recent results in which techniques
originating in the area known as additive combinatorics have been applied to
give results in other areas, such as group theory, number theory and
theoretical computer science. We begin with a discussion of the notion of an
approximate group and also that of an approximate field, describing key results
of Freiman-Ruzsa, Bourgain-Katz-Tao, Helfgott and others in which the structure
of such objects is elucidated. We then move on to the applications. In
particular we will look at the work of Bourgain and Gamburd on expansion
properties of Cayley graphs on SL_2(F_p) and at its application in the work of
Bourgain, Gamburd and Sarnak on nonlinear sieving problems.Comment: 25 pages. Survey article to accompany my forthcoming talk at the
Current Events Bulletin of the AMS, 2010. A reference added and a few small
changes mad
