Georgia Institute of Technology College of Sciences

Sum-Product inequalities originated in the early 80's with the work of Erdos and Szemeredi, who showed that there exists a constant c such that if A is a set of n integers, n sufficiently large, then either the sumset A+A = {a+b : a,b in A} or the product set A.A = {ab : a,b in A}, must exceed n^(1+c) in size. Since that time the subject has exploded with a vast number of generalizations and extensions of the basic result, which has led to many very interesting unsolved problems (that would make great thesis topics). In this talk I will survey some of the developments in this fast-growing area.