Mathematical Conversations

In 2004 Jean Bourgain proved, with Netz Katz and Terry Tao, the "sum-product theorem in finite fields". He referred to this result (and proof technique) as a "goose which lays golden eggs". Indeed, in subsequent years, he has published a couple of dozen papers on consequences and applications to numerous areas: Analysis, Number Theory, Group Theory, Dynamical Systems, Combinatorial Geometry, Graph expansion, Randomness extractors and more. His works were naturally followed by many others, creating what Jean liked calling "the sum-product phenomena", and making clear how fundamental this initial work was.

I will only have time to explain what is a "sum-product theorem", talk about the history of the problem, perhaps give a couple of applications. These may change your view on the interaction of the two operations we all learn in grade school and take for granted since: addition and multiplication.