Sums, products and growth

Habilitation Colloquium

Oliver Roche-Newton

Symbolic Computation Group

Sum-product theory revolves around the fundamental idea that additive and multiplicative structures cannot coexists in sets of numbers. The most famous manifestation of this principle is the Erdős-Szemerédi sum-product conjecture, which says that any set of integers must determine very many distinct sums or products. This is a very important problem in additive number theory which remains wide open.

There are many other problems which have the same idea of additive/multiplicative disharmony at their core. These include

• item transferring the Erdős-Szemerédi conjecture to other fields
• focusing on extreme cases where one of the sets is particularly small,
• proving that sets defined by a combination of additive and multiplicative operations are
• always large,
• various beautiful geometric problems which turn out to be secretly about sums and products.

In this Habilitation defense, I will give a survey of this area of research and focus on some of what I consider to be my favourite contributions to the field.

Habilitation Colloquium Roche-Newton.pdf