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Tuesday, September 29, 2015

**Abstract:** A number is said to be $y$-smooth if all of its prime factors are at most $y$. Exponential sums over the $y$-smooth numbers less than $x$ have been widely investigated, but existing results were weak for $y$ too small compared with $x$. For example, if $y$ is a power of $\log x$ then existing results were insufficient to study ternary additive problems involving smooth numbers, except by assuming conjectures like the Generalised Riemann Hypothesis. I will try to describe my work on bounding mean values of exponential sums over smooth numbers, which allows an unconditional treatment of ternary additive problems even with $y$ a (large) power of $\log x$. There are connections with restriction theory and additive combinatorics.