The last years have seen a series of breakthroughs in the understanding of mean values related to exponential sums, that give rise to new improved estimates on the number of solutions to diagonal equations. This includes the proof of Vinogradov's Mean Value Theorem by Wooley and Bourgain, Demeter and Guth, as well as close-to-perfect mean value estimates for systems of diagonal cubic equations due to Bruedern and Wooley.

I will give an account of recent progress regarding mixed systems consisting of both cubic and quadratic equations. Building on the methods of Wooley and Bruedern-Wooley, we establish asymptotic estimates for the number of solutions of such systems, provided that the number of variables is not much larger than what is required by square root cancellation, and in a few cases we achieve bounds of this quality. This work is partly joint with Scott Parsell