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Thursday, September 14, 2017 — 1:30 PM EDT

Julia Brandes, Department of Pure Mathematics, University of Waterloo

"Optimal mean value estimates beyond Vinogradov's mean value theorem"

Mean values for exponential sums play a central role in the study of diophantine equations. In particular, strong upper bounds for such mean values control the number of integer solutions of the corresponding systems of diagonal equations. Since the groundbreaking resolution of Vinogradov's mean value theorem by Wooley and Bourgain, Demeter and Guth, we can now prove optimal upper bounds for mean values connected to 
translation-dilation-invariant systems. This has inspired Wooley's call for a "Big Theory of Everything", a challenge to establish optimal mean value estimates for any mean values associated with systems of diagonal equations.

We establish optimal bounds for a family of mean values that are not of Vinogradov type. This is the first time bounds of this quality have been obtained for non-translation-dilation-invariant systems. As a consequence, we establish the analytic Hasse principle for the number of solutions of certain systems of quadratic and cubic equations in fewer variables than hitherto thought necessary. This is joint work with Trevor Wooley.

MC 5501

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