Larry Guth: Reflections on the proof(s) of the Vinogradov mean value conjecture (NTWS 114)
Channel: Number Theory Web Seminar
Date: 2023-09-02

Abstract: The Vinogradov mean value conjecture concerns the number of solutions of a system of diophantine equations.  This number of solutions can also be written as a certain moment of a trigonometric polynomial.  The conjecture was proven in the 2010s by Bourgain-Demeter-Guth and by Wooley, and recently there was a shorter proof by Guo-Li-Yang-Zorin-Kranich. The details of each proof involve some intricate estimates.  The goal of the talk is to try to reflect on the proof(s) in a big picture way.  A key ingredient in all the proofs is to combine estimates at many different scales, usually by doing induction on scales.  Why does this multi-scale induction help?  What can multi-scale induction tell us and what are its limitations?

Link to slides: https://drive.google.com/file/d/1pEgq0GU6AFS2lQedx9IjfjKhDDGZQT8J

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: Thursday, January 27, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, January 28, 2022 (12am CST, 3am AEDT, 5am NZDT)