Location:

Zoom: register here: https://www.ntwebseminar.org/

## Number Theory Web Seminar: Larry Guth (MIT)

**Reflections on the proof(s) of the Vinogradov mean value conjecture**

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?

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