The Paucity of Knowledge Concerning Weyl Sums and Their Mean Values

This lecture by Trevor Wooley examines the limited understanding of Weyl sums and their mean values despite a century of mathematical progress. Explore how advances related to Vinogradov's mean value theorem, the delta-function method, and L-function conjectures have shaped our knowledge of exponential sums with polynomial arguments. Learn why only linear and quadratic Weyl sums are well-understood, while cubic Weyl sums remain problematic with circular reasoning in current conjectures, and high-degree exponential sums fall significantly short of theoretical predictions. Discover the common approaches used to study Weyl sums, the obstacles preventing further advancement, and connections to important mathematical concepts including congruences to large moduli, arithmetic stratification, and L-function conjectures.