Infinitely Many Non-hypergeometric Local Systems

Explore advanced research in algebraic geometry and differential equations through this mathematical lecture examining the Bombieri-Dwork conjecture and its implications for G-operators. Delve into the conjecture's prediction that irreducible differential operators with G-function solutions originate from geometric sources, specifically encoding period variations in algebraic variety pencils. Learn about Dwork's proposed strategy from the early 1990s for establishing the conjecture for second-order G-operators through pullbacks of Gauss's hypergeometric differential operators. Examine sporadic counterexamples discovered by Kraamer (1996) and Bouw-Möller (2010), then discover groundbreaking joint research with Josh Lam and Yichen Qin demonstrating that most second-order G-operators from geometry are not pullbacks of hypergeometric differential operators. Understand how the construction of infinitely many counterexamples relies on the André-Pink-Zannier conjecture for Shimura varieties, particularly in cases recently established by Richard and Yafaev, connecting deep themes in arithmetic geometry and transcendental number theory.