Mumford-Tate and André-Oort Conjectures in Characteristic p

Explore the characteristic p versions of the Mumford—Tate and André—Oort conjectures in this advanced mathematical lecture from the Workshop on Special Cycles and Related Topics at the Institute for Advanced Study. Delve into how these fundamental conjectures in algebraic geometry and number theory behave differently in positive characteristic compared to their characteristic zero counterparts. Learn about the concept of formal linearity and discover how it creates unexpected connections between the Mumford—Tate conjecture, André—Oort conjecture, and a third conjecture called modpAx—Lindemann, making them more closely intertwined than in characteristic zero. Examine how this interconnectedness reduces the Mumford—Tate conjecture in characteristic p to an "unlikely intersection problem," providing new pathways for understanding these deep mathematical questions. Follow the speaker's groundbreaking work establishing all three conjectures for products of orthogonal Shimura varieties, and gain insights into recent extensions beyond the orthogonal case that push the boundaries of current research in arithmetic geometry and the theory of Shimura varieties.