Functoriality: ubiquity and progress

Abstract: Questions in automorphic forms and number theory often get tied up with the magnificent, largely conjectural, edifice of functoriality, a simple instance being the desire to know if certain four-dimensional Galois representations occurring inside the cohomology of Siegel modular threefolds are symplectic. Of particular importance, besides base change, is the transfer of automorphic forms from orthogonal and symplectic groups to the general linear group, which sheds light on many prob- lems. Crucial progress has been made of late in the work of Arthur via the twisted trace formula, extending the earlier results known for generic cusp forms, which had relied on the elegant converse theorem insight of Piatetski-Shapiro. Part of what makes Arthur’s approach work is the incredible recent progress on the (different guises of) fundamental lemma due to Ngô, Waldspurger and others. This talk will try to introduce the basic global statements, a few ideas, and applications.