Functoriality and Beyond Endoscopy

Wednesday, November 14, 2018 - 2:00pm - 3:00pm
Keller 3-180
Salim Altug (Boston University)
An overarching goal of the Langlands program is the principle of functoriality, first formulated in the famous 1967 letter from Langlands to Weil. It roughly states that if the L-groups of two (reductive) groups are related then so are their automorphic spectrum. It has many important consequences including the Ramanujan conjecture for GL(n) as well as general forms of the Sato-Tate conjecture. Since its introduction, the principle of functoriality has seen lots of progress (especially in the so-called endoscopic cases), however, in their full generality the conjectures are still wide open. Aiming to attack the functoriality conjectures in their general form, in early 2000s, Langlands introduced a strategy, known as Beyond Endoscopy. In its core, the proposed strategy is to use the trace formula in a novel way to study the analytic properties of automorphic L-functions. This strategy has attracted ample attention and more recently evolved into various approaches to functoriality and related problems.

In this talk, I will give an overview of beyond endoscopy, describe Langlands' approach, and the role of the trace formula. I will discuss various results and obstacles in executing the bond endoscopy strategy. Time permitting, I will also talk about further research and other problems stemming from beyond endoscopy.