In 2002, R. Langlands put forward a new strategy to prove the general functoriality conjecture in the Beyond endoscopy paper. The main purpose of this strategy is to detecting the automorphic representations of G$G$, where the L$L$-function has a pole at s = 1$s = 1$.
In 2009, Ngo Bao Chau proved fundamental lemma.
A year later Ngo and Frenkel wrote an article about geometrization of trace formula. Arthur-Selberg trace formula is a fundamental tool in the theory of automorphic forms. If we want to prove functoriality conjecture, we first need to establish an identity between the orbital integrals on the groups g$g$ and h$h$ for the f_g$f_g$ and f_h$f_h$ test functions, and then use the trace formula to get the identity on the spectral side. The following article Ngo and Frenkel geometrizes the orbital side of trace formula and using this strategy, they formulate conjecture about the relationship between cohomology of two modular stacks. (M_G, M_H) G=SL2$(M_G, M_H)$ $G=SL_2$, H$H$: non-split one-dimensional torus..
https://arxiv.org/abs/1004.5323
Edward Frenkel, Ngo Bao Chau, Geometrization of Trace Formulas, Bulletin of Mathematical Sciences 1 (2011) no 1 pp 129–199. doi:10.1007/s13373-011-0009-0, arXiv:1004.5323
And as far as I can tell, Lafforgue is working on functoriality and developed a different strategy of attack.
What is the status of the functoriality in 2017? Or what are your thoughts about the future of functoriality?
