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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 $G$, where the $L$-function has a pole at $s = 1$.

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 $g$ and $h$ for the $f_g$ and $f_h$ 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. $(M_G, M_H)$ $G=SL_2$, $H$: non-split one-dimensional torus.

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?

David Roberts
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    "The principle of functoriality awaits the efforts of future Fields medallists." - Jim Arthur – Arun Debray Aug 12 '17 at 14:13
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    I think it might be better to suggest a year-independent edit to https://mathoverflow.net/questions/1252/where-stands-functoriality-in-2009?rq=1 instead of writing a new question. Also, both questions appear to be subsumed by https://mathoverflow.net/questions/161820/current-status-on-langlands-program?rq=1 – S. Carnahan Aug 12 '17 at 14:51
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    Possible duplicate of Current Status on Langlands Program – Harry Gindi Aug 12 '17 at 15:57
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    As for me I advocate a periodic reevaluation of the status of LP instead of a dense and difficult to go through unique question, though not too often. Why not every four years, before each ICM ? – Sylvain JULIEN Aug 12 '17 at 17:20
  • I don't think the final question is appropriate. For the titular question, see also https://mathoverflow.net/q/227171/6518 In terms of classical functoriality, that together with some recent work on unitary groups is essentially the most we know in general now. – Kimball Aug 13 '17 at 12:07
  • A dumb question : is it fair to say the problem of Langlands program (that I don't understand in its main parts) is about a general way to construct automorphic forms from Euler products ? @Kimball – reuns Aug 13 '17 at 15:39
  • @reuns It would be better to say something like: Show "all" L-functions are automorphic. Functoriality is part of this, and it basically means that whatever you can do to Galois representations you can do to automorphic representations. – Kimball Aug 14 '17 at 00:31
  • I don't think much has changed in terms of the big questions since 2014 when the last question of this type was asked. – Will Sawin Sep 20 '17 at 12:23

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