The MSRI is organising a programme with the above title from Aug 11, 2014 to Dec 12, 2014. Here is a short description from their website :
The branches of number theory most
directly related to the arithmetic of
automorphic forms have seen much
recent progress, with the resolution
of many longstanding conjectures.
These breakthroughs have largely been
achieved by the discovery of new
geometric techniques and insights. The
goal of this program is to highlight
new geometric structures and new
questions of a geometric nature which
seem most crucial for further
development. In particular, the
program will emphasize geometric
questions arising in the study of
Shimura varieties, the $p$-adic
Langlands program, and periods of
automorphic forms.
Question Which new geometric structures, techniques and insights have been crucial for this recent progress ?
Best Answer
Knowing the organizers well and working in the field, I can try an answer, but this is nothing more than an educated guess.
First, the breakthroughs in question include
(i) The construction and study of Galois representations attached to self-dual cohomological automorphic forms for $Gl_n$ (satisfying local-global compatibility, etc.) This is the work of many people, based on the fundamental work of Arthur and Ngo, including Shin, Morel, Harris, Clozel, Labesse, and many many others. This can be considered as done, even if the four-volume Paris' book edited by Harris that should contain every detail is not completely ready.
(ii) The construction and study of Galois representations attached to not necessary self-dual cohomological automorphic forms for $Gl_n$, announced last year by Lan, Harris, Taylor and Thorne (the preprint has yet to be released).
(iii) The proof by Kisin and also by Emerton of large parts of the Fontaine-Mazur conjecture for $Gl_2$.
(iv) The proof of Sato-Tate by many people with various multiplicity, the two highest being Taylor and Harris.
(v) The progresses on the p-adic Langlands program, especially on the Breuil-Mezard conjecture.
(vi) The progress on the theory of Shimura varieties, including the proof of two major conjectures of the subject by Kisin (one has an older, not universally accepted, proof by Vas): the conjectures of Milne and of Langlands-Rapoport.
At first I thought I should include (0) the proof of the fundamental lemma by Ngo, but since none of the organizers is a specialist of this area, I am not sure.
Now why the emphasis on the "geometric methods", and what are those? Well, there is a même saying that along the traditional tripartite division of mathematicians as "algebraists", "analysts", and "geometers" (see e.g. Recoltes et Semailles), while people like Breuil (or Fontaine) are more on the algebraic side, and perhaps Colmez on the analytic side, people like Kisin and Emerton are really on the geometric side, and that their geometric intuition played a crucial role in their recent successes. Whatever you think of this même (or even of the tripartite classification) it is quite possible that it made its way to the mind of one or more of the organizer. The geometric insights and methods include
(a) the use of "eigenvarieties": families of automorphic or/and Galois representations that have a geometric structure, and whose geometric properties, local and global illuminate the properties of the individual objects that compose them. For example, this plays a crucial role in Emerton's proof of Fonntaine-Mazur's conjecture (iii), in constructing Galois representations by "passage to the limit" (i) and (ii), and also in recent progress toward Bloch-Kato and Birch-Swinnerton-Dyer conjecture (work of Chenevier and myself, Urban and Skinner), and also in the work on the Breuil-Mezard conjecture (Kisin first, then others)
(b) The better understanding of certain Shimura varieties, in particular the ones attached to unitary groups, in particular in connection with Rapoport-Zink spaces etc. For example one can cite the thesis work of Mantovan, which is used in Shin's subsequent work on (i). Also whatever Kisin uses to prove the conjecture about Shimura variety (at this point I don't know what it is, but I am organizing a seminar at Yale to learn this eventually)
(c) Also, the better understanding and the use of the boundary components of non-compact Shimura varieties, including in cases (this is mainly speculative so far) where this components has only the structure of a differentiable manifold, not of an algebraic variety. I am not sure, but idea like that plays a role in (iv).
(d) Study of cycle on Shimura's varieties, in particular in connection to periods and p-adic L-function (how to define them in higher ranks? that is very hard and important).
(e) if (0) is included (which as I have said, I am not sure of), the geometric methods of Ngo (and before him Laumon, Goreski, MacPherson: balloons, Hitchin's vibrations, etc.) used in proving the fundamental lemma, and perhaps also the ones of Laurent Lafforgue. But I think this might be the subject of another conference.
I hope that helps... Sorry to anyone I forgot to mention, my list of people having a part in the recent breakthroughs is far from complete.