New Geometric Methods in Number Theory and Automorphic Forms 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 ?
 A: 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.
A: To complement Joel's wonderful and (as far as I understand) very much on point answer, let me quote from the proposal for the parallel program on Geometric Representation Theory, which touches on several related themes:
Representation theory is the study of the basic symmetries of
mathematics and physics. The primary aim of the subject is to
understand concrete linear models for abstract symmetry groups. A
signature triumph of the past century is our understanding of compact
Lie groups. At the foundation, there is Cartan's classification of Lie
algebras and Borel-Weil-Bott's uniform construction of all
representations in the cohomology of line bundles on flag
varieties. Thus we have a list of every compact Lie group we could
ever encounter and every way in which it could appear concretely as a
matrix group. Furthermore, there is a deep combinatorial theory of
other key structures such as the tensor product of
representations. The ideas and results of this subject are the basic
input to diverse areas from number theory to quantum field theory.
Though mysteries still remain, the theory of compact Lie groups is a
representative model for what we would like to achieve with other
symmetry groups.  Because of their universal importance, symmetry
groups come in many different flavors: finite groups, Lie groups,
$p$-adic groups, loop groups, ad\'elic groups,...  and the list will
only increase with the discovery of new important structures.  For the
above examples, our understanding is still very coarse though the last
decades have witnessed breathtaking advances.  The Langlands program
along with its geometric spinoffs provide a visionary roadmap for
where the subject could go in the coming years.  In particular,
current developments such as the recent proof of the Fundamental Lemma
create great optimism that geometric techniques will have a deep
impact.
Geometric representation theory seeks to understand groups and
representations as a consequence of more subtle but fundamental
symmetries. A groundbreaking example of its success is
Beilinson-Bernstein's uniform construction of all representations of
Lie groups via the geometry of $\mathcal D$-modules on flag varieties.
The result is not difficult to state and prove but has the
Borel-Weil-Bott theorem and the Kazhdan-Lusztig multiplicity
conjectures as immediate consequences. A reasonable reaction is to
wonder what allows one to prove deep results about representations of
Lie groups with so little effort. One answer is that the true focus of
the Beilinson-Bernstein theory is not representations but rather the
symmetries of flag varieties. The geometric notion of $\mathcal
D$-module allows one to localize symmetries and to apply the
sheaf-theoretic techniques of algebraic geometry.  In this way, our
understanding of representations of Lie groups follows from that of
infinitesimal symmetries of algebraic varieties.
There are now many instances where difficult questions about
representations can be translated into more tractable questions about
geometry.  Other famous examples include the Deligne-Lusztig theory of
representations of finite groups of Lie type, the Springer theory of
representations of Weyl groups, the Kazhdan-Lusztig theory of modules
for Hecke algebras, and Lusztig and Nakajima's theory of
representations of Kac-Moody algebras and quantum groups via quiver
varieties. In the theory of automorphic forms, one of the fundamental
tools is the realization of representations of ad`elic groups in the
cohomology of Shimura varieties (in the case of number fields) and
Drinfeld modular varieties (in the case of function fields). The
recent proof of the Fundamental Lemma exploits the fact that computing
orbital integrals in $p$-adic groups can be reduced to calculating
cohomology of fibers of Hitchin's integrable system.
The twin modern goals of geometric representation theory are to
explore the above dictionaries and to discover new unexpected ones.
As an important consequence, the geometric realization of
representations often reveals deeper layers of structure in the form
of categorification. Categorification typically turns numbers (for
example, the coefficients of Kazhdan-Lusztig polynomials) into the
dimensions of vector spaces (in this case, the Ext groups of
intersection cohomology sheaves).  It is a primary explanation for
miraculous integrality and positivity properties in algebraic
combinatorics.  At the center of geometric representation theory is
Grothendieck's categorification of functions by $\ell$-adic
sheaves. An important example is Lusztig's theory of character
sheaves: it provides a uniform geometric source for the characters of
all finite groups of Lie type.  Another important example is the
theory of canonical bases: here categorification replaces
representation spaces with linear categories equipped with canonical
generating objects.  Another broad example is the geometric Langlands
program: it provides a categorification of the Langlands program in
the setting of function fields, providing new insights into many
classical constructions.
Geometric representation theory has close and profound connections to
many fields of mathematics, which we expect to play a significant role
in the program. Perhaps the most significant are to number theory, via
the theory of automorphic forms, L-functions and modularity. Much
current activity in the field is motivated either directly by problems
in number theory or by more tractable geometric analogues
thereof. Another major influence on the subject comes from physics, in
particular gauge theory, integrable systems, and recently topological
string theory.  There is a significant interaction with the theory of
$C^*$-algebras through the Baum-Connes conjecture, an instance of
which provides an organizing principle for representation theory of
real and $p$-adic groups. Finally, geometric representation theory is
closely entwined with very active areas in combinatorics  such as
Schubert calculus, its affine analogue, and the theory of
Macdonald polynomials.
The proposed program has two primary goals: 


*

*To bring together researchers working in the
arithmetic and geometric Langlands programs so as to discover 
new relations between the objects appearing in the two
subjects.

*To explore new principles and paradigms within 
geometric representation theory.
A particular emphasis will be placed on geometric methods in
representation theory over local fields. This is an area which has advanced
dramatically in the past decade. More importantly, it holds great promise for
future discoveries of interest to a diverse collection of researchers.
