(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 many people, based on 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 Paris' book edited by Harris that should contain every detail is not completely ready.

(iii) The proof by Kisin and also by Emerton of large part of the Fontaine-Mazur 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, controversed, proof by Vasiu as well): the conjecture of Milne and of Langlands-Rapoport.

At first I thought I should include (0) the proof of the fundamental by Ngo, but since none of the organizer 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 "algebraist", "analyst", 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 illuminated the properties of the individual object 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)

(c) Also, the better understanding and the use of the boundary component of non-compact Shimura's 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).

(e) if (0) is included (which as I have said, I am not sure of), the geometric method of Ngo (and before him Laumon, Goreski, MacPherson) used in proving the fundamental Lemma, and perhaps also the one 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 breakthrough is far from complete.

(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.

(iii) The proof by Kisin and also by Emerton of large parts of the Fontaine-Mazur conjecture for $Gl_2$.

(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)

(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).

(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.