To prove that a mathematical object is “modular” is to link it to an automorphic representation. In the past few years and months, three outstanding modularity conjectures have been settled in a large number of cases: Serre's conjectures on mod p Galois representations, the Fontaine-Mazur conjecture for p-adic Galois representations, and the Sato-Tate conjecture for elliptic curves. The aim of this conference is to summarize the results and techniques in these directions and to sketch out a research program that will take us from GL(2) to unitary groups of higher rank. Clay Mathematics Institute Invited Speakers Include: N. Katz (Princeton University), K. Ribet (University of California, Berkeley), C. Breuil (Université de Paris), M. Emerton (Northwestern University), T. Gee (Imperial College, London), M. Kisin (University of Chicago), R. Taylor (Harvard University) D. Blasius (University of California, Los Angeles), M. Harris (Université de Paris), N. Shepherd-Barron (Cambridge University), R. Langlands (School of Mathematics , Princeton), J. Bellaiche (Columbia University) and T. Yoshida (Harvard University). SCHEDULE Monday, October 30 These lectures are aimed at non-specialists. In particular the morning lectures will provide a colloquium style introduction to the main results to be discussed in the workshop. 9:30-10:30 Ken Ribet, Berkeley: Recent advances in modular forms. This will be a colloquium style introduction to recent developments in the theory of modular forms, particularly Serre's conjecture and applications to the modularity of rank two motives, the Fontaine-Mazur conjecture in dimension 2 and perhaps the Sato-Tate conjecture. 11:00-12:00 Nick Katz, Princeton: The Sato-Tate Conjecture This will be a colloquium style introduction to the Sato-Tate conjecture and its relation to L-functions and automorphicity. 2:00- 3:15 Richard Taylor, Harvard: Galois representations and automorphic forms: basic conjectures. This will explain some of the basic concepts in the study of Galois representations and state some motivational conjectures about Galois representations and particularly their relationship to automorphic forms. 3:45-5:00 Richard Taylor, Harvard: Automorphic forms and Galois representations: examples and techniques. Illustrations of the conjectures discussed in the previous talk. Also a discussion of the known techniques to attack them and their limitations. Tuesday, October 31 These talks will aim to explain recent developments in modularity lifting theorems for GL_2/Q to graduate students and people whose main interest is in automorphic forms. 9:00-10:00 Richard Taylor, Harvard: Modularity lifting theorems I 10:30-11:30 Richard Taylor, Harvard: Modularity lifting theorems II Modularity lifting theorems assert that a suitable l-adic lift of an automorphic mod l representation must itself be automorphic. This will describe how to prove such theorems in the simplest possible case: GL_2/Q, weight less than l and level prime to l. It will not follow the original arguments of Wiles and Taylor-Wiles; rather it will incorporate subsequent simplifications due to Diamond/Fujiwara, Kisin and Taylor. In particular it will not use arithmetic algebraic geometry nor Ihara's lemma. 1:30-2:30 Mark Kisin, Chicago: Modularity of potentially Barsotti-Tate representations This will explain the modularity lifting theorems in weight 2 for GL_2 over a totally real field. There will be no restriction on the power of l in the level. 2:45-3:45 Toby Gee, Imperial College: The weight in Serre's conjecture This will explain the conjecture of Diamond et al on the weight in Serre's conjecture for GL(2) over a totally real field. It will also discuss Gee's proof of this conjecture in many cases. 4:00- 5:00 Teruyoshi Yoshida, Harvard: Potential modularity Will discuss potential modularity theorems for GL_2 and their applications. In particular the meromorphic continuation of the L-function of regular rank two motives and the embedding of a two dimensional odd mod l representation of G_Q into a compatible system of l-adic representations with prescribed types. Wednesday, November 1 These talks will summarise the main facts about automorphic forms on unitary groups and the associated Galois representations. They will be aimed at graduate students and experts on elliptic modular forms. 9:00-10:30 Don Blasius, UCLA: Automorphic forms on unitary groups I 11:00-12:30 Joel Bellaiche, Columbia: Automorphic forms on unitary groups II These talks will cover the following topics. The classification of unitary groups over local fields and number fields. The spectral decomposition of automorphic forms on an anisotropic unitary group in terms of cusp forms on general linear groups, including multiplicity formulas: conjectures and known cases. Shimura varieties attached to unitary groups as moduli spaces and their zeta functions; compatibility of the associated Galois representations with the local Langlands correspondence. Afternoon: freetime. We hope to arrange a hike. Thursday, November 2 This will be a continuation of Tuesday’s program but will also discuss the beginnings of p-adic Langlands conjectures for GL_2 and recent work on Serre's conjecture for GL_2. It will be aimed at graduate students and experts on automorphic forms. 9:00-10:00 Christophe Breuil, IHES: Deformations of p-adic representations of p-adic Galois groups This will explain the existence of local deformation rings for De Rham representations of G_Q_l with fixed Hodge-Tate numbers and given type. (Probably without any proof.) It will then explain the Breuil-Mezard conjecture. 10:15-11:15 Mark Kisin, Chicago: p-adic local Langlands A discussion of the p-adic local Langlands conjecture for GL_2(Q_p). 11:30-12:30 Mark Kisin: The Breuil-Mezard conjecture and modularity lifting theorems Will discuss the proof of many cases of the Breuil-Mezard conjecture and its application to general modularity lifting theorems for GL_2(Q). 2:30-3:30 Matt Emerton, Northwestern: The p-adic completion of cohomology This lecture will give the statement of a p-adic global Langlands conjecture for GL_2, and sketch a proof of this conjecture in many cases. Applications of the conjecture to the Fontaine-Mazur conjecture for two-dimensional Galois representations, as well as a related conjecture of Kisin, will also be described. 4:00- 5:00 Ken Ribet, Berkeley: Serre's conjecture Will discuss the Khare-Wintenberger proof of many cases of Serre's conjecture for GL_2(Q). 4:30-5:30: Tea, Sherry and Library Tours; Austine McDonnell Hearst Library Friday, November 3 9:00-10:00 Michael Harris, Paris: Modularity lifting theorems for unitary groups Discussion of the generalization of modularity lifting theorems from GL_2 to higher rank unitary groups. 10:30-11:30 Nick Shepherd-Barron, Cambridge: Geometry of the Dwork family Discussion of the geometric properties of the Dwork family needed for potential modularity results. In particular discussion of its monodromy both locally and globally. [As in section 1 of Harris, Shepherd-Barron, Taylor.] 1:30- 2:30 Michael Harris, Paris: Potential modularity for n-dimensional representations The use of the Dwork family and modularity lifting theorems for unitary groups to prove potential modularity results for higher dimensional representations. Also, arithmetic applications (e.g. the Sato-Tate conjecture and L-functions for elements of the Dwork family). 3:00- 4:00 Robert Langlands, IAS: Whither the trace formula Whither the trace formula and why: relation between trace formula and functoriality -- brief review of history, recent progress, and of scarcely explored approaches; functoriality and reciprocity between automorphic forms and motives -- questions for the audience.
A block of rooms has been reserved at the Hotel Durant. Please mention the workshop name and reference the following code when making reservations via phone, fax or e-mail: K40000. The cut-off date for reservations is September 30, 2006. A block of rooms has been reserved at the Rose Garden Inn. Reservations may be made by calling 1-800-992-9005 OR directly on their website. Click on Corporate at the bottom of the screen and when prompted enter code MATH (this code is not case sensitive). By using this code a new calendar will appear and will show MSRI rate on all room types available. The following Group Code should be entered in the "Comment" field when booking through our Website. Group Code = CGMS35. Please note: The Original Queen room type does not have an in-room T1 connection, but we do have WIFI available in our main lobby area. The cut-off date for reservations is September 30, 2006.

To prove that a mathematical object is “modular” is to link it to an automorphic representation. In the past few years and months, three outstanding modularity conjectures have been settled in a large number of cases: Serre's conjectures on mod p Galois representations, the Fontaine-Mazur conjecture for p-adic Galois representations, and the Sato-Tate conjecture for elliptic curves. The aim of this conference is to summarize the results and techniques in these directions and to sketch out a research program that will take us from GL(2) to unitary groups of higher rank. *Clay Mathematics Institute* **Invited Speakers Include:** N. Katz (Princeton University), K. Ribet (University of California, Berkeley), C. Breuil (Université de Paris), M. Emerton (Northwestern University), T. Gee (Imperial College, London), M. Kisin (University of Chicago), R. Taylor (Harvard University) D. Blasius (University of California, Los Angeles), M. Harris (Université de Paris), N. Shepherd-Barron (Cambridge University), R. Langlands (School of Mathematics , Princeton), J. Bellaiche (Columbia University) and T. Yoshida (Harvard University). **SCHEDULE** **Monday, October 30** These lectures are aimed at non-specialists. In particular the morning lectures will provide a colloquium style introduction to the main results to be discussed in the workshop. 9:30-10:30 Ken Ribet, Berkeley: Recent advances in modular forms. This will be a colloquium style introduction to recent developments in the theory of modular forms, particularly Serre's conjecture and applications to the modularity of rank two motives, the Fontaine-Mazur conjecture in dimension 2 and perhaps the Sato-Tate conjecture. 11:00-12:00 Nick Katz, Princeton: The Sato-Tate Conjecture This will be a colloquium style introduction to the Sato-Tate conjecture and its relation to L-functions and automorphicity. 2:00- 3:15 Richard Taylor, Harvard: Galois representations and automorphic forms: basic conjectures. This will explain some of the basic concepts in the study of Galois representations and state some motivational conjectures about Galois representations and particularly their relationship to automorphic forms. 3:45-5:00 Richard Taylor, Harvard: Automorphic forms and Galois representations: examples and techniques. Illustrations of the conjectures discussed in the previous talk. Also a discussion of the known techniques to attack them and their limitations. **Tuesday, October 31** These talks will aim to explain recent developments in modularity lifting theorems for GL_2/Q to graduate students and people whose main interest is in automorphic forms. 9:00-10:00 Richard Taylor, Harvard: Modularity lifting theorems I 10:30-11:30 Richard Taylor, Harvard: Modularity lifting theorems II Modularity lifting theorems assert that a suitable l-adic lift of an automorphic mod l representation must itself be automorphic. This will describe how to prove such theorems in the simplest possible case: GL_2/Q, weight less than l and level prime to l. It will not follow the original arguments of Wiles and Taylor-Wiles; rather it will incorporate subsequent simplifications due to Diamond/Fujiwara, Kisin and Taylor. In particular it will not use arithmetic algebraic geometry nor Ihara's lemma. 1:30-2:30 Mark Kisin, Chicago: Modularity of potentially Barsotti-Tate representations This will explain the modularity lifting theorems in weight 2 for GL_2 over a totally real field. There will be no restriction on the power of l in the level. 2:45-3:45 Toby Gee, Imperial College: The weight in Serre's conjecture This will explain the conjecture of Diamond et al on the weight in Serre's conjecture for GL(2) over a totally real field. It will also discuss Gee's proof of this conjecture in many cases. 4:00- 5:00 Teruyoshi Yoshida, Harvard: Potential modularity Will discuss potential modularity theorems for GL_2 and their applications. In particular the meromorphic continuation of the L-function of regular rank two motives and the embedding of a two dimensional odd mod l representation of G_Q into a compatible system of l-adic representations with prescribed types. **Wednesday, November 1** These talks will summarise the main facts about automorphic forms on unitary groups and the associated Galois representations. They will be aimed at graduate students and experts on elliptic modular forms. 9:00-10:30 Don Blasius, UCLA: Automorphic forms on unitary groups I 11:00-12:30 Joel Bellaiche, Columbia: Automorphic forms on unitary groups II These talks will cover the following topics. The classification of unitary groups over local fields and number fields. The spectral decomposition of automorphic forms on an anisotropic unitary group in terms of cusp forms on general linear groups, including multiplicity formulas: conjectures and known cases. Shimura varieties attached to unitary groups as moduli spaces and their zeta functions; compatibility of the associated Galois representations with the local Langlands correspondence. Afternoon: freetime. We hope to arrange a hike. **Thursday, November 2** This will be a continuation of Tuesday’s program but will also discuss the beginnings of p-adic Langlands conjectures for GL_2 and recent work on Serre's conjecture for GL_2. It will be aimed at graduate students and experts on automorphic forms. 9:00-10:00 Christophe Breuil, IHES: Deformations of p-adic representations of p-adic Galois groups This will explain the existence of local deformation rings for De Rham representations of G_Q_l with fixed Hodge-Tate numbers and given type. (Probably without any proof.) It will then explain the Breuil-Mezard conjecture. 10:15-11:15 Mark Kisin, Chicago: p-adic local Langlands A discussion of the p-adic local Langlands conjecture for GL_2(Q_p). 11:30-12:30 Mark Kisin: The Breuil-Mezard conjecture and modularity lifting theorems Will discuss the proof of many cases of the Breuil-Mezard conjecture and its application to general modularity lifting theorems for GL_2(Q). 2:30-3:30 Matt Emerton, Northwestern: The p-adic completion of cohomology This lecture will give the statement of a p-adic global Langlands conjecture for GL_2, and sketch a proof of this conjecture in many cases. Applications of the conjecture to the Fontaine-Mazur conjecture for two-dimensional Galois representations, as well as a related conjecture of Kisin, will also be described. 4:00- 5:00 Ken Ribet, Berkeley: Serre's conjecture Will discuss the Khare-Wintenberger proof of many cases of Serre's conjecture for GL_2(Q). 4:30-5:30: Tea, Sherry and Library Tours; Austine McDonnell Hearst Library **Friday, November 3** 9:00-10:00 Michael Harris, Paris: Modularity lifting theorems for unitary groups Discussion of the generalization of modularity lifting theorems from GL_2 to higher rank unitary groups. 10:30-11:30 Nick Shepherd-Barron, Cambridge: Geometry of the Dwork family Discussion of the geometric properties of the Dwork family needed for potential modularity results. In particular discussion of its monodromy both locally and globally. [As in section 1 of Harris, Shepherd-Barron, Taylor.] 1:30- 2:30 Michael Harris, Paris: Potential modularity for n-dimensional representations The use of the Dwork family and modularity lifting theorems for unitary groups to prove potential modularity results for higher dimensional representations. Also, arithmetic applications (e.g. the Sato-Tate conjecture and L-functions for elements of the Dwork family). 3:00- 4:00 Robert Langlands, IAS: Whither the trace formula Whither the trace formula and why: relation between trace formula and functoriality -- brief review of history, recent progress, and of scarcely explored approaches; functoriality and reciprocity between automorphic forms and motives -- questions for the audience.

A block of rooms has been reserved at the Hotel Durant. Please mention the workshop name and reference the following code when making reservations via phone, fax or e-mail: **K40000**. The cut-off date for reservations is September 30, 2006. A block of rooms has been reserved at the Rose Garden Inn. Reservations may be made by calling 1-800-992-9005 OR directly on their website. Click on Corporate at the bottom of the screen and when prompted enter code **MATH** (this code is not case sensitive). By using this code a new calendar will appear and will show MSRI rate on all room types available. The following Group Code should be entered in the "Comment" field when booking through our Website. Group Code = **CGMS35**. Please note: The Original Queen room type does not have an in-room T1 connection, but we do have WIFI available in our main lobby area. The cut-off date for reservations is September 30, 2006.

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