Summer School 2009

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Clay Mathematics Institute 2009 Summer School

Galois Representations

June 15 - July 10, 2009

University of Hawaii at Manoa Honolulu, Hawaii

Many advances on the algebraic side of number theory in the last 15 years (such as the solutions of the Shimura-Taniyama conjecture, Sato-Tate conjecture and Serre's conjecture, as well as decisive progress on the Fontaine-Mazur conjecture and Main Conjectures for modular forms) have relied in an essential way on improvements in the theory of Galois representations. For example, such improvements have enabled the local and global aspects of modularity lifting theorems to be extended far beyond the traditional 2-dimensional case over the rational numbers, and have led to generalizations of the "classical" theory of p-adic modular forms in a way that makes more effective use of representation theory and geometry to obtain results on the arithmetic of L-values.

The aim of the three main courses is to present an overview of many of these ideas and applications, aimed at advanced graduate students and postdocs with a strong background in number theory, Galois cohomology, and basic algebraic geometry. One course will focus entirely on local problems (p-adic representations of Galois groups of p-adic fields), a second course will have a more global flavor (Galois deformation theory and global applications), and a third (on L-values) will rely on the other two courses. During the final week of the school there will be mini-courses on some more specialized topics.

Foundational Courses

• p-padic Hodge Theory
  Olivier Brinon, Brian Conrad
  (Φ, Γ)-modules, applications to potentially semi-stable deformation rings and families of Galois representations
• Deformation of Galois Representations and Modular Forms
  Jacques Tilouine, Mark Kisin
  Deformation theory of Galois representations, pseudo-representations, with applications to eigenvarieties, construction of Galois representations, Taylor-Wiles method
• Iwasawa Theory and Automorphic Applications
  Joel Bellaiche, Chris Skinner
  Iwasawa theory and Hida theory with applications to Selmer groups, automorphic forms, and the arithmetic of special values of L-functions

Scientific Committee

• Brian Conrad (Stanford University)
• David Alexandre Ellwood (CMI)
• Mark Kisin (University of Chicago)
• Chris Skinner (Princeton University)

Lecturers

• Joel Bellaiche (Brandeis University)
• Olivier Brinon (Université Paris 13)
• Brian Conrad (Stanford University)
• Mark Kisin (University of Chicago)
• Chris Skinner (Princeton University)
• Jacques Tilouine (Université Paris 13)

Funding

Funding is available to graduate students and postdoctoral fellows who are within five years of receipt of their Ph.D. Standard support amounts will include funds towards accommodation and economy travel.

Application

Interested participants should complete the online application form as well as upload a copy of their CV/list of publications. A letter of recommendation from either a senior mathematician or mathematics advisor is required. Please note that only complete applications will be considered. The application deadline is February 15, 2009.

Online Application Form

University of Hawaii, Mathematics Department