The Langlands philosophy predicts a connection between Galois representations and automorphic forms, which vastly generalizes the class field theory. This has become a central research area in arithmetic geometry, number theory and representation theory ever since he made public his program in 1960ˇ¦s. In establishing correspondences from automorphic forms to Galois representations Shimura varieties played an essential role. The remarkable breakthrough by Wiles and Taylor in reverse direction resulted in the settlement of Fermatˇ¦s Last Theorem. Since then, tremendous progress has been made in this area in recent years, as evidenced by the resolution of Serreˇ¦s conjecture and progress on the conjectures of Artin, Sato-Tate and Fontaine-Mazur, and the proof of the fundamental lemma. Meanwhile, the study of arithmetic of modular forms for noncongruence forms is revitalized; through the attached Galois representations and Langlands philosophy, congruence relations between Fourier coefficients of noncongruence forms and those of automorphic forms can be obtained.
The purpose of this conference is to showcase the multi-facets of Galois representations and their interconnections. Recent advances in and applications of Galois representations, automorphic forms and arithmetic of Shimura varieties, from theoretical and computational aspects, will be addressed by the invited experts.