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and its Applications, and the Number Theory Foundation. Over 100 mathematicians attended, and there were 23 one-hour lectures. The conference focused on several aspects of the Langlands program, including some exposition of Shahidi's work, recent progress, and future avenues of investigation. Far from being a retrospective, the conference emphasized the vast array of significant problems ahead. All lecturers were invited to contribute material for this volume. In addition, some important figures who were unable to attend or deferred on speaking at the conference were invited to submit articles as well. We hope this resulting volume will serve as a modest tribute to Shahidi's legacy to date, but should not be considered the final word on this subject.The editors wish to thank all of the authors for their willingness to contribute manuscripts of such high quality in honor of our colleague. We also wish to thank the anonymous referees for their conscientious reading of these manuscripts and their helpful comments to authors which have improved the contents. We wish to express our deep gratitude to

Functoriality is one of the most central questions in the theory of automorphic forms and representations [1,2,35,36]. Locally and globally, it is a manifestation of Langlands' formulation of a non-abelian class field theory. Now known as the Langlands correspondence, this formulation of class field theory can be viewed as giving an arithmetic parameterization of local or automorphic representations in terms of admissible homomorphisms of (an appropriate analogue) of the WeilDeligne group into the Langlands dual group or L-group. When this conjectural parameterization is combined with natural homomorphisms of the L-groups it predicts a transfer or lifting of local or automorphic representations of two reductive algebraic groups. As a purely automorphic expression of a global non-abelian class field theory, global functoriality is inherently an arithmetic process.In this paper we establish global functoriality from the split classical groups G n = SO 2n+1 , SO 2n , or Sp 2n to an appropriate general linear group GL N , associated to the natural embedding of L-groups, for globally generic cuspidal representations π of G n (A) over a number field k. We had previously presented functoriality for the case G n = SO 2n+1 in [6], but were limited at that time by a lack of suitable local tools in the other cases. The present paper is by no means a simple generalization of [6]. There were serious local problems to be overcome in the development of the tools that now allow us to cover all three series of classical groups simultaneously and that will be applicable to other cases of functoriality in the future. In addition, we have completely determined the associated local images of functoriality and as a result are able to present several new applications of functoriality, including both global results concerning the Ramanujan conjecture for the classical groups and various applications to the local representation theory of the classical groups.There are several approaches to the question of functoriality: the trace formula, the relative trace formula, and the Converse Theorem. In this work we use the Converse Theorem, which is an L-function method. The Converse Theorem itself states that if one has an irreducible admissible representation Π ⊗ Π v of GL N (A), then Π is in fact automorphic if sufficiently many of its twisted L-functions L(s, Π × τ), with τ cuspidal automorphic representations of smaller GL m (A), are nice [7,9]. As a vehicle for establishing functoriality from cuspidal representations π = ⊗π v of some G n (A) to an automorphic representation of GL N (A), there are three main steps. The first is to construct a candidate lift Π = ⊗ Π v . This is done by locally lifting each local component representationin such a way that twisted local L-and ε-factors are matched. At the archimedean places and the finite places where π v is unramified we may accomplish this local lift by using the local Langlands correspondence, i.e., the local arithmetic Langlands classification. At the remaining finite set of places where π...

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