James Arthur's work on the Langlands program has fundamentally shaped the modern theory of automorphic forms and representations. His development of the Arthur-Selberg trace formula provided a rigorous framework connecting the geometric and spectral aspects of automorphic representations. This analytical tool, though technically demanding, has become essential for studying the discrete spectrum representations of reductive groups. Arthur's mathematical journey began at Yale under Robert Langlands, with his thesis on discrete series for semisimple Lie groups. His subsequent work focused on developing and refining the trace formula, establishing its convergence and invariance properties through careful analysis. The introduction of Arthur parameters and Arthur packets provided a structured approach to understanding the discrete automorphic spectrum for classical groups. This culminated in his work on endoscopic classification, detailed in his 2013 book "The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups." This classification addressed many of his own conjectures and resolved long-standing questions about the structure of automorphic representations for classical groups. 

At the University of Toronto, Arthur's research program has contributed to a strong Canadian presence in automorphic forms and representation theory. His mathematical legacy includes not only the trace formula but also significant advances in the theory of endoscopy, which connects representations of certain different reductive groups. His current thought provides key insights into Langlands functoriality more generally. The applications of Arthur's work extend across several mathematical domains. The Arthur-Selberg trace formula has found applications in base change, period relations, and various aspects of the Langlands program. His ideas have influenced developments in p-adic groups, motivic Galois theory, and geometric Langlands theory. Arthur's contributions have been recognized through major awards including the Wolf Prize and the Steele Prize, and he is a fellow of the Royal Society. His research continues to generate new directions in representation theory and automorphic forms. The stabilization of the trace formula and the theory of Arthur packets remain active areas of investigation, with ongoing applications in number theory and representation theory. The mathematical legacy of James Arthur reaches around the globe, as reflected in this highly international conference to celebrate his work and to discuss recent developments that are central to the Langlands program.