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Explore the fascinating mathematical journey from Ramanujan's 1916 paper to the resolution of Serre's modularity conjecture in this Foundation Day lecture by renowned number theorist Chandrashekhar Khare from UCLA. Trace the development of mathematical ideas spanning nearly a century, beginning with Ramanujan's groundbreaking work and culminating in the formulation and eventual proof of Jean-Pierre Serre's influential conjecture from the 1970s. Discover how Serre's modularity conjecture served as a catalyst for mathematical developments that proved crucial to Andrew Wiles's historic 1994 proof of Fermat's Last Theorem, and learn about the remarkable irony that while Wiles initially considered his methods orthogonal to Serre's conjecture, the eventual proof of Serre's conjecture relied fundamentally on Wiles's techniques. Delve into the intricate interplay between Galois and Ramanujan symmetries, the two mathematical symmetries that form the central narrative of this mathematical tale. Gain insights into the collaborative breakthrough achieved by Khare and Jean-Pierre Wintenberger in 2009, when they successfully resolved this long-standing conjecture in number theory. The lecture provides a mathematical introduction to these complex ideas, offering a more technical perspective on concepts explored in Khare's mathematical memoir "Chasing A Conjecture: Inside the Mind of a Mathematician," while making the sophisticated interplay of modular forms and Galois representations accessible to a mathematical audience.
