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Explore the foundational concepts of harmonic Maass forms in this comprehensive lecture delivered at the International Centre for Theoretical Sciences. Delve into the mathematical framework that emerged from Ramanujan's mysterious mock theta functions, which he described in his famous deathbed letter to G. H. Hardy in 1920. Learn how these "modular-like" functions remained enigmatic until breakthrough work by Zwegers, Bruinier, and Funke provided the proper theoretical foundation for understanding them. Discover how Ramanujan's mock theta functions are actually the holomorphic parts of nonholomorphic functions called harmonic Maass forms. Understand the connections between harmonic Maass forms and mock modular forms, and their wide-ranging applications across diverse mathematical fields including partition theory, elliptic curves, black hole physics, and representation theory. Gain insight into why these mathematical objects have become such active areas of research over the past 25 years, building upon the rich tradition of modular forms and their generalizations that appear throughout number theory, representation theory, discrete geometry, and theoretical physics.
