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Explore the foundational concepts of mock theta functions in this conference lecture delivered at the International Centre for Theoretical Sciences. Discover the historical origins of these mysterious mathematical objects, first described by Ramanujan in his famous deathbed letter to G. H. Hardy in 1920, where he characterized them as having "modular-like" properties. Learn how these enigmatic functions remained poorly understood for decades until breakthrough work by Zwegers, Bruinier, and Funke provided the proper mathematical framework for their study. Understand the modern perspective that reveals Ramanujan's mock theta functions as holomorphic parts of nonholomorphic functions called harmonic Maass forms. Examine the fundamental properties and characteristics that distinguish mock theta functions from classical modular forms while exploring their connections to harmonic Maass forms and mock modular forms. Gain insight into the diverse applications of these mathematical objects across multiple fields including partition theory, elliptic curves, black hole physics, and representation theory. This introductory lecture serves as part of a comprehensive discussion meeting focused on harmonic Maass forms, mock modular forms, and their wide-ranging applications in mathematics and theoretical physics.
