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Explore the intricate connections between modular forms and differential equations in this comprehensive lecture. Delve into the origins of automorphic forms theory from the late 19th and early 20th centuries, tracing its development through the works of Klein, Fricke, Poincaré, and others. Discover how this aspect has evolved over time and its relevance in modern mathematics. Examine various types of differential equations, including linear equations used in Apéry's renowned proof of the irrationality of ζ(2) and ζ(3), as well as modular linear differential equations significant in conformal field theory and vertex operator algebras. Investigate non-linear differential equations, such as the Chazy differential equation in Painlevé equations theory and operators from Frobenius manifolds. Gain insights into the applications of these connections and their impact on contemporary mathematical research.
