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Focusing on a new theory of modular forms, this monograph aims to integrate existing theories from various fields, including congruence groups and automorphic forms on Hermitian symmetric domains. It explores advanced concepts like Calabi-Yau modular forms and their applications, such as Yukawa couplings and topological string partition functions. A key element of this theory is the innovative use of the "Gauss-Manin connection in disguise," which serves as a foundational component to extend the current understanding of modular forms.

Offers an examination of the precursors of Hodge theory: first, the studies of elliptic and abelian integrals by Cauchy, Abel, Jacobi, and Riemann; and then the studies of two-dimensional multiple integrals by Poincare and Picard. The focus turns to the Hodge theory of affine hypersurfaces given by tame polynomials.