Explore a 38-minute lecture from the 15th ALGA (Commutative Algebra and Algebraic Geometry) conference, delivered by Jorge Vitório Pereira at the Instituto de Matemática Pura e Aplicada in Rio de Janeiro. Delve into topics such as differential forms, Riemann surfaces, twisted connections, closed metamorphic one-forms, foliations, and rational transformations. Gain insights into the Bot's Connection, Residue Formula, and simple singularities. Learn about a simple criterion for local singularities and understand the motivation behind these advanced mathematical concepts. Part of a larger conference series focusing on Algebraic Geometry, Commutative Algebra, and related topics in Number Theory, Code Theory, and Foliation, this lecture offers a deep dive into cutting-edge mathematical research.