Explore the intricate concepts of modularity and resurgence in Calabi-Yau compactifications through this comprehensive lecture delivered by Sergey Alexandrov at the Galileo Galilei Institute. Delve into advanced topics in theoretical physics and mathematics as the speaker presents a detailed analysis of these complex phenomena within the context of string theory and algebraic geometry. Gain insights into the mathematical structures underlying Calabi-Yau manifolds and their role in compactification scenarios. Examine the interplay between modular forms, resurgent functions, and their applications in understanding the geometry and physics of these multidimensional spaces. Enhance your understanding of cutting-edge research in mathematical physics and its implications for our comprehension of the fundamental structure of the universe.