Explore a 39-minute lecture on the formalization of Carleson's theorem in Fourier analysis. Delve into the fundamental question of when Fourier series converge to the original function, examining Carleson's 1966 proof of convergence for almost all points on the real line. Learn about the challenges in understanding Carleson's complex proof and recent developments in generalizing the theorem. Discover the 2024 work by Christoph Thiele et al. on extending the boundedness of Carleson's operator to doubling metric measure spaces. Gain insight into an ongoing project to formalize this theorem using the Lean theorem prover, based on a detailed blueprint by Thiele and collaborators.