Explore the fourth lecture in a series on birational invariants derived from Gromov-Witten theory, delivered by renowned mathematician Maxim Kontsevich at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the intricate world of Gromov-Witten invariants and their application to smooth projective varieties, examining the infinite tower of cohomology classes they generate and the constraints encoded in the quantum connection. Discover Hiroshi Iritani's groundbreaking theorem on controlling the quantum connection of blowups along smooth centers. Investigate the development of new birational invariants based on Iritani's work, and witness how these concepts can be applied to prove the non-rationality of a generic cubic 4-fold with remarkable ease. Gain insights from Kontsevich's expertise in this advanced mathematical exploration that bridges algebraic geometry and quantum cohomology.