Explore a 42-minute lecture from the 15th ALGA (Commutative Algebra and Algebraic Geometry) conference, delivered by Renzo Cavalieri at IMPA in Rio de Janeiro. Delve into topics such as modular space of curves, intersection theory, dual graphs, and pseudostable curves. Gain insights into Matthews theorem and its applications in algebraic geometry. Learn about the long-standing tradition of the ALGA meeting, which focuses on algebraic geometry, commutative algebra, and related topics in number theory, code theory, and foliation. Discover the work of the organizing committee and access additional resources, including the event page and video playlist, to further enhance your understanding of these advanced mathematical concepts.