Explore advanced concepts in commutative algebra and algebraic geometry through this 44-minute lecture by Pedro Montero, presented at the 15th ALGA (Commutative Algebra and Algebraic Geometry) conference held at IMPA in Rio de Janeiro. Delve into cutting-edge research and discussions on algebraic geometry, commutative algebra, and related topics in number theory, code theory, and foliation. Gain insights from this prestigious event, which has been a cornerstone in the field for over two decades, featuring renowned mathematicians and researchers from around the world.