Explore symmetries in complex algebraic varieties and the Cremona group in this 59-minute math seminar. Delve into the concept of automorphisms and their limitations in birational geometry, leading to the study of birational self-maps. Examine examples of automorphism groups of projective varieties and investigate the structure of the Cremona group, particularly in dimensions 2 and higher. Learn about recent work on constructing special subgroups of the Cremona group that preserve certain meromorphic volume forms. Gain insights into the classification problem, minimal model program, and other key concepts in algebraic geometry through this comprehensive lecture by Carolina Araujo from IMPA.