Mumford–Tate Groups in Hodge Theory - Aula 04

Explore advanced concepts in algebraic geometry through this lecture focusing on Mumford-Tate groups within Hodge theory. Delve into the intricate mathematical structures that connect algebraic cycles, motives, and representation theory as Professor Andrey Soldatenkov from UNICAMP presents the fourth installment of this specialized minicourse. Examine the fundamental role of Mumford-Tate groups in understanding the Galois-theoretic properties of Hodge structures, their applications to periods of algebraic varieties, and their significance in the broader context of arithmetic geometry. Learn about the classification of Hodge structures through their associated Mumford-Tate groups, the relationship between these groups and special subvarieties of Shimura varieties, and recent developments in the field. Gain insights into how these mathematical objects provide a bridge between complex geometry and number theory, with particular emphasis on their role in conjectures such as the Hodge conjecture and the André-Oort conjecture. This advanced mathematical content is part of the 2026 Summer School program at Instituto de Matemática Pura e Aplicada, designed for graduate students and researchers working in algebraic geometry, number theory, and related fields.