The Power of Partitions and Partitions into Powers

Explore the fascinating world of mathematical partitions in this colloquium lecture delivered by renowned number theorist Don Zagier from the Max Planck Institute for Mathematics. Discover how the study of partitions began with Euler's foundational work on generating functions, which led to beautiful identities and recursions that later emerged as expansions of the Dedekind eta-function - one of the earliest examples of modular forms. Learn about the revolutionary contributions of Hardy and Ramanujan in the 20th century, who developed the circle method to derive approximate formulas with error terms smaller than 1/2, effectively yielding exact formulas for partition functions. Delve into the more challenging generalization of partitions into squares and higher powers, which introduces fascinating new mathematical aspects including functional equations that replace classical modularity. Gain insights into the subtle numerical experiments underlying these results, which reveal numerous mathematical surprises and demonstrate the deep connections between partition theory, modular forms, and advanced number theory techniques.