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Watch this 20-minute conference presentation from CPP 2026 that explores the formalization of Roby's 1965 construction of the universal divided power algebra using the Lean/Mathlib mathematical library. Learn about the ongoing formalization work by Antoine Chambert-Loir and MarÃa Inés de Frutos-Fernández, which tackles a crucial algebraic structure analogous to classical polynomial algebras in divided power theory. Discover how this universal divided power algebra serves as an essential tool in crystalline cohomology development and p-adic Hodge theory for defining crystalline period rings. Explore the main challenge of constructing a divided power structure on the augmentation ideal through Roby's identification of graded pieces with homogeneous polynomial laws. Understand the formalization of polynomial laws theory and see how this foundational work enables future completion of the divided power structure formalization. Gain insights into the technical difficulties encountered during formalization, including universe management, extending Mathlib library aspects to semirings, and addressing instances of "invisible mathematics" that complicate formal verification efforts.
