Wieslava Niziol - Hodge theory of p-adic varieties - Faltings' legacy

Wiesława Nizioł (CNRS, Sorbonne University) — Hodge Theory of p-adic Varieties: Faltings' Legacy | Abel Lecture 2026 How do you relate the étale cohomology of a variety to its differential forms when the ground field is p-adic rather than complex? In this lecture, Wiesława Nizioł traces the development of p-adic Hodge theory — the area that answers this question — and the central role played throughout by the ideas of Gerd Faltings. Starting from the contrast between the real/complex and the p-adic worlds, she builds the story up through classical Hodge theory and the notion of a period (the p-adic analogue of 2πi), Galois representations arising from geometry, and Fontaine's period rings. She explains the Hodge–Tate, de Rham, crystalline and semi-stable comparison theorems, and shows how Faltings' almost purity theorem and his theory of almost étale extensions make the key computations possible — ideas later recast by Scholze through perfectoid spaces and tilting. The lecture closes with applications across number theory, commutative algebra, and algebraic geometry in mixed characteristic. Wiesława Nizioł is director of research at CNRS, Sorbonne University. She completed her PhD at Princeton University under Gerd Faltings, has held positions at Chicago, Minnesota and Utah, was an invited speaker at the 2006 International Congress of Mathematicians, and was elected to Academia Europaea in 2021. Learn more about the Abel Prize: https://abelprize.no Subscribe to follow the worlds highest ranked mathematics prize. 0:00 Introduction 0:42 The p-adic world: numbers and non-archimedean norms 4:17 Periods and the p-adic analogue of 2πi 7:18 Classical Hodge theory: Betti and de Rham cohomology 10:15 The Hodge decomposition and the p-adic goal 12:49 Galois representations from geometry 15:13 Period rings and the search for the ring B 17:38 The Hodge–Tate decomposition 20:15 The de Rham comparison and Fontaine's ring B_dR 25:15 Crystalline and semi-stable periods (B_cr, B_st) 25:50 Fontaine on inventing the period rings 28:24 The semi-stable comparison theorem 31:54 Faltings' almost purity theorem 36:43 The key computation in dimension one 40:29 Tilting and perfectoid spaces 45:15 Applications 46:19 Questions from the audience