Uploaded June 2026 | Updated September 2026, 3 weeks ago
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
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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
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
