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research is at the intersection of number theory and algebraic geometry, and deals with the basic questions and structures that underly our modern understanding of diophantine equations. This concerns on the one hand the Langlands program, which connects objects from arithmetic geometry, such as elliptic curves over number fields, with analytic or topological objects, such as modular forms or the cohomology of hyperbolic manifolds. On the other hand, this concerns the cohomological invariants attached to arithmetic varieties themselves, which forms the subject of Hodge theory, or more particularly p-adic Hodge theory.
My own contributions to these questions include the construction of Galois representations associated for example with torsion classes on hyperbolic 3-manifolds, and the development of a geometric framework underlying much of p-adic Hodge theory, in the form of perfectoid spaces. In the future, I plan to pursue these questions. In particular, building on perfectoid spaces, I want to transport some machinery from the geometric Langlands program to the case of p-adic fields, and I hope to use this to obtain new results on the local Langlands conjecture for general groups over p-adic fields. On the other hand, in work with Bhatt and Morrow, we found new cohomological invariants of arithmetic varieties, which lead to a surprising q-deformation of de Rham cohomology, which I plan to study further.
2018
Fields Medal
My own contributions to these questions include the construction of Galois representations associated for example with torsion classes on hyperbolic 3-manifolds, and the development of a geometric framework underlying much of p-adic Hodge theory, in the form of perfectoid spaces. In the future, I plan to pursue these questions. In particular, building on perfectoid spaces, I want to transport some machinery from the geometric Langlands program to the case of p-adic fields, and I hope to use this to obtain new results on the local Langlands conjecture for general groups over p-adic fields. On the other hand, in work with Bhatt and Morrow, we found new cohomological invariants of arithmetic varieties, which lead to a surprising q-deformation of de Rham cohomology, which I plan to study further.
2018
Fields Medal
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Kestutis Cesnavicius,Peter Scholze
ANNALS OF MATHEMATICSno. 1 (2024): 51-180
Oberwolfach Reportsno. 2 (2021): 1023-1082
arxiv(2021)
Princeton University Press eBookspp.191-206, (2020)
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Berkeley Lectures on p-adic Geometrypp.169-181, (2020)
Berkeley Lectures on p-adic Geometrypp.149-160, (2020)
Princeton University Press eBookspp.41-48, (2020)
Princeton University Press eBookspp.149-160, (2020)
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