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Applied Mathematics Lab (Fukuhiro Ueda)
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The lab studies arithmetic geometry and number theory, centered on p-adic Hodge theory.
AboutAbout the lab
Connecting number theory and geometry through p-adic Hodge theory
The lab studies arithmetic geometry and number theory.
Its main subjects are p-adic Hodge theory, Galois representations, and K-theory.
The lab has worked on one of the central questions of the Langlands program: which Galois representations arise from algebraic geometry.
Currently, it is focusing on computing the étale cohomology of arithmetic schemes using K-theory and higher class field theory.
Key terms
- p-adic Hodge theory
- A theory that compares objects of number theory with objects of geometry in the p-adic numbers, the number system attached to each prime p
- Galois representation
- A description, in the form of matrices, of the permutations of the solutions of an equation (the Galois group)
- Étale cohomology
- An extension of the tools that extract information about the "holes" of a shape so that they can also be used for objects in number theory
ResearchResearch themes
The lab studies arithmetic geometry and number theory, in particular p-adic Hodge theory, Galois representations, and K-theory. It works on comparison theorems in p-adic Hodge theory and their applications to automorphic forms.
*Illustrative imageThe Fontaine-Mazur conjecture
In the Langlands program, one of the central questions is which Galois representations come from algebraic geometry. The two-dimensional case over the rational numbers has been resolved through research over the past several decades.
*Illustrative imageComparison theorems in p-adic Hodge theory
On comparison theorems between p-adic étale cohomology and crystalline cohomology, the lab used methods based on the pro-étale site to prove comparisons with nontrivial coefficients and in relative settings.
*Illustrative imageApplications to families of automorphic forms
p-adic Hodge theory can also be applied to automorphic forms (and their families). Using p-adic Hodge theory, the lab constructed two-dimensional eigenvarieties over arbitrary number fields.
MembersFaculty
- LecturerFukuhiro Ueda
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Eisenstein 14” Jan Homann / Public domain (cropped)
- The Fontaine-Mazur conjecture: “KleinjHSV” Empetrisor / CC BY-SA 4.0 (cropped)
- Comparison theorems in p-adic Hodge theory: “3-adic integers” LightbulbMEOW / CC BY-SA 4.0 (cropped)
- Applications to families of automorphic forms: “Mathematical analysis, Linear algebra, Mathematics lecture notes, Rostov-on-Don, Russia” Vyacheslav Argenberg / CC BY 4.0 (cropped)
- Related lab (Hikita Lab): “Young tableau for partitions of number 10 stacked” Unmismoobjetivo / CC BY 4.0 (cropped)
- Related lab (Applied Mathematics Division (Kawakita) Lab): “Du Val singularities” Aizenr / CC BY-SA 4.0 (cropped)
- Related lab (Applied Mathematics Lab (Kazuhisa Makino)): “B functies 2var versie02” Bob.v.R / CC BY-SA 4.0 (cropped)
- Related lab (Applied Mathematics Division (Kobayashi) Lab): “Graph-matroid” Didia / CC BY-SA 4.0 (cropped)
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