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Abe Lab
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The lab uses p-adic cohomology to study the forms hidden in equations of numbers.
AboutAbout the lab
Seeing equations of numbers through the eyes of geometry
The Abe Lab studies arithmetic geometry, in particular p-adic cohomology.
Arithmetic geometry is a field that uses the methods of geometry to study number-theoretic equations and, more generally, arithmetic varieties.
The lab is interested in the relationships among several cohomology theories, such as étale cohomology, that extract topological information from equations.
The lab is also interested in bringing into number theory the intuition behind the relationship between geometric Langlands and S-duality, which physicists have pointed out.
Key terms
- Cohomology
- A tool that extracts information such as the holes and connectedness of a shape, in the language of algebra
- Étale cohomology
- A cohomology theory defined by Grothendieck that reflects the arithmetic properties of equations
- Arithmetic D-module
- An object of a theory that handles the idea of differential operators in the world of p-adic numbers; one of the frameworks of p-adic cohomology
ResearchResearch themes
The lab's field is arithmetic geometry, with a specialization in p-adic cohomology. From the viewpoint of the Langlands program, it investigates the relationships among various cohomology theories.
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MembersFaculty
- ProfessorTomoyuki Abe
VoicesStudent voices
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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: “3-adic integers” LightbulbMEOW / CC BY-SA 4.0 (cropped)
- p-adic cohomology: “Elliptic curve y^2=x^3-x on finite field Z 71” Technopop.tattoo / CC0 (cropped)
- Related lab (Imai Lab): “A Klein Quartic embedding in 3D” Ferkel / CC BY-SA 4.0 (cropped)
- Related lab (Shiho Lab): “Elliptic curve y^2=x^3-x on finite field Z 89” Technopop.tattoo / CC0 (cropped)
- Related lab (Tsuji Lab): “Riemann zeta surface” SteJaes / CC BY-SA 4.0 (cropped)
- Related lab (Ito Lab): “Tetrahedroid” Salix alba / CC BY-SA 4.0 (cropped)
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