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Kawazumi Lab
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The lab studies the shape of Riemann surfaces and their moduli spaces.
AboutAbout the lab
Exploring the shape of the space of Riemann surfaces
The Kawazumi Lab works in topology, studying the topology of Riemann surfaces themselves and of the moduli spaces of Riemann surfaces.
In its research over the past few years, the (generalized) Magnus expansion of free groups has played a central role.
The lab constructed Magnus expansions by geometric methods and extended the Johnson homomorphisms to the whole mapping class group in a geometrically meaningful way.
It also constructed a Lie algebra homomorphism from the Goldman Lie algebra, and as an application gave an explicit description of the action of Dehn twists on the completed group ring of the fundamental group of a surface.
Key terms
- Moduli space
- A space formed by collecting all figures of a certain kind and regarding each one as a point.
- Mapping class group
- A group formed by collecting the transformations that map a surface to itself, where two transformations are regarded as the same if one can be continuously deformed into the other.
- Dehn twist
- An operation that cuts a surface along a loop, twists it once, and glues it back together.
ResearchResearch themes
The lab studies the topology of Riemann surfaces and their moduli spaces. Using Magnus expansions of free groups as a central tool, it investigates the actions of the mapping class group and of Dehn twists.
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MembersFaculty
- ProfessorNariya KawazumiPublications ↗
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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: “Riemann sqrt” Wlongqi / CC BY-SA 3.0 (cropped)
- Moduli Spaces of Riemann Surfaces: “Dehn twist” Orthografer / Public domain (cropped)
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