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Geometry Lab (Akita)
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Research on group cohomology, which spans algebra and topology.
AboutAbout the lab
Linking algebra and topology through groups.
Prof. Akita's main research subject is the cohomology of groups.
Group cohomology is defined algebraically, yet it also coincides with the cohomology of the space determined by the group, placing it at the intersection of topology and algebra.
The professor studies the cohomology of Coxeter groups, Artin groups, and mapping class groups of closed surfaces.
Recently, the professor has also been working on algebraic systems closely related to topology, such as quandles.
Key terms
- Cohomology
- A tool that extracts information such as holes and twisting from geometric or algebraic objects and expresses it in a form such as a group
- Mapping class group
- A group formed by collecting the ways of mapping a surface onto itself, treating maps that can be continuously deformed into one another as the same
- Quandle
- An algebraic structure that arose from the rules of knot diagrams. It also serves as a tool for distinguishing knots
ResearchResearch themes
The lab studies group cohomology, which is defined algebraically and can also be understood as the cohomology of a space. Its main objects are Coxeter groups, Artin groups, and mapping class groups, and the research extends to algebraic systems closely related to topology.
*Illustrative imageCohomology of Groups
Group cohomology is defined algebraically, yet it coincides with the cohomology of the space determined by the group (the classifying space or Eilenberg-MacLane space). The lab studies this object, which spans topology and algebra.
*Illustrative imageCoxeter Groups, Artin Groups, and Mapping Class Groups
The lab studies the cohomology of Coxeter groups, Artin groups, and mapping class groups of closed surfaces, among other main objects.
*Illustrative imageAlgebraic Systems Related to Topology
Recently, in addition to group cohomology, the lab has also been studying algebraic systems closely related to topology, such as crossed modules and quandles.
MembersFaculty
- ProfessorToshiyujki Akita
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: “Omnitruncated tiling on conformal square” Powerslide / CC0 (cropped)
- Cohomology of Groups: “Surface Boy1” Theon / CC BY-SA 3.0 (cropped)
- Coxeter Groups, Artin Groups, and Mapping Class Groups: “Orange Genus 2 Surface” Gelada / CC BY-SA 3.0 (cropped)
- Algebraic Systems Related to Topology: “Bowline Knot sculpture - geograph.org.uk - 5242366” Michael Dibb / CC BY-SA 2.0 (cropped)
- Related lab (Algebra (Matsumoto) Lab): “Color complex plot” Claudio Rocchini / CC BY 2.5 (cropped)
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- Related lab (Geometry Lab (Kobayashi)): “Chen-Gackstatter-Thayer surfaces” Anders Sandberg / CC BY-SA 3.0 (cropped)
- Related lab (Algebra (Travis Scrimshaw) Lab): “E10 e8coxeterplane height120” Teake Nutma / CC BY 3.0 (cropped)
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