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Geometry Lab (Kobayashi)
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Research exploring where the geometry of surfaces meets integrable systems.
AboutAbout the lab
Curved spaces, seen through integrable systems.
Prof. Kobayashi studies the area where differential geometry and integrable systems overlap.
In differential geometry, 'curved spaces' such as curves, surfaces, and manifolds are investigated using curvature, connections, differential forms, Lie groups, and differential equations.
In certain special geometric problems, a parameter enters Cartan's structure equations, allowing them to be deformed, and they become part of an integrable system.
The professor works on themes where geometry and differential equations intersect, including constant mean curvature surfaces, constant Gaussian curvature surfaces, harmonic maps, statistical manifolds, and convex optimization.
Key terms
- Constant mean curvature surface
- A surface whose bulging (mean curvature) takes the same value everywhere. A typical example is a shape like a soap bubble
- Integrable system
- A family of differential equations with a 'mechanism of solvability.' They also appear in physical phenomena such as solitons
- Harmonic map
- Among maps from one space to another, one for which the energy of distortion is at an extremum (a stationary point)
ResearchResearch themes
The foundation is differential geometry in the tradition of Élie Cartan, which uses moving frames and structure equations to study geometric structures based on Lie groups and homogeneous spaces. The central viewpoint is that in special geometric problems, a spectral parameter enters the structure equations and turns them into an integrable system.
*Illustrative imageIntegrable Systems and the Geometry of Surfaces
The lab studies constant mean curvature surfaces and constant Gaussian curvature surfaces using loop group methods, the DPW construction, and harmonic maps. This is the central thread of the research, tracing how integrable systems arise in the theory of surfaces.
*Illustrative imageTeichmüller Theory and Harmonic Maps
The lab works on complex landslide flows and harmonic maps into hyperbolic space. It also investigates their connections to geometric structures.
*Illustrative imageMinimal Lagrangian Submanifolds
The lab studies minimal Lagrangian surfaces in complex space forms, product spaces, and quadric hypersurfaces.
*Illustrative imageStatistical Manifolds and Affine Geometry
The lab deals with statistical manifolds, α-connections, affine differential geometry, and Hessian geometry, and connects them with convex optimization.
MembersFaculty
- ProfessorShimpei Kobayashi
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: “Chen-Gackstatter-Thayer surfaces” Anders Sandberg / CC BY-SA 3.0 (cropped)
- Integrable Systems and the Geometry of Surfaces: “Triunduloid” Anders Sandberg / CC BY-SA 3.0 (cropped)
- Teichmüller Theory and Harmonic Maps: “Order 7-3 rhombic tiling in the Band Model” TheKing44 / CC BY-SA 4.0 (cropped)
- Minimal Lagrangian Submanifolds: “Minimalfläche mit Seifenhaut Würfel 0328” Bin im Garten / CC BY-SA 3.0 (cropped)
- Statistical Manifolds and Affine Geometry: “Dirichlet-3d” Empetrisor / CC BY-SA 4.0 (cropped)
- Related lab (Geometry Lab (Inoguchi)): “Hopf Fibration” Niles Johnson / CC BY 4.0 (cropped)
- Related lab (Analysis Lab (Hamamuki)): “Real snowflake macro photo” Alexey Kljatov / CC BY-SA 4.0 (cropped)
- Related lab (Honda Lab): “Essential singularity” Functor Salad / CC BY-SA 3.0 (cropped)
- Related lab (Mathematical Sciences (Kuroda) Lab): “Newly fallen snow crystals accumulating on a window sill” W.carter / CC0 (cropped)
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