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Differential Geometry Lab
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Differential geometry is a field that explores the geometric properties of shapes through how they curve.
AboutAbout the lab
Exploring the properties of shapes through how they curve
Differential geometry is a field that seeks to understand the geometric properties of shapes through how they curve (curvature).
It has one of its origins in Gauss's theory of surfaces, and it is connected with many other fields, including topology, integrable systems, and information theory.
The faculty's research includes the geometry of Alexandrov spaces, the Oka principle in complex analysis, and compact quotients of homogeneous spaces.
Key terms
- Curvature
- A measure of how much a shape or surface bends. In differential geometry, it is used to capture the properties of shapes.
- Alexandrov space
- A space that generalizes Riemannian manifolds. It is one of the objects studied in this field.
- Oka principle
- A homotopy principle in complex analysis. It states that analytic problems on Stein manifolds face only topological obstructions.
- Conley index
- A homotopy invariant that appears in the study of topological dynamical systems.
ResearchResearch themes
Differential geometry is a field that has one of its origins in Gauss's theory of surfaces, and it seeks to understand the geometric properties of shapes through how they curve (curvature). It is connected with many other fields, including topology, integrable systems, and information theory.
*Illustrative imageModuli of Alexandrov Spaces
The lab studies the geometric structure of individual spaces and the structure of their moduli spaces at the same time. With particular focus on moduli spaces of Alexandrov spaces, which generalize Riemannian manifolds, it has introduced a method that examines the statistical behavior of the discrete Laplacian.
*Illustrative imageThe Oka Principle and Ellipticity
The lab studies the Oka principle, a homotopy principle in complex analysis, and the area around "Gromov ellipticity," a property that is the exact opposite of Kobayashi hyperbolicity. It is currently examining the Oka principle from a broader perspective that includes algebraic geometry and differential geometry.
*Illustrative imageCompact Quotients of Non-Riemannian Homogeneous Spaces
The lab investigates quotient spaces (Clifford-Klein forms) obtained when a discrete subgroup acts properly and freely on a homogeneous space, focusing on the question of when such a space admits compact quotients. The main approach is through cohomological methods.
*Illustrative imageA Simpler Formulation of the Conley Index
The lab has proposed a simpler formulation of the Conley index, a homotopy invariant that appears in the study of topological dynamical systems. The framework does not use index pairs, and it handles both discrete time and continuous time.
MembersFaculty
- ProfessorLin Weng
- ProfessorHirofumi Sasahira
- ProfessorTakaharu Yaguchi
- Associate ProfessorYukio Otsu
- Associate ProfessorYuta Kusakabe
- Associate ProfessorYosuke Morita
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: “Schwarz H Surface” Anders Sandberg / CC BY-SA 3.0 (cropped)
- Moduli of Alexandrov Spaces: “Metric circles” Jacob Rus / CC BY-SA 4.0 (cropped)
- The Oka Principle and Ellipticity: “Surface torus” Hellingspaul / CC BY-SA 3.0 (cropped)
- Compact Quotients of Non-Riemannian Homogeneous Spaces: “Lie transport” Phidus / CC BY-SA 4.0 (cropped)
- A Simpler Formulation of the Conley Index: “Lorenz Attractor from Gauss-Legendre” Mgolden96 / CC BY-SA 4.0 (cropped)
- Related lab (トポロジー): “Klein bottle green” Aizenr / CC BY-SA 4.0 (cropped)
- Related lab (数理統計): “08fig-empirical” ASpiegler / CC BY-SA 4.0 (cropped)
- Related lab (代数幾何): “Heike Kamerlingh Onnes - 29 - Plaster model of a Ψ surface of a mixture of two substances” 作者不詳 / Public domain (cropped)
- Related lab (表現論・調和解析): “Worm's-eye view of Tokyo Skytree with vertical symmetry impression, a sunny day, in Japan” Basile Morin / CC BY-SA 4.0 (cropped)
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