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Differential Geometry Lab
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The lab studies curved spaces and discrete groups from the viewpoint of differential geometry.
AboutAbout the lab
Seeing discrete shapes through the eyes of differential geometry.
Prof. Nayatani studies geometry from the viewpoint of differential geometry.
He is mainly interested in conformal geometry and the geometry of spaces of negative curvature.
Starting from research on the rigidity of discrete groups, he also studies discrete objects such as buildings, and combinatorial harmonic maps, which are a discrete version of harmonic maps.
In the 2026 undergraduate thesis research, students will read a book on spectral geometry, which studies eigenvalues of the Laplacian in connection with properties such as the curvature of a shape, in a group reading format.
Key terms
- Discrete group
- A group whose elements are separated from one another, like the set of all integers. Here, groups with countably infinitely many elements are considered.
- Building
- A discrete geometric object formed by gluing flat pieces (apartments) together in a regular way. A tree is one example.
- Harmonic map
- A map that sends one shape to another so as to make a certain kind of energy as small as possible.
- Laplacian
- A differential operation that measures how much a function "bends." Its eigenvalues correspond to the pitch of a drum's sound.
ResearchResearch themes
From the viewpoint of differential geometry, the lab studies conformal geometry and the geometry of spaces of negative curvature. Starting from research on the rigidity of discrete groups, it also works on discrete objects such as buildings and combinatorial harmonic maps.
*Illustrative imageActions and Rigidity of Discrete Groups
The lab studies how discrete groups act isometrically on spaces that have properties similar to those of Riemannian manifolds of nonpositive curvature. Depending on the choice of group and space, good actions may be extremely limited or may not exist at all.
*Illustrative imageSpectral Geometry
This is a field that seeks to understand the eigenvalues and eigenfunctions of the Laplacian on Riemannian manifolds and graphs in relation to geometric properties such as curvature. In the 2026 small-group graduate class, problems such as maximizing eigenvalues by varying the metric are also kept in mind.
MembersFaculty
- ProfessorShin Nayatani
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: “Surfaces of constant positive Gaussian curvature” Salix alba / CC BY-SA 4.0 (cropped)
- Actions and Rigidity of Discrete Groups: “Cayley graphs of quaternion group” PSL27 / CC BY-SA 4.0 (cropped)
- Spectral Geometry: “Chladni plate 10” Exhibit made by Estes Objethos Atelier, photo by Rodrigo Tetsuo Argenton / CC BY-SA 4.0 (cropped)
- Related lab (Combinatorics, Theoretical Computer Science, Algorithm Theory, and Optimization Theory Lab (Hiroshi Hirai)): “Imperialist-competitive-algorithm-flowchart-bg” Original author: Icasite Author of the derivative work: Spiritia / CC BY-SA 3.0 (cropped)
- Related lab (Nakashima Lab): “Galton-watson(poisson(1),11gen, 47noeud)” Ipipipourax, S. Roger / CC0 (cropped)
- Related lab (Fourier Analysis and Partial Differential Equations Lab): “Ocean waves water” Jon Sullivan / Public domain (cropped)
- Related lab (Topology, Differential Geometry, and Global Analysis Lab (Hitoshi Moriyoshi)): “Surface of revolution illustration” Oleg Alexandrov / Public domain (cropped)
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