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Geometry Lab (Jun O'Hara)
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Geometry that gives knots energy and searches for their nicest shapes.
AboutAbout the lab
Determining the nicest shape of a knot with mathematics.
Prof. O'Hara studies the shapes of knots and surfaces as maximum and minimum problems in geometry.
The professor considers an energy similar to electric repulsion on a knot of string, and searches for the shapes that make it smaller.
Möbius geometry, which understands shapes through properties that do not change under inversions in spheres, is also one of the lab's pillars.
For general audiences, the work is also introduced as 'how to find the best positions for streetlights in a park.'
Key terms
- Knot
- A loop made by joining the two ends of a piece of string. Shapes that can be moved onto each other without cutting are regarded as the same
- Möbius geometry
- Geometry that understands shapes by focusing on properties that do not change under transformations such as inversions in spheres
- Potential
- A quantity that represents the magnitude of energy determined by position
- Regularization
- A method of extracting a meaningful finite value from an integral that would otherwise diverge to infinity
ResearchResearch themes
The lab studies the shapes of figures such as knots and surfaces as maximum and minimum problems in geometry. The main subjects are knot energy, Möbius geometry, and potentials of submanifolds.
*Illustrative imageKnot energy
The lab defines a 'knot energy' that generalizes the electrostatic energy of a charged knot, and considers whether the shape that minimizes this energy can be given as the 'best shape' of each knot.
*Illustrative imageMöbius geometry of curves and surfaces
In joint research with Remi Langevin, the lab studies knot energies that do not change under Möbius transformations. Such energies are expressed using the 'infinitesimal cross ratio,' which is built from sets of points on the knot.
*Illustrative imagePotentials and the center of a figure
The lab defines a potential on a figure by regularizing a divergent integral, and calls the points that give its maximum and minimum the 'center' of the figure. This generalizes the center of mass, and the lab has shown cases in which the center of a convex figure is determined uniquely.
*Illustrative imageEnergy of submanifolds
The lab defines the energy of closed submanifolds and solid bodies in Euclidean space by extracting finite values from divergent integrals. This is joint research with Gil Solanes.
MembersFaculty
- ProfessorJun O'Hara
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: “(2,13)-Torus Knot” Bostelk2 / CC BY-SA 4.0 (cropped)
- Knot energy: “Versatackle Knot Step4” No machine-readable author provided. Starfoxy~commonswiki assumed (based on copyright claims). / CC BY-SA 3.0 (cropped)
- Möbius geometry of curves and surfaces: “Earth as riemann sphere large 500mio 254dpi” Rolf Böhm / CC0 (cropped)
- Potentials and the center of a figure: “EB1911 - Geometry Fig. 9” 作者不詳 / Public domain (cropped)
- Energy of submanifolds: “TorusSystoleLoop” RobHar / Public domain (cropped)
- Related lab (Geometry (Sanda) Lab): “3D-Cylinder with handle and torus with hole” 作者不詳 / Public domain (cropped)
- Related lab (Algebra Lab (Kenta Sato)): “Chatelet surface” Claudio Rocchini / CC BY-SA 3.0 (cropped)
- Related lab (Geometry (Futaki) Lab): “Expansion de l'univers dans une variété de Calabi-Yau” N738139 / CC0 (cropped)
- Related lab (Geometry Lab (Hiroshige Kajiura)): “Kleineigenspace” Joroco246 / CC BY-SA 4.0 (cropped)
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