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Topology, Differential Geometry, and Global Analysis Lab (Hitoshi Moriyoshi)
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The lab studies manifolds, centered on index theorems that connect analysis and geometry.
AboutAbout the lab
Where analysis and geometry meet.
Prof. Moriyoshi is interested in the study of manifolds in general, and works on topology and differential geometry.
The professor is particularly interested in generalizations of the Atiyah–Singer index theorem, which connects the analytic invariants and the geometric invariants of manifolds.
The professor believes the appeal of the index theorem lies in the way it brings together fields of mathematics with entirely different backgrounds.
The seminar for undergraduate thesis research takes "Introduction to Geometry and Manifolds" as its theme. In a reading format where students take turns presenting, they learn the basic knowledge needed to study manifolds.
Key terms
- Manifold
- A shape that looks like flat space near every point. It is a generalization of curves and surfaces to arbitrary dimensions.
- Invariant
- A quantity that does not change when a shape is deformed. It serves as a clue for telling whether two shapes are the same.
- Foliated manifold
- A manifold divided without gaps into surfaces of the same dimension, stacked like thin layers (leaves).
ResearchResearch themes
The lab studies the topology of manifolds, and is particularly interested in generalizations of the index theorem, which connects analytic invariants of manifolds to topological invariants. It also works on the framework of noncommutative geometry, which arose with the index theorem as one of its motivations.
*Illustrative imageAtiyah–Singer Index Theorem
This theorem arises where analysis and geometry meet on manifolds, and it connects two invariants of entirely different nature. One of its consequences is that invariants of closed manifolds such as the L-genus and the Â-genus are integers. The lab has a deep interest in generalizing this theorem.
*Illustrative imageNoncommutative Geometry and Foliated Manifolds
Noncommutative geometry, which arose with the index theorem as one of its motivations, is a framework spanning differential geometry, topology, operator algebras, and more. Using methods from K-theory and cyclic cohomology, the index theorem can also be extended to foliated manifolds, such as the Kronecker foliation on a torus.
MembersFaculty
- ProfessorHitoshi Moriyoshi
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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: “Surface of revolution illustration” Oleg Alexandrov / Public domain (cropped)
- Atiyah–Singer Index Theorem: “Big bang manifold (it)” Big_bang_manifold.png: Jim Pivarski derivative work: F l a n k e r (talk) / Public domain (cropped)
- Noncommutative Geometry and Foliated Manifolds: “Orange Torus” Gelada / CC BY-SA 3.0 (cropped)
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