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Manifold Theory Lab
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A mathematics lab that studies the calculus of variations, algebraic analysis, geometric analysis, representation theory, and related fields.
AboutAbout the lab
From gradient flows of curves to representation theory of quantum groups
The Manifold Theory Lab is a research group in the Department of Mathematics, Graduate School of Science, that studies nonlinear analysis and the calculus of variations, algebraic analysis and singularity theory, geometric analysis, algebraic geometry, complex geometry, representation theory, and mathematical physics.
Faculty profile pages list research subjects including the dynamics of curves and surfaces, D-module theory, quantum walks, canonical metrics on Kähler manifolds, and quantum groups.
Prof. Yamaki specializes in algebraic geometry, non-Archimedean geometry, and Arakelov geometry.
Key terms
- Gradient flow
- A flow that deforms the shape of a curve or surface over time in the direction that decreases the value of a functional
- Quantum walk
- A unitary operator defined on a graph (a network of points and the lines connecting them), whose properties and long-time behavior are studied
- Quantum group
- An algebraic system that extends Lie groups, which are algebraic systems that capture symmetry
ResearchResearch themes
In the Manifold Theory Lab, faculty members work in nonlinear analysis and the calculus of variations, algebraic analysis and singularity theory, geometric analysis, algebraic geometry, complex geometry, representation theory, and mathematical physics. Below are the research subjects described on the faculty profile pages of the Department of Mathematics.
Gradient Flows of Curves and Surfaces and Variational Problems
Prof. Okabe studies the dynamics of curves and surfaces governed by gradient flows of functionals that have geometric meaning. Prof. Okabe also investigates qualitative properties of solutions to the corresponding variational problems, and has recently become interested in difference schemes that construct solutions of evolution equations from solutions of variational problems, as well as obstacle problems for geometric evolution equations.
D-Modules and Algebraic Analysis
Prof. Takeuchi has applied the theory of D-modules and the microlocal analysis of sheaves, both developed in algebraic analysis, to topological Radon transforms, motivic Milnor fibers, and A-hypergeometric differential equations. Recently, Prof. Takeuchi has been studying the Riemann–Hilbert correspondence for irregular singularities and the Fourier transform of D-modules.
*Illustrative imageGeometric Asymptotic Analysis and Canonical Metrics
Prof. Tate has studied asymptotic-analysis problems that arise from quantum theory and are formulated in a geometric setting. At present, the main focus is on the properties of quantum walks, which are unitary operators on graphs, and on the long-time asymptotic behavior of their matrix components. Associate Professor Takahashi studies the existence problem for canonical metrics on Kähler manifolds and related topics.
*Illustrative imageRepresentation Theory and Mathematical Physics
Associate Professor Hasegawa specializes in representation theory and integrable systems, and studies algebraic systems called quantum groups, which extend Lie groups, along with related solvable systems of difference equations. Assistant Professor Kuroki applies the representation theory of infinite-dimensional Lie algebras and quantum groups to mathematical physics, such as conformal field theory.
MembersFaculty
- ProfessorShinya Okabe
- ProfessorKiyoshi Takeuchi
- ProfessorTatsuya Tate
- ProfessorKazuhiko Yamaki
- Associate ProfessorRyosuke Takahashi
- Associate ProfessorKoji Hasegawa
- Assistant ProfessorGen Kuroki
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: “Soap Films” Ryan Somma / CC BY-SA 2.0 (cropped)
- Geometric Asymptotic Analysis and Canonical Metrics: “Undirected NetworkX Graph” KalebD07 / CC0 (cropped)
- Representation Theory and Mathematical Physics: “Representation of SU(3)” Arkadipta Sarkar at English Wikipedia / CC BY-SA 3.0 (cropped)
- Related lab (代数学講座): “A row of mathematics books at Jinja Public Library, Jinja City Centre” AKibombo / CC BY-SA 4.0 (cropped)
- Related lab (幾何学講座): “Riemann Minimal Surface” Anders Sandberg / CC BY-SA 3.0 (cropped)
- Related lab (解析学講座): “Slide Rule Mathematics - DPLA - afb7ca99683dc5a3f82ec3981f6eb04e” 作者不詳 / Public domain (cropped)
- Related lab (相関数理解析講座
（協力講座） ): “Mathematical and Simulation Modeling in Neuroscience (NeuroMat 2018) 04” Sturm / CC BY-SA 4.0 (cropped)
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