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Kuwagaki Lab
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The lab uses sheaf theory to study symplectic geometry and algebraic analysis.
AboutAbout the lab
Connecting geometry and differential equations through sheaf theory.
The Kuwagaki Lab uses sheaves to study topics related to symplectic geometry and algebraic analysis.
It is especially interested in problems related to mathematical physics, such as mirror symmetry and, more broadly, field theory.
Its main tool is microlocal sheaf theory, and it carries out research with methods motivated by microlocal sheaf theory and its interpretation in terms of Fukaya categories.
In its research description, the lab says that students do not particularly need to work on the same research as the professor. It asks anyone who wants to do research in a neighboring field and to study here as a graduate student to get in touch first.
Key terms
- Microlocal sheaf theory
- A theory that mimics the microlocal analysis of differential equations in the language of topology. It describes the directions in which a sheaf changes using the language of the cotangent bundle.
- Fukaya category
- A category defined for a symplectic manifold. It is built from objects such as Lagrangian submanifolds.
- Homological mirror symmetry
- A conjecture that the Fukaya category and the derived category of coherent sheaves defined in algebraic geometry are equivalent as categories.
ResearchResearch themes
The lab uses sheaves to study topics related to symplectic geometry and algebraic analysis. With methods motivated by microlocal sheaf theory and its interpretation in terms of Fukaya categories, it works on problems related to mathematical physics, such as mirror symmetry and field theory.
*Illustrative imageMicrolocal Sheaf Theory
Microlocal sheaf theory, the lab's main tool, is (in a certain sense) a topological analogue of the microlocal analysis of differential equations. Kashiwara and Schapira's Sheaves on Manifolds is one of its foundational references, and the theory is still developing today.
*Illustrative imageFukaya Categories and Sheaf Quantization
It has become clear that the Fukaya category of a special class of symplectic manifolds can be described in terms of microlocal sheaf theory through the “sheaf quantization” of Lagrangian submanifolds. The lab works to strengthen this correspondence.
*Illustrative imageHomological Mirror Symmetry
Homological mirror symmetry is a conjecture about an equivalence of categories between the Fukaya category and the derived category of coherent sheaves, which is defined in algebraic geometry. This was carried out for toric varieties in a doctoral thesis.
*Illustrative imageGeometric Algebraic Analysis and Quantization
The lab has shown that assembling the solution data obtained from exact WKB analysis into a sheaf yields a “sheaf quantization” of a Lagrangian submanifold.
MembersFaculty
- Associate ProfessorTatsuki Kuwagaki
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: “Whitney umbrella” Jlrodri / CC BY-SA 3.0 (cropped)
- Microlocal Sheaf Theory: “Commutative diagram used in proof of cellular chain complex” Dom walden / CC BY-SA 4.0 (cropped)
- Fukaya Categories and Sheaf Quantization: “Lemniscate of Bernoulli NURBS” Eastnman / CC BY-SA 4.0 (cropped)
- Homological Mirror Symmetry: “Toric lens surface 2” HHahn / CC BY-SA 3.0 (cropped)
- Geometric Algebraic Analysis and Quantization: “Wave packet (no dispersion) instance” User:Dicklyon / CC BY-SA 3.0 (cropped)
- Related lab (基礎数理研究部門
（入江 慶） ): “Overtwisted contact structure” Pmassot / CC BY-SA 3.0 (cropped) - Related lab (応用数理研究部門
（疋田 辰之） ): “Young tableau for partitions of number 10 stacked” Unmismoobjetivo / CC BY 4.0 (cropped) - Related lab (無限解析研究部門
（並河 良典） ): “Consani-Scholten” Ursulageorges / CC BY 4.0 (cropped) - Related lab (無限解析研究部門
（緒方 芳子） ): “Thousands of Atoms Swap 'Spins' with Partners in Quantum Square Dance ( )” National Institute of Standards and Technology / Public domain (cropped)
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