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Algebraic Geometry, Representation Theory, Infinite-Dimensional Integrable Systems, and Mathematical Physics Lab (Shintaro Yanagida)
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The lab studies representation theory, algebraic geometry, and mathematical physics.
AboutAbout the lab
Studying representation theory, mathematical physics, and algebraic geometry.
Prof. Yanagida's lab studies representation theory, mathematical physics, and algebraic geometry as its specialties.
The main themes are the algebraic representation theory of vertex algebras and quantum algebras, the algebraic geometry of moduli, and quantum integrable systems.
As a mathematical physics topic at the intersection of the algebraic geometry of moduli and the representation theory of quantum algebras, the lab is also pursuing research on the geometric representation theory of vertex algebras.
In the graduation research seminar for fourth-year undergraduates, students learn the basics of algebraic representation theory, centered on Lie algebras and vertex algebras.
Key terms
- Representation theory
- A branch of mathematics that studies algebraic objects through how they act on matrices and linear spaces.
- Moduli
- A single space formed by collecting all geometric objects of a certain kind.
- Quantum integrable system
- A model of quantum mechanics that has the property of being exactly solvable.
ResearchResearch themes
The research areas are representation theory and algebraic geometry. The main themes are the algebraic representation theory of vertex algebras and quantum algebras, the algebraic geometry of moduli, and quantum integrable systems.
*Illustrative imageQuantum Algebras and Macdonald Polynomials
The lab investigated the relationship between the quantum integrable system associated with Macdonald symmetric functions and the representation theory of the quantum toroidal gl1 algebra. This led to studies of the double affine Hecke algebra and Macdonald polynomials, and recently the lab has been working on Koornwinder polynomials as well.
*Illustrative imageModuli Spaces and Stability Conditions
The lab is interested in moduli problems and Bridgeland stability. It has studied the birational geometry of moduli spaces of Gieseker stable sheaves on abelian surfaces, and the spaces of stability conditions on abelian surfaces and K3 surfaces.
*Illustrative imageGeometric Representation Theory of Vertex Algebras
The lab treats vertex algebras from the viewpoint of derived algebraic geometry. It introduced the Li filtration for supersymmetric vertex algebras and used it to study the relationship between the representation theory of superconformal algebras and Poisson geometry. Most recently, it has been carrying out collaborative research on Zhu algebras.
*Illustrative imageSupersymmetric Vertex Algebras and Hopf Superalgebras
In collaborative research, the lab introduced an operad that describes the algebraic structure of supersymmetric vertex algebras, and computed the cohomology complex that governs their deformation theory. In another collaboration, it introduced combinatorial Hopf superalgebras.
MembersFaculty
The members of Prof. Yanagida's lab consist of faculty and students at the doctoral, master's, and undergraduate levels.As of 2025
- ProfessorShintaro Yanagida
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: “Truncated icosahedron cherry model by George W. Hart” George W. Hart / CC BY-SA 4.0 (cropped)
- Quantum Algebras and Macdonald Polynomials: “Elementary symmetric functions (1)” Daviddaved / CC BY 3.0 (cropped)
- Moduli Spaces and Stability Conditions: “1965 International Algebraic Geometry Symposium” Photographic studio "Reporters Foto", Rome / Public domain (cropped)
- Geometric Representation Theory of Vertex Algebras: “Dynkin Diagram Triangle” Pacman 2.0 / CC BY-SA 3.0 (cropped)
- Supersymmetric Vertex Algebras and Hopf Superalgebras: “Hopf algebroid structure diagram encoding composition of an arrow and its inverse in both directions” Wundzer / CC BY-SA 4.0 (cropped)
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