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Algebra (Ouchi) Lab
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The lab studies high-dimensional shapes defined by polynomials using the language of derived categories.
AboutAbout the lab
Shapes that cannot be drawn, described in the language of mathematics.
Prof. Ouchi studies algebraic geometry, which investigates algebraic varieties.
A space that can be described as the common zero set of several polynomials is called an algebraic variety.
High-dimensional shapes are difficult to build physically or draw directly, but they can still be written down and reasoned about in the language of mathematics.
In particular, he prefers to study Calabi-Yau varieties in a broad sense, such as K3 surfaces and holomorphic symplectic varieties.
Key terms
- Algebraic variety
- A shape or space that can be written as the common zero set of several polynomials.
- Derived category of coherent sheaves
- A category associated with an algebraic variety. It is a tool for describing relationships between varieties and their symmetries.
- K3 surface
- A type of algebraic variety with rich symmetries. It belongs to the family of Calabi-Yau varieties.
ResearchResearch themes
The lab studies algebraic varieties defined by polynomials, especially K3 surfaces and holomorphic symplectic varieties. Using derived categories of coherent sheaves as its main method, it describes the symmetries and dualities hidden in these varieties.
*Illustrative imageSymmetries of Calabi-Yau varieties
K3 surfaces, holomorphic symplectic varieties, and Fano varieties of K3 type have rich symmetries. They have been studied from perspectives such as sporadic finite simple groups (the Mathieu and Conway groups), dynamical systems and Salem numbers, and geometric group theory.
*Illustrative imageDerived categories of coherent sheaves
To each algebraic variety one can associate a category called the derived category of coherent sheaves. Using Fourier-Mukai transforms, one can describe relationships between different algebraic varieties and explore in depth the symmetries of a single variety.
*Illustrative imageMirror symmetry
Mirror symmetry, which originated in superstring theory, is a duality between algebraic geometry and symplectic geometry. It is believed that this too can be understood in the language of derived categories.
MembersFaculty
- Associate ProfessorGenki Ouchi
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: “CalabiYau5” Andrew J. Hanson / CC BY-SA 3.0 (cropped)
- Symmetries of Calabi-Yau varieties: “Sarti surface” Claudio Rocchini / CC BY-SA 3.0 (cropped)
- Derived categories of coherent sheaves: “CY5-6D-lattice” Andrew J. Hanson / CC BY-SA 3.0 (cropped)
- Mirror symmetry: “Quintic 1” The original uploader was Floriang at German Wikipedia. / CC BY-SA 3.0 (cropped)
- Related lab (Algebra (Daisuke Matsushita) Lab): “K3 surface” BTotaro / CC BY-SA 4.0 (cropped)
- Related lab (Geometry (Nozaki) Lab): “Trefoil knot cgi” Altapedia / CC BY-SA 4.0 (cropped)
- Related lab (Kawasaki Lab): “Chaos Theory & Double Pendulum - 1” Cristian V. / CC BY-SA 4.0 (cropped)
- Related lab (Kasuya Lab): “Standard contact structure r 3” Pmassot / CC BY-SA 3.0 (cropped)
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