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Algebra (Matsumoto) Lab
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The lab studies functions defined by integrals from a geometric point of view.
AboutAbout the lab
Seeing geometry in functions defined by integrals.
Prof. Matsumoto studies the theory of special functions, using geometric methods to investigate functions defined by integrals.
Many functions have integral representations, including the gamma function, the beta function, zeta functions, and hypergeometric functions.
By viewing these integrals as pairings between cohomology groups and homology groups, the lab understands the functions geometrically.
The lab also works on finding new automorphic functions and automorphic forms through the study of period maps for families of algebraic varieties.
Key terms
- Hypergeometric function
- A type of special function defined by integrals. It allows many fundamental functions to be treated in a unified way.
- Cohomology group
- An algebraic way of capturing properties of a shape, such as holes and twists.
- Period map
- A map that assigns to each member of a family of shapes the values (periods) of integrals over it.
ResearchResearch themes
The lab studies functions defined by integrals geometrically, viewing them as pairings between cohomology groups and homology groups. Another goal is to discover new automorphic functions from period maps.
*Illustrative imageGeometry of functions defined by integrals
The lab studies functions defined by integrals or having integral representations, such as the gamma function, the beta function, zeta functions, and hypergeometric functions, from a geometric viewpoint by regarding them as pairings between certain cohomology groups and homology groups.
*Illustrative imageFrom period maps to automorphic functions
Elliptic modular functions arise as the inverse of the period map for families of elliptic curves. By studying period maps for families of more general algebraic varieties, the lab aims to find new automorphic functions and automorphic forms.
MembersFaculty
- Project ProfessorKeiji Matsumoto
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: “Color complex plot” Claudio Rocchini / CC BY 2.5 (cropped)
- Geometry of functions defined by integrals: “Jacobi theta 1” 唐戈 / CC BY-SA 4.0 (cropped)
- From period maps to automorphic functions: “Bers-Einbettung” Café Bene / CC BY-SA 3.0 (cropped)
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