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Number Theory Lab
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A field that explores various mathematical objects related to numbers using algebra, geometry, and analysis.
AboutAbout the lab
Exploring the structures hidden in numbers through algebra, geometry, and analysis.
The field of number theory studies various mathematical objects related, directly or indirectly, to numbers.
Its objects of study include algebraic varieties and motives, automorphic forms and automorphic representations, and the associated Galois representations and zeta functions and L-functions.
Analogies between number-theoretic objects and topological objects (especially in knot theory) are also studied.
The methods draw on algebra, geometry, and analysis alike; the field has close ties to representation theory, and computers can be actively used.
Key terms
- Automorphic form
- A complex function satisfying special symmetries. An object that appears frequently in number theory.
- Galois representation
- A representation of the ways of permuting the roots of an equation (the Galois group) as the action of matrices.
- Zeta function
- A function that packs the properties of primes and integers into a single function. The Riemann zeta function is the representative example.
ResearchResearch themes
In number theory, various mathematical objects related to numbers are studied with methods of algebra, geometry, and analysis. The research deals with automorphic forms, Galois representations, and zeta functions and L-functions, and also investigates analogies with knot theory.
*Illustrative imageVarious Mathematical Objects Related to Numbers
Research covers algebraic varieties and motives, automorphic forms and automorphic representations, their associated Galois representations, zeta functions and L-functions and their special values, multiple zeta values, and periods. Analogies between number-theoretic objects and knot theory are also addressed.
*Illustrative imageNumber-Theoretic Study of Automorphic Forms
One of the faculty members in this field investigates automorphic forms from the viewpoint of number theory. Three themes are under consideration: refinement of the trace formula, representation theory of reductive groups over p-adic fields, and endoscopy.
*Illustrative imageMathematical Crystallography, Lattices, and Quadratic Forms
One of the faculty members in this field works on developing theories and methods related to packing and to computations in algebra and harmonic analysis, from the viewpoint of applying algebra and number theory and solving problems in mathematical crystallography. This faculty member has also developed measurement and analysis methods for crystallography using tools such as lattices, quadratic forms, and semidefinite programming.
*Illustrative imageIntegrals Associated with Graphs and Topological Invariants
One of the faculty members in this field studies algebra and geometry related to representations of fundamental groups and Galois groups and to configuration spaces. Recently, the research has focused on periods expressed as integrals associated with graphs, and on topological invariants.
MembersFaculty
- ProfessorWeng Weng
- ProfessorShinichi Kobayashi
- ProfessorYasuro Gon
- ProfessorRyoko Tomiyasu
- ProfessorKentaro Nakamura
- ProfessorMasanori Morishita
- Associate ProfessorAde Irma Suriajaya
- Associate ProfessorYasuhiko Ikematsu
- Associate ProfessorTakuya Konno
- Associate ProfessorShingo Saito
- Associate ProfessorYuichiro Takeda
- Associate ProfessorKazuhiro Hikami
- Associate ProfessorToshiki Matsusaka
- Assistant ProfessorKarin Ikeda
- Assistant ProfessorYasuhiro Ishitsuka
- Assistant ProfessorHisatoshi Kodani
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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: “SinRiemannSiegelTheta” Chrgue / CC BY-SA 3.0 (cropped)
- Various Mathematical Objects Related to Numbers: “Spirale Ulam 150” Généré avec la librairie GD de PHP, par Cortexd / Public domain (cropped)
- Number-Theoretic Study of Automorphic Forms: “Hyperbolic domains 932 black” Tomruen / Public domain (cropped)
- Mathematical Crystallography, Lattices, and Quadratic Forms: “Stone Flower (Кам’яна квітка)” Яна Сычикова, Сергей Ковачев / CC BY-SA 4.0 (cropped)
- Integrals Associated with Graphs and Topological Invariants: “Graph example (Graph theory)” Römert / CC BY-SA 3.0 (cropped)
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