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Ozawa Lab
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A lab that studies discrete groups using analytic methods from operator algebras.
AboutAbout the lab
Analyzing the symmetry of groups through operator algebras
The Ozawa Lab studies the relationship between operator algebras and discrete groups.
A group is a mathematical language for describing the symmetry of an object. For example, the set of all transformations, such as rotations and reflections, that leave the structure of a crystal unchanged forms a group.
The lab treats the elements of a group as operators and investigates problems about groups using analytic techniques from operator algebra theory, while also pursuing applications in the opposite direction.
In addition to traditional operator algebra theory, the lab also studies operator theory, Banach algebra theory, Banach space theory, and discrete metric geometry.
Key terms
- Operator algebra
- An algebraic system generated by operators on a vector space, completed with respect to a topology. The two main kinds are C*-algebras and von Neumann algebras.
- Discrete group
- A group consisting of isolated (discrete) elements. Like the symmetry of a crystal, it describes the symmetry of objects.
- Kazhdan's property (T)
- A rigidity property of groups: a representation that has almost invariant vectors in fact has a genuinely invariant vector.
ResearchResearch themes
The lab studies the relationship between operator algebras and discrete groups. In recent years, under the banner of "analytic group theory," it has focused on using functional-analytic methods to investigate the algebraic and geometric structure of groups.
*Illustrative imageThe relationship between operator algebras and discrete groups
The lab deals both with applications of operator algebra theory to mathematical physics, group theory, ergodic theory, and related areas, and with applications in the opposite direction.
*Illustrative imageAnalytic group theory
The lab obtained a purely algebraic characterization of Kazhdan's property (T). In joint work using a computer, it has shown with full mathematical rigor that Aut(F5), the automorphism group of the free group of rank 5, has property (T).
*Illustrative imageCoarse geometry and problems on von Neumann algebras
Using techniques from C*-algebra theory, the lab resolved in the negative Roe's problem in coarse geometry, which asks whether quasi-local operators can be approximated by operators of finite propagation.
MembersFaculty
- ProfessorNarutaka Ozawa
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: “Cayley graph of Alternating Group A5” Kathryn Cramer / CC BY-SA 4.0 (cropped)
- The relationship between operator algebras and discrete groups: “Kelli graph free prod” Tosha / Public domain (cropped)
- Analytic group theory: “Dutch national supercomputer "Huygens"” Dennis van Zuijlekom / CC BY-SA 2.0 (cropped)
- Coarse geometry and problems on von Neumann algebras: “Hyperbolic 3d order 4 hexagonal tiling” Claudio Rocchini / CC BY-SA 3.0 (cropped)
- Related lab (応用数理研究部門
（磯野 優介） ): “Bernoullishift” Andreas Mandl / CC BY-SA 3.0 (cropped) - Related lab (応用数理研究部門
（SEHNEM, Camila） ): “Hyperbolic tiling 7 7-2” Double sharp (talk), based on this picture by Jim McNeill, but with the remaining overlapping heptagons left as outlines instead of deleted. A green background to the tiling has also been added. Created using Tyler and Paint. / Public domain (cropped) - Related lab (応用数理研究部門
（岸本 展） ): “Rogue waves breaking behavior at different crossing angles, McAllister 2019” McAllister et al 2019 / CC BY 4.0 (cropped) - Related lab (応用数理研究部門
（牧野 和久） ): “B functies 2var versie02” Bob.v.R / CC BY-SA 4.0 (cropped)
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