Tsukasa ISHIBASHI【石橋 典】,  Asst. Professor

Research Field

Geometric Structures on Surfaces, Lie Groups and Representation Theory, Mathematical Physics


Research Interests

 I’m interested in geometric structures on surfaces, the geometry and the quantization of their moduli spaces.

1) Having classified the 2-dimensional closed differentiable manifolds (i.e. closed surfaces), one can further ask how many additional geometric structures they admit, e.g. complex structures. The space of complex structures on a fixed closed surface is called the ``Moduli Space of Riemann Surfaces”. Its universal cover is the Teichmüller space, and its covering group is the mapping class group. When the genus is at least two, the complex structures are equivalent to hyperbolic structures. Therefore these structures can be studied from the viewpoint of hyperbolic geometry.

2) Hyperbolic structures correspond to group homomorphisms (representations) from the fundamental group into the Lie group PSL(2,R). Generalizing this, the higher Teichmüller theory studies the geometric structures associated with representations of the fundamental group into a Lie group G. These representations are equivalently described by flat G-bundles on the surface, and their moduli also arise naturally in two-dimensional gauge theory.

3) Motivated by these connections with mathematical physics, the problem of quantization of the (higher) Teichmüller space arises. The deformation quantization of the moduli space of flat G-bundles gives rise to various quantum algebras, such as quantum groups and skein algebras. The theory of (quantum) cluster algebras, introduced around 2000, provides a powerful framework for explicit coordinate descriptions of these quantum theories. More broadly, I’m interested in related topics connected with cluster algebras and their interactions with geometry and mathematical physics.

Keywords: (higher/quantum) Teichmüller theory, mapping class groups, skein algebras, (quantum) cluster algebras.


Remarks

Personal Web Page Tsukasa ISHIBASHI
Laboratory Mathematics Building 513
Telephone  
E-mail tsukasa.ishibashi.a6[at]tohoku.ac.jp

Back to Top