Our research areas are in both sides of non-commutative algebra and low-dimensional topology. I have mainly studied representation categories of Hopf algebras, quantum groups, subfactors and quivers, in connection with knots and 3-manifolds.

In the past, graduate and undergraduate students in our laboratory researched topics on group theory, quiver representations, knots and braids, fundamental groups, frieze patterns, spatial graphs, Pythagorean numbers, q-deformations of rational numbers, and so on. We would like to continue to study representation theory from a wide viewpoint and understand how algebra, low-dimensional topology, and mathematical physics are related.

Since groups and rings are highly abstract, it is difficult to study them head-on depending on conditions. In such cases, by representing them as matrices, one may use linear algebraic techniques. Though often little information can be obtained from one representation, by considering several representations, and furthermore all of them, one may know more detailed information on groups and rings. "Representation theory" in mathematics is a main field studying properties of algebraic systems including groups and rings by methods as described above. Currently, representation theory much progress related to other fields, as especially topology and mathematical physics.