I am studying topology, geometry and combinatorics concerning the braid groups, orderable groups and 3-manifolds. The braid group is illustrated by strands in 3-space and is intuitively easy to understand,
but it plays an important role in many branch of mathematics, like quantum topology or contact geometry.
An orderable group is a group having a total ordering that is invariant under the multiplication of the group itself. It is also an interesting object related to one-dimensional dynamics and various other fields. I am mainly interested in an application of orderings to topology,
and studying an explicit construction of orderings having strange properties such as isolated orderings.
Recently I am studying topology and contact structure of 3-manifolds via open book decomposition,
which can be seen as a generalization of closed braids.