This research project aims to extend the theory of integrable systems and special functions. We have focused mainly on applications to differential geometry and combinatorics, and obtained the following results;
(1) Goldstein and Petrich introduced a hierarchy of equations that governs isoperimetric deformation of plane curves. We have revealed that the Goldstein-Petrich hierarchy is indeed a reduced case of the Toda lattice hierarchy. (2) We have studied rigged configurations from the viewpoint of soliton cellular automaton. We have obtained alternative proof of the linearization property of the time-evolution, which is simpler than the known proofs.
We have obtained still other preliminary results that are related to applications of integrable systems to differential geometry and combinatorics. We will investigate them further in near future.