Mathematics in ultradiscrete integrable systems

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1.We have determined asymptotic behaviour of the fundamental cycle of periodic box-ball systems (PBBSs) of type (A_1^<(1)>)in the limit of the system size N, N → ∞. Due to the integrability of the PBBS, their orbits are located in a small region of the phase space the volume of which is proportional to exp[N]. We have proved that the maximum fundamental cycle is of order of exp[N ^<1/2>], but that almost all the fundamental cycle is less than exp[(log N)^2].2.We constructed the geometric crystal, which was proposed by Berenstein and Kazhdan, for the affine Lie algebra Using the realization by matrices with spectral parameters, the birational transformation (tropical R matrix), which intertwines the tensor product of the geometric crystals, is obtained. We also proved its uniqueness.The tropical R matrix is comutable with the geometric Kashiwara operators and satisfies the Yang-Baxter relation.It does not preserve the inverse operation to addition formulae and produce piecewise linear equations for ultradiscrete systems.3.We have constructed the combinatorial R matrices for B_n^<(1)>, D_n^<(1)>, A_<2n>^<(2)> and D_<n+1>^<(2)>. The soliton scattering in the lattices with Boltzmann weight given by these R matrices are expressed as the action by Wyle group. We also constructed an analogue to the inverse scattering methods.