Direct methods for ultradiscrete systems

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Studying the algebraic structure of so-called discrete integrable systems-which can be obtained from continuous integrable systems by appropriate discretisation techniques-it proved possible to elucidate the symmetry structure of certain cellular automata known as "ultradiscrete integrable systems". As a result I obtained a novel technique for solving the so-called ultradiscrete KdV equation, which is the prototypical ultradiscrete integrable system. Moreover, the classification of certain types of discrete integrable systems not only led to the discovery of unexpected relations among them, but also, in the lower dimensional case, to extensions of the geometric structures of previously known discrete integrable systems to several new classes of discrete systems.