Study on the integrable structure of ultradiscrete systems

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An ultradiscrete system denotes a cellular automaton constructed from a continuous equation through a limiting procedure. We investigated the ultradiscrete systems which are considered to have integrability. The principal results in this research are; (1) the correlation functions of the box-ball systems, which are typical integrable ultrtadiscrete systems, are obtained, (2) the mathematical structure of the so-called negative solitons in the ultradiscrete KdV equation is clarified, (3) solutions to ultradiscrete Painlevé equations are constructed and a series of formulae for partition numbers are obtained, (4) integrable systems over finite fields are shown to have a property called AGR (almost good reduction).