Introduction The articles in this volume form the proceedings of the conference ‘Tropical Geometry and Integrable Systems’ that was held in July 2011 at the University of Glasgow. Over the five days of the conference over twenty talks were given on a wide range of topics, from ultra-discrete soliton equations and cellular automata to curve counting problems in enumerative geometry. While the original motivations for the subjects of these talks were different, common themes were the idea of the max-plus algebra and ideas originating from both tropical geometry and integrable systems. The historical origins of integrable systems theory lie in three main areas: the 19th century treatment of certain dynamical systems due to Weierstrass and Ko- valevskaya the Cauchy theory of integrability of ordinary differential equations in the complex plane developed in the late 19th and early 20th century by Painl´eve, Garnier and others and the geometry of surfaces studied by acklund, Darboux and Goursat. In the late 20th century, particularly under the seminal influence of Gardner, Greene, Kruskal and Miura in the West and of the Russian and Japanese schools in the East the area of integrable systems acquired a momentum that has since kept it centre-stage in the interplay of pure and applied mathematics, drawing on and influencing a plethora of pure subjects such as algebraic geometry, repre- sentation theory and combinatorics, while at the same time finding applications in such areas as nonlinear optics and plasma physics. The first examples of integrable systems were continuous: the dependent fields were continuous and satisfied differential equations. However it was soon realized that the same ideas could be applied to certain discrete systems, where deriva- tive are replaced with difference operators. One may go further and discretize the dependent variables as well, arriving at so-called ultra-discrete integrable systems, while still retaining the notion of integrability. In the most extreme case the depen- dent variable can take only two values 0 and 1 and since the independent variable is also discrete one arrives at discrete dynamics - a ‘box and ball’ system - where the movement of balls (where the fields take the value 1) are described by simple sets of rules. Remarkably the essential features of integrability still remain for such sys- tems. In fact such discrete integrable systems can be regarded as more fundamental as many continuous integrable systems may be obtained by taking the continuum limit of discrete systems. Algebraic geometry begins with the fundamental problem of solving systems of polynomial equation in several variables. Even as early as the 19th century the connection between algebraic curves and solutions to differential equations was well vii
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