Workshop on Triangulations and Mutations

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In many branches of mathematics, a recurring theme is to find a combinatorial object that can be used as a handle to describe a deeper, more fundamental mathematical theory and turn certain questions about it into problems that can actually be calculated. One example is triangulations of surfaces. If we divide a surface, say a sphere or a donut, into triangular regions, then the geometry of the surface (which is potentially complicated) can be understood in terms of which triangular regions are incident to each other (and this is much simpler).Combinatorial objects also occur in the context of generating objects of categories such as strong exceptional sequences or tilting objects, crepant resolutions of singularities, and vanishing cycles of Lefschetz fibrations. Recently, it has been observed that these combinatorial objects have remarkably similar properties which are independent from the branch of mathematics in which the were developed. On one hand, they can be constructed by finding maximal cliques of subobjects that satisfy certain compatibility relations. On the other hand, once one has obtained such a combinatorial object, it is possible to generate more by a process that swaps one of its subobjects for another one.We use mutation as a generic term for these processes which go by different names in different branches of mathematics: Flops, Fomin-Zelevinsky mutations, twists etc. In the case of triangulations of surfaces, we have a particularly simple description: If we remove the border between two neighbouring triangular regions, then we create a quadrangular region, and there is a unique alternative border (linking the two other corners) which divide it into two new triangular regions. This is the fundamental instance of mutation.Clearly, these observations suggest that there must be an underlying mathematical theory that ties all this neatly together, and the purpose of the workshop is to bring together researchers to discuss these questions.