Workshop on Triangulations and Mutations

在数学的许多分支中，一个反复出现的主题是找到一个组合对象，它可以作为一个句柄来描述一个更深入、更基本的数学理论，并将有关它的某些问题转化为实际上可以计算的问题。一个例子是曲面的三角剖分。如果我们把一个表面，比如说一个球体或一个甜甜圈，分成三角形区域，那么表面的几何形状（这可能很复杂）可以根据哪些三角形区域彼此入射来理解（这要简单得多）。组合对象也出现在生成类别对象的上下文中，例如强例外序列或倾斜对象，奇异性的crepant分辨率，和莱夫谢茨纤维化的消失周期。最近，已经观察到这些组合对象具有非常相似的性质，这些性质独立于它们发展的数学分支。一方面，它们可以通过寻找满足某些相容关系的子代数的最大团来构造。另一方面，一旦我们得到了这样一个组合对象，就有可能通过将它的一个子对象交换为另一个子对象的过程来生成更多的子对象。我们使用突变作为这些过程的通用术语，这些过程在不同的数学分支中有不同的名称：Flops，Fomin-Zelevinsky突变，twists等。如果我们删除两个相邻三角形区域之间的边界，那么我们创建一个四边形区域，并且有一个唯一的替代边界（连接其他两个角）将其分为两个新的三角形区域。这是突变的基本实例，显然，这些观察结果表明，必须有一个基本的数学理论将所有这些紧密联系在一起，而研讨会的目的就是将研究人员聚集在一起讨论这些问题。

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.