The critical group of a topological graph: an approach through delta-matroid theory

沙堆模型是物理学中一个被广泛研究的模型，在经济学中被用来模拟当其中一家银行违约时银行网络的后果。它有一个简单直观的描述如下。假设我们把沙粒堆在一些地方。在某个时刻，一个桩变得太大而无法保持直立，它就会倒塌，雪崩发生，沙子被转移到邻近的地方。接下来会发生什么？转移到邻近地点的沙子数量是否足以使它们倒塌？这个过程要持续多久，直到所有的地点再次稳定下来，我们最终会得到什么样的沙粒？这种情况背后的系统和模式是什么？现在假设我们有一个网络（例如，由计算机服务器或通过电缆连接的风力发电场组成）。在网络断开之前，必须有多少电缆失效？如果每条电缆都有一定的故障概率，那么网络断开的概率是多少？这种情况背后的系统和模式是什么？事实证明，这两种情况以及其他许多情况下的模式与称为“临界群”的代数结构密切相关。在数学中，网络被称为“图”。每个图都有一个与之关联的临界组。临界群以许多不同的方式出现，在数学和物理中图形的许多不同应用中出现；例如，通过芯片发射或沙堆模型，流动和切割空间，计数生成树，甚至停车的数学模型。到目前为止，在我们所描述的情况下，没有几何关系：所有重要的是邻接性，当另一个站点倒塌时，哪些站点会收到额外的沙子，哪些对计算机通过电缆连接。网络的样子对于我们所描述的模型来说并不重要。然而，在许多情况下，图的几何结构是通过在一个表面上嵌入它而提供的（例如，一个网络可能画在一个环面上），几何结构和邻接关系决定了图的关键属性。例子包括模拟酶的作用或某些形式的DNA链的图形。最近的一些研究暗示了在图的几何结构和临界群之间有很深的联系的可能性。当人们远离可以在平面上绘制的图时，与图相关的一些基本对象会发生深刻的变化：特别是人们研究生成拟树而不是生成树。到目前为止，对于嵌入在曲面上的图的临界群的研究还没有考虑到这些变化的基本结构。我们的目标是探索这个空间，通过开发考虑嵌入的临界群理论来实现几何结构的全部潜力。

The Sandpile Model is a widely studied model in physics and used in economics to model the consequences for a network of banks when one of them defaults. It has a simple intuitive description as follows. Suppose we pile up grains of sand on a number of sites. At some point one pile becomes too large to stay upright, it topples, an avalanche occurs and sand gets transferred to adjacent sites. What happens next? Is the amount of sand transferred to the adjacent sites enough to make them topple? For how long does the process continue until all the sites become stable again and what configuration of grains of sand do we end up with? What are the systems and patterns underlying this situation? Now suppose we have a network (comprising, for example, computer servers or wind farms connected by cables). How many cables must fail before the network becomes disconnected? If each cable has a certain probability of failure, what is the probability that the network becomes disconnected?What are the systems and patterns underlying this situation?It turns out that the patterns underlying these two situations and many others are closely related to algebraic structures called 'Critical Groups'. Networks are modelled in mathematics by objects called 'graphs'. Each graph has a Critical Group associated with it. Critical Groups arise in many different ways and in many different applications of graphs in mathematics and physics; for example, through Chip-Firing or the Sandpile Model, Flow and Cut Spaces, counting spanning trees and even mathematical models of car parking. In the situations we have described so far there is no geometry: all that matters is adjacency and which sites receive extra sand when another topples and which pairs of computers are linked by a cable. The way that the networks look is not important for the models we have described.However, in many settings a graph comes equipped with geometric structure provided by an embedding of it in a surface (for example, a network may come drawn on a torus), and the geometric structure as well as the adjacencies determine the key properties of the graph. Examples include graphs that model the actions of enzymes or certain forms of DNA strands. Some recent work has hinted at the possibility that there are deep links between the geometric structure of graphs and Critical Groups. When one moves away from graphs that can be drawn on a plane, some of the fundamental objects associated with the graph change profoundly: specifically one studies spanning quasi-trees rather than spanning trees. So far, the study of Critical Groups for graphs embedded on surfaces has not taken into account these changed fundamental structures. We aim to explore this space, realising the full potential of the geometric structure by developing a theory of the Critical Group that takes into account the embedding.