Representation stability and the homology of simplicial complexes of graphs

在这个项目中，我们计划从不同的角度研究图复形的同调性。一个图复形是一个单纯复形，它的面对应于一个固定顶点集上的所有图，这些图共享一些属性。一个例子是所有不连通图的复形。一方面，我们将考虑对称群在同调群上的表示，特别是从表示稳定性的角度。另一方面，我们将研究具有有界投影维数或有界Castelnuovo-Mumford正则性的图复形，这是由这些类与复杂性理论的自然联系所激发的。特别是，我们计划使用复杂性理论的结果，以确定可能的反例的斯坦利猜想之间的图complex.在该项目的最后一部分，我们计划研究更一般的理想链的作用的对称群的观点表示稳定性.

In the proposed project, we plan to study the homology of graph complexes from different perspectives. A graph complex is a simplicial complex whose faces correspond to all graphs on a fixed set of vertices, which share some property. One example is the complex of all disconnected graphs.On the one hand, we will consider the representations of the symmetric group on the homology groups, in particular from the point of view of representation stability. On the other hand, we will study graph complexes with bounded projective dimension or bounded Castelnuovo-Mumford regularity.This is motivated by a conjectural connection of these classes to complexity theory. In particular, we plan to use complexity theoretic results to identify possible counterexample to the Stanley conjecture among graph complexes.In the last part of the project, we plan to study more generally chains of ideals with an action of the symmetric group from the point of view of representation stability.