Representation stability and the homology of simplicial complexes of graphs

• Applicant: Dr. Lukas Katthän
• Subject Area: Mathematics
• Term: from 2015 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 282402286

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.

• DFG Programme: Research Fellowships
• International Connection: USA
• Host: Professor Victor Reiner, Ph.D.