SIR Mathematics Investigation Abstract
IDENTITY CONFIGURATIONS OF THE SANDPILE GROUP
Presenter:
William B. Chen, Illinois Mathematics and Science Academy
Advisors:
Professor Laszlo Babai, University of Chicago
Mr. Travis Schedler, University of Chicago
Abstract:
The abelian sandpile model on a connected graph yields a finite abelian group G of recurrent configurations that is closely related to the combinatorial Laplacian. We consider the identity configuration of the sandpile group on graphs with large edge multiplicities, called "thick" graphs. We explicitly compute the identity configuration for all thick trees and extend the result to include certain symmetrical thick cycle graphs. We introduce the amputated path graph, the computation of whose identity is shown equivalent to that of the thick cycle. An algorithm for this computation is devised and related to explicit formulas provided for identity configurations of the three- and four-cycle graphs.