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Sandpile torsors are all about elevating an enumerative graph theoretic result about spanning trees to an algebraic relationship using group theory.
The term sandpile torsor was introduced in a MathOverflow post by Jordan Ellenberg. The general idea is to understand simply transitive actions of something called the sandpile group of a graph on its set of spanning trees. I am especially interested in situations where the action is in some sense canonical.
Contraction(clip).aviOne major challenge in the subject of sandpile torsors is making precise the notion of canonical. In joint work with Ankan Ganguly, I introduced a deletion-contraction property called consistency. We went on to show that all consistent sandpile torsor algorithms on plane graphs have the same structure. The animation above is part of a longer video I made to accompany a talk about the paper. (Note that the animation has no sound and includes some pauses for my explanations).
In addition to a deep exploration of sandpile torsor ideas, I am also interested in making the topic more accessible. The diagram above is from a paper I wrote for American Mathematics Monthly, where I describe actions on spanning trees in terms of mathematicians and coffee cups moving around the vertices of the graph. The article will be published in 2026, but you can email me for an early copy.
The study of sandpile torsors can be explored through the language of chip-firing. One great resource is The Mathematics of Chip-Firing, a textbook written by my advisor, Caroline Klivans. On the left above, I illustrated the material of the textbook as a leisurely train ride through a variety of beautiful topics. The goal for my American Mathematics Monthly paper was chart a direct path to the sandpile torsor material I'm most interested in. Below is a silent animation meant to explain some of the main ideas of the construction.
While sandpile torsor ideas are easiest to visualize when working with spanning trees of graphs, many of the same concepts generalize to bases of regular matroids. In joint work with Changxin Ding, Lilla Tóthmérész, and Chi Ho Yuen, we showed that a known matroidal sandpile torsor algorithm is consistent. At the time of writing, it is an open question whether this sandpile torsor algorithm is unique.
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