Mathematics and Statistics Colloquium – Joe Chen, Colgate University
Sandpiles in a Fractal Labyrinth
Monday, April 9, starting at 4:00 pm, Davis 301
Refreshments at 3:30 pm, outside of Davis 216
A sandpile on a graph is a configuration of indistinguishable particles ("chips") occupying the vertices of the graph.
Whenever the number of chips at vertex $x$ s greater than or equal to the degree of $x$, we topple the vertex by emitting one chip to each neighboring vertex of $x$. The procedure continues until every vertex has fewer number of chips than its degree, in which case the configuration is said to be stable.
A classic problem in this area is to describe the stable sandpile cluster when there are initially $N$ chips at a single vertex.
As $N$ grows, what can we say about the shape and patterns of the cluster?
On the square lattice, it has been recently proved that the cluster contains many copies of the Apollonian gasket, a famous self-similar fractal. So in this sense we see a "fractal in a sandpile."
On the other hand, I will describe a growing "sandpile on a fractal," the fractal being the Sierpinski gasket.
It turns out that results on the cluster shapes and patterns are unusually sharp.
I will explain how these results arise from a combination of geometric and algebraic origins.
PS: There will be many pictures and animations in my talk.