Sandpiles and Dominos

Start: 02/18/2015 - 4:15pm
End  : 02/18/2015 - 5:15pm


David Perkinson, Reed College


The Abelian Sandpile Model (ASM) is a mathematical model devel- oped by physicists around 1990 to elucidate self-organized criticality, a phe- nomenon claimed to be ubiquitous in nature. Roughly, self-organized criti- cality describes a system that naturally evolves into a state at the border of stability, with instabilities over time characterized by scale invariance. The Gutenberg-Richter law in geophysics and Zipf’s law in linguistics are often cited as real-world examples. The ASM has been shown to have connections to algebraic geometry and commutative algebra, combinatorics, potential theory, and number theory.
In this talk, I will present work done with undergraduate students con- necting the sandpile model with domino tilings. We will be interested in tiling an m × n checkerboard (m rows and n columns) with dominos. A domino covers exactly two squares of the checkerboard, and a tiling consists of covering the checkerboard with non-overlapping dominos.
As warm-up for the talk you may want to answer the following two ques- tions: (i) How many ways are there of tiling a 4 × 4 checkerboard with dominos? (ii) Take a flexible 4 × 4 checkboard and glue one of its edges to the opposite edge with a twist to get a Mo ̈bius band. How many ways are there of tiling this twisted checkerboard?

Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460

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