Seminars & Colloquia Calendar

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Tiling problems, old and new

James Propp (UMass, Lowell)

Location:  Zoom
Date & time: Wednesday, 30 March 2022 at 3:30PM - 4:30PM

Abstract: A classic puzzle is to show that an 8-by-8 square cannot be tiled by 1-by-2 and 2-by-1 rectangles if two opposite 1-by-1 corner-squares are removed. A less famous puzzle is to count how many tilings there are if you DON’T remove those corners. This is an example of a dimer problem, first considered by physicists, at the intersection of graph theory and enumerative combinatorics. I’ll review some highlights in the theory of dimers, displaying elegant formulas for the number of tilings and striking pictures of the kinds of long-range order that can arise spontaneously.
Then I’ll discuss beautiful work of Conway, Lagarias, and Thurston applying combinatorial group theory to the study of tiling problems, yielding criteria for distinguishing between doable and non-doable tiling problems. I’ll apply the method to obtain a surprising result about a class of tiling problems that are ALMOST never doable.
Finally, bringing the two strands together, I’ll discuss my recent empirical work counting tilings related to trimer models. None of the methods that I know of for obtaining exact formulas for dimers apply here, yet the data show a wealth of patterns suggesting that there are deep theorems to be proved. Especially mysterious are some patterns involving the 2-adic numbers.
This talk should be understandable to undergraduates. No background in graph theory, enumerative combinatorics, combinatorial group theory, or p-adic analysis will be required, and you won’t need to know what a dimer or a trimer is when you arrive (though you will when you leave).

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