« Squags, Sloops, Quasigroups and Loops: Using made-up words to find Latin Squares
February 19, 2020, 12:15 PM - 1:15 PM
Location:
Mathematics Graduate Student Lounge -- 7th Floor
Rutgers University
Hill Center
Mathematics Department
110 Frelinghuysen Road
Piscataway, NJ 08854
Brian Pinsky, Rutgers University
What do groups, semirings, lattices, and magmas all have in common? Well, if you squint hard enough, all of their definitions are essentially the same: they're sets with a bunch of operations that have to satisfy a bunch of equations. Moreover, all of these structures have a lot in common: all of them have a 0 object, a direct product, a notion of homomorphism, etc. In fact, they have so much in common, you could make an entire field of math just studying just how similar they are. They did that, and they called it universal algebra. Like any worthwhile field of math, universal algebra has tons of pun-tastic and ridiculous words, surprising and powerful theorems, and almost no useful applications whatsoever. But while it's generally useless, it turns out there are some combinatorics topics where it's incredibly useful, and that's what this talk will be about. Specifically, we'll talk about latin squares and some special kinds of hypergraphs. Latin squares are grids like sudoku puzzles; each number must occur in each row and column exactly once. They seem friendly enough, but studying latin squares is generally quite hard. Looking at this with universal algebra gives a nice reason why it might be hard: latin squares correspond precisely to cayley tables of quaisigroups (a universal variety), so the theory of latin squares should be at least as hard as group theory, plus whatever complications this "quasi-" prefix adds. Orthogonal latin squares are n by n latin squares which, when stacked on top of each other, contain all the n^2 possible pairs of numbers as columns. It turns out, these correspond to another universal variety. My main goal this talk is to use that characterization to construct as many orthogonal latin squares as I can. Euler conjectured that it was impossible for n=2 mod 4, and we're going to prove he was wrong.