You only get to say “absolutely nothing” if you sing it.
Today I’m headed to Hampshire College to talk about ideals, varieties, and their applications. If you were lucky enough to sit through my job talk, you already know (or at least already dozed through) what I’ll talk about. Starting with a 4 by 4 sudoku puzzle, I’ll define ideals and varieties and explain how they’re useful for solving problems (like… a 4 by 4 sudoku puzzle).
Now, I’m a commutative algebraist, so I love polynomials almost as much as life itself. The first question you might have is, “What the hell do polynomials have to do with sudoku?” It turns out that you can work in a polynomial ring with 16 variables (or 81 variables for the 9 by 9 case) and model the “game space” for all legitimate game boards. Have a specific board in mind? Then add in a few more polynomials that describe the given clues. This isn’t going to give you a local ring, but that’s okay. Using primary decomposition, you can find the primary decomposition of your game board ideal, and this will tell you about the number of solutions (whether there are any at all, whether there is exactly one, or whether there are many).
It’s really cool stuff. I admit that the talk gets pretty hand-wavy right around primary decomposition (analogy to factoring prime numbers), but I think it’s pretty compelling.
Anyway, my friend and former officemate Amanda Croll and I are going to work on writing up a nice, rigorous version of the paragraphs above. I think it would make a nice piece for an undergraduate-friendly journal along the lines of Math Horizons.
But I digress a little. One of the exciting trends in algebra right now is that, hey, lots of statistical models are actually polynomial models. These models answer questions in evolutionary genetics, economics, quantum physics, and other cool areas. Why use all the calculus machinery when you have lovely, classical commutative algebra and algebraic geometry at your fingertips?
A big thanks to Alex Kunin for inviting me out to Hampshire College’s HCSSiM program. I’m so excited to talk about algebra! To a captive audience!