Recently, the card game Set has been a trending topic in math circles. A breakthrough came on the topic of finding an upper bound on the number of cards one could lay down without any players finding a set. The problem is a sort of combinatorial problem (it falls under the umbrella of Ramsey theory), but was translated to the language of group theory and algebraic geometry where the authors used the “polynomial method” to get much, much better bounds than ever before, and a very, very short number of pages. See these papers for more information on the math. According to the first linked article, the popularized proof technique has already been used to solve other “cap set” problems, which tend to have applications in fields like computer science and information theory.
The reason I’m interested in the news is that I didn’t realize how many applications the “mathematics of Set” had — both inside and outside of mathematics. I have spent time playing it as a logic game with many students for simply that, that it practices “logical reasoning” which is the foundation on which mathematics is built. To learn that all along, I had been playing a game with these deep connections — and that I could have been telling the students about them! — I felt both excitement and regret.
As mathematicians we get excited about the chase of new ideas, new puzzles, new theoretical structures to explore. But there’s something really special about being able to communicate this excitement to the “layperson” in the terms that they’re already familiar with, like card games. It’s a real struggle in the field of operator algebras to find such connections, since there’s such an enormous amount of structure just built up to define the questions we want to tackle. Often times the connections I do know of seem superficial, in the sense that the applications don’t often drive the problems I’m interested in or the hypotheses I impose. Even so, and even while some of the “technical speak” might be out of grasp for Average Alex, I believe that with enough thought and perseverance I can find ways to phrase the math I do in ways that are intuitive and relate-able. Maybe not ways to get across the finer points, or give those easily-digestible-but-unrealistic-“real-world” applications. What I want is something like Set is to Ramsey theory — a way that will get at the heart of the matter.