in autumn In 2017, then-Massachusetts Institute of Technology undergraduate Matab Thorney joined a graduate reading group to study a single paper for a semester. But by the end of the semester, baffled by the complexity of the proof, Thorney recalls, they decided to take the next step. “It was amazing,” he said. “It looked perfectly there.”
This paper is by Peter Kiebash of the University of Oxford. Its subject is a mathematical object called design.
The study of design dates back to 1850. At the time, Thomas Kirkman, a parish minister in the north of England who was also dabbled in mathematics, raised a seemingly simple problem in a magazine called The Magazine. Ladies and Gentlemen’s Diaries. Suppose 15 schoolgirls line up three lines each day for her week to go to school. Can you put two girls in the same row for her 7 days so that she doesn’t line up more than once?
Soon mathematicians began asking a more general version of Kirkman’s question. n Can the elements in the set (15 schoolgirls) always be sorted into size groups? k (3 columns) for each smaller size set t Do (all girl pairs) appear exactly in one of those groups?
Such a configuration (n, k, t) designs have since been used to develop error-correcting code, design experiments, test software, win sports slots and lottery tickets, and more.
But they also become very difficult to build, so k and t growing. In fact, mathematicians have not yet found a design with the following values: t So it was a big surprise when Keevash showed in 2014 that even if you don’t know how to build such a design, it will always exist as long as it lasts. n is large enough to satisfy some simple conditions.
Now, Keevash, Thorney and MIT graduate student Ashwin Sur have shown that there is always an even more elusive object called subspace design. “They proved the existence of objects whose existence was completely unknown,” said Caltech mathematician David Conlon.
To do that, I needed to refine Kevash’s original approach (a magical combination of randomness and careful construction) to work in a more restrictive setting. And Thorney, now pursuing his Ph.D. at MIT, is faced with a paper that stumbled upon him just a few years ago. “It was really fun to fully understand the technique, to really think about it, work on it and develop it,” he said.
Illustrated by Merrill Sherman/Quanta Magazine
“Beyond the imagination”
For decades, mathematicians have transformed problems about sets and subsets, such as design problems, into problems about so-called vector spaces and subspaces.
A vector space is a special kind of set whose elements (vectors) are related to each other in a much more rigorous way than a simple set of points. A dot shows where you are. The vector tells you how far and in what direction you moved. You can add or subtract, grow or shrink.
