in autumn in 2017, Matab SawneyAs an undergraduate at the Massachusetts Institute of Technology at the time, he joined a graduate reading group that worked on a single paper for one 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.”
The paper is Peter Keebash of Oxford University. 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 northern 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 arrange So in those 7 days, can’t two girls stand in the same line 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) has been used ever since to develop error-correcting code, design experiments, test software, and win sports slots and lotteries.
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 Thus, in 2014 Keevash Indicated Even if you don’t know how to build such a design, they always exist,as long as n is large enough to satisfy some simple conditions.
Well, Keevash, Thorny, and Ashwin SirAn MIT graduate student, he says that even more elusive objects called subspace designs can always exist the same. “They proved the existence of an object whose existence was completely unknown,” he said. David Conlona mathematician at the California Institute of Technology.
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 PhD 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, really worry 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.
