In the fall of the year 2017, Mehtaab Sawhney, then an undergraduate at the Massachusetts Institute of Technology, joined a graduate reading group to research a paper for a semester. But by the end of the semester, Sawhney recalled, they decided to move on, confused by the complexity of the evidence. “It was really great,” he said. “It just seems to be completely out there.”
The article belongs to Peter Keevash of the University of Oxford. Its subject: mathematical objects called design.
The study of designs can be traced back to 1850, when Thomas Kirkman, a parish priest in the north of England who was passionate about mathematics, posed a seemingly simple problem in a journal. even named Ladies and Gentlemen’s Diary. Suppose 15 girls walk to school in rows of three every day for a week. you can sort them so that during those seven days no two girls find themselves in the same line more than once?
Soon, mathematicians were asking a more general version of Kirkman’s question: If you had WOMEN elements in a set (15 of our girls), can you arrange them into always sized groups k (row three) such that every set of sizes is less than t (each pair of girls) appear in exactly one of those groups?
Such configurations, called (WOMEN, k, t) design, which has since been used to help develop error-correcting codes, test designs, test software as well as win sports and lottery matches.
But they are also extremely difficult to build because k And t larger development. In fact, mathematicians have yet to come up with a valid design t older than 5. And so it was a surprise that in 2014, Keevash shows that even if you don’t know how to build such designs, they always existas long as WOMEN is large enough and satisfies some simple conditions.
Now Keevash, Sawhney and Ashwin Saha graduate student at MIT, has shown that even more elusive objects, called subspace design, always exist. “They have proven the existence of objects whose existence is not obvious,” he said David Kunluna mathematician at the California Institute of Technology.
To do so, they had to refine Keevash’s original approach—which involved an almost magical blend of randomness and careful structure—so that it worked in a much more restrictive environment. And so Sawhney, now pursuing a PhD at MIT, found himself confronted with the paper that had puzzled him just a few years earlier. “It was really exciting to fully understand the techniques, really suffer and work with them and develop them,” he said.
Illustration: Merrill Sherman/Quanta Magazine
“Beyond what is beyond our imagination”
For decades, mathematicians have translated sets and subsets problems—like design questions—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 stricter way than a simple set of points can. A point that tells you where you are. The vector tells you how far and in what direction you have moved. They can be added and subtracted, made larger or smaller.
