“They’ve proved the existence of objects whose existence is not at all obvious.” Via QuantaMagazine
, you end up working with millions of subspaces. The complexity of such systems “isn’t just beyond our imagination; it’s beyond what is beyond our imagination,” saidBut do subspace designs always exist, for any? Some mathematicians conjectured that, by and large, such objects are impossible. Others, heartened by work done over the years on designs, figured that “it may be hard to prove, but if there isn’t an obvious reason for them not to exist, then they should exist,” Keevash said.
The two young mathematicians were intrigued that it had been so hard to write down just one explicit example of a subspace design, and they saw the problem as a perfect way to explore the boundaries of important techniques in combinatorics. They followed the same overall strategy that Keevash had laid out in his designs work—but because of the tighter constraints at hand, “in practice, all of the steps ended up being very different in their implementation,” Keevash said. First, they set aside a carefully chosen set of subspaces, called a template. The template would later act as an island of structure in an ocean of randomness.
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