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Surfaces beyond imagination are discovered after decades-long search

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https://www.quantamagazine.org/impossibl...-20220602/

INTRO: Last July, two mathematicians from Durham University, Will Hide and Michael Magee, confirmed the existence of a much sought-after sequence of surfaces: each more complicated than the last, ultimately becoming so intricately connected with themselves that they nearly reach the limits of what’s possible.

At first, it wasn’t obvious these surfaces existed at all. But since the question of their existence first arose in the 1980s, mathematicians have come to realize that these surfaces may actually be commonplace, even if they’re exceedingly difficult to pinpoint — a perfect example of how mathematics can subvert human intuition. The new work is a step forward in a quest to move beyond intuition to understand the myriad ways surfaces can manifest.

“It’s a brilliant piece of mathematics,” said Peter Sarnak, a mathematician at the Institute for Advanced Study in Princeton, New Jersey.

Surfaces include all kinds of two-dimensional objects — the outer shell of a sphere, doughnut or cylinder; a Möbius strip. They’re essential to mathematics and physics. But although mathematicians’ relationship with surfaces goes back centuries, they don’t know these objects well at all.

Simple surfaces are not the problem. Simple in this case means the surface has a small number of holes, or a low “genus.” A sphere, for example, has no holes, and thus has a genus of zero; a doughnut has one.

But when the genus is high, intuition fails us. When Alex Wright, a mathematician at the University of Michigan, tries to visualize a high-genus surface, he ends up with holes arranged in a tidy row. “If you wanted me to be a bit more creative, I could wrap it around into a circle with many holes. And I’d be hard-pressed to come up with any mental picture fundamentally different from those,” he said. But in high-genus surfaces, holes overlap with one another in complicated ways that make them difficult to grasp. A simple approximation is “as far from representative as it could ever be, in every sense,” Wright said.

This struggle is to be expected, said Laura Monk, a mathematician at the University of Bristol. “You can often make things which are bad. However, making things which are good, which are like the thing we expect typically to be true, is a bit more difficult,” she said.

That means mathematicians wishing to truly understand the space of surfaces need to find ways to uncover objects that they don’t even know exist.

In their July paper, Hide and Magee did just that, confirming the existence of surfaces that mathematicians had wondered about for decades. The conjecture they proved, and the history surrounding it, took inspiration from a different field of math altogether: graph theory... (MORE - details)
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