But the methods were no longer effective when mathematicians only wanted to find, say, the number of real solutions to the equations in an enumerative geometry problem, or the number of integer solutions. If they asked an enumerative geometry problem in any number system other than the complex one, inconsistencies cropped up again. In these other number systems, mathematicians couldn’t address enumerative questions systematically.
At the same time, the mysterious, shifting answers that mathematicians encountered when they limited themselves to the integers, or to the real numbers, made enumerative questions a great way to probe those other number systems — to better understand the differences between them, and the objects that live inside them. Mathematicians thought that developing methods to deal with these settings would open up new, deeper areas of mathematics.
Among them was the mathematical great David Hilbert. When he penned a list of what he considered the most important open problems of the 20th century, he included one about making the techniques for solving enumerative geometry questions more rigorous.
In the 1960s and ’70s, Alexander Grothendieck and his successors developed novel conceptual tools that helped resolve Hilbert’s problem and set the foundation for the field of modern algebraic geometry. As mathematicians pursued an understanding of those concepts, which are so abstract that they remain impenetrable to nonspecialists, they ended up leaving enumerative geometry behind. Meanwhile, when it came to enumerative geometry problems in other number systems, “our techniques hit a brick wall,” Katz said. Enumerative geometry never became the beacon that Hilbert had imagined; other threads of research illuminated mathematicians’ way instead.
Enumerative geometry no longer felt like a central, lively area of study. Katz recalled that as a young professor in the 1980s, he was warned away from the subject “because it was not going to be good for my career.”
But a few years later, the development of string theory temporarily gave enumerative geometry a second wind. Many problems in string theory could be framed in terms of counting: String theorists wanted to find the number of distinct curves of a certain type, which represented the motion of strings — one-dimensional objects in 10-dimensional space that they believe form the building blocks of the universe. Enumerative geometry “became very much in fashion again,” Katz said.
But it was short-lived. Once physicists answered their questions, they moved on. Mathematicians still lacked a general framework for enumerative geometry problems in other number systems and had little interest in pursuing one. Other fields seemed more approachable.
That was the case until the mathematicians Kirsten Wickelgren and Jesse Kass came to a sudden realization: that enumerative geometry might provide the exact kind of deep insights that Hilbert had hoped for.
A Bird’s-Eye View
Kass and Wickelgren met in the late 2000s and soon became regular collaborators. In many ways their demeanors couldn’t be more different. Wickelgren is warm, but restrained and deliberate. Whenever I asked her to confirm that I’d understood a given statement correctly, she’d pause for a moment, then answer with a firm “Yes, please” — her way of saying “Exactly, you’ve got it!” Kass, on the other hand, is nervously enthusiastic. He’s easily excited and talks at a rapid-fire pace.
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