New Math Revives Geometry’s Oldest Problems
Kristina Armitage/Quanta Magazine
Introduction
In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight.
Such questions, which ask for the number of solutions that satisfy a set of geometric conditions, were a favorite of the ancient Greeks. And they’ve continued to entrance mathematicians for millennia. How many lines lie on a cubic surface? How many quadratic curves lie on a quintic surface? (Twenty-seven and 609,250, respectively.) “These are really hard questions that are only easy to understand,” said Sheldon Katz, a mathematician at the University of Illinois, Urbana-Champaign.
As mathematics advanced, the objects that mathematicians wanted to count got more complicated. It became a field of study in its own right, known as enumerative geometry.
There seemed to be no end to the enumerative geometry problems that mathematicians could come up with. But by the middle of the 20th century, mathematicians had started to lose interest. Geometers moved beyond concrete problems about counting, and focused instead on more general abstractions and deeper truths. With the exception of a brief resurgence in the 1990s, enumerative geometry seemed to have been set aside for good.
That may now be starting to change. A small cadre of mathematicians has figured out how to apply a decades-old theory to enumerative questions. The researchers are providing solutions not just to the original problems, but to versions of those problems in infinitely many exotic number systems. “If you do something once, it’s impressive,” said Ravi Vakil, a mathematician at Stanford University. “If you do it again and again, it’s a theory.”
That theory has helped to revive the field of enumerative geometry and to connect it to several other areas of study, including algebra, topology and number theory — imbuing it with fresh depth and allure. The work has also given mathematicians new insights into all sorts of important number systems, far beyond the ones they’re most familiar with.
At the same time, these results are raising just as many questions as they answer. The theory spits out the numbers that mathematicians are seeking, but it also gives additional information that they’re struggling to interpret.
That mystery has inspired a new generation of talent to get involved. Together, they’re bringing counting into the 21st century.
Counting Forward
All enumerative geometry problems essentially come down to counting objects in space. But even the simplest examples can quickly get complicated.
Consider two circles some distance apart on a piece of paper. How many lines can you draw that touch each circle exactly once? The answer is four:
Mark Belan/Quanta Magazine
You can slide these circles further apart, or shrink one to half its size, and the answer won’t change. But move one circle so that it intersects the other like a Venn diagram, and suddenly the answer does change — from four to two. Slide whichever circle is smaller entirely inside the bigger one, and now the answer is zero: You can’t draw any lines that touch each circle only once.
Such inconsistencies are a real pain. In this example, there were only three different configurations to consider, but often the problem is too complicated for researchers to work through every possible case. You might find the answer for one case, but you’ll have no idea how it will change when you move things around.
In practice, mathematicians try to write the problem’s geometric constraints as a collection of equations, then figure out how many solutions satisfy all those equations simultaneously. But even though they know that the number of solutions won’t always stay consistent, there’s nothing in the nature of the equations they write down that indicates whether they’ve stumbled on a new configuration that will yield a different answer.
There’s one exception — when the problem is defined in terms of complex numbers. A complex number has two parts: a “real” part, which is an ordinary number, and an “imaginary” part, which is an ordinary number multiplied by the square root of −1 (what mathematicians call i).
In the example above with the circles and lines, if you ask for the number of complex solutions to your equations, you always get four as your answer, no matter what arrangement you look at.
By around 1900, mathematicians had developed techniques to solve any enumerative geometry problem in the complex realm. These techniques didn’t have to take different configurations into account: No matter what answer mathematicians got, they knew it had to be true for every configuration.
Mark Belan/Quanta Magazine
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.
Kirsten Wickelgren has been using a sophisticated set of mathematical techniques to probe the fundamental nature of numbers.
Joseph Rabinoff
But Kass and Wickelgren worked well together and shared many interests — including a love for extending geometry’s reach into other fields.
In 2015, Kass was passing through Atlanta, where Wickelgren lived, and decided to approach her with his latest obsession: He wanted to revisit enumerative questions in restricted number systems, that long-abandoned endeavor.
He brought along a bunch of loose ideas and old papers that seemed relevant. “I realized this was a kind of pie-in-the-sky project,” Kass said. “She very politely explained to me that all my answers were nonsense.” Then he mentioned a result from 1977, and suddenly “a light bulb went off.”
In that 1977 paper, the mathematicians Harold Levine and David Eisenbud were working out a proof that involved counting. They ended up with a special type of expression called a quadratic form — a simple polynomial where each term’s exponents always add up to 2, such as x2 + y2, or z2 − x2 + 3yz.
Eisenbud and Levine realized that the count they were interested in was hidden in plain sight. The answer lay in the form’s “signature”: the number of positive terms minus the number of negative terms. (For example, the quadratic form z2 − x2 + 3yz has two positive terms, z2 and 3yz, and one negative term, x2, so its signature is 2 − 1, or 1.)
This was Wickelgren’s light bulb. In the decades since Eisenbud and Levine had published their proof, mathematicians had devised a seemingly unrelated framework called motivic homotopy theory. That framework, which treated solutions to equations as special mathematical spaces and studied the relationships between them, was both sophisticated and powerful. Among other things, it gave mathematicians a way to describe those relationships using particular kinds of quadratic forms.
Jesse Kass, while pursuing what he called “a kind of pie-in-the-sky project,” helped revive interest in one of math’s oldest types of questions.
Caroyn Lagattuta
Listening to Kass, Wickelgren immediately recognized that Eisenbud and Levine had come up with one of these forms. The mathematicians had been doing motivic homotopy theory without realizing it — and it had given them the answer they’d been seeking.
And while Eisenbud and Levine weren’t working on an enumerative geometry problem, it was similar enough in flavor — it involved counting, after all — that it got Kass and Wickelgren thinking. Perhaps they could solve their own counting problems using the framework of motivic homotopy theory, too. And since motivic homotopy theory could be broadly applied to any number system, perhaps it would unlock the enumerative geometry questions in those settings that had eluded mathematicians for so long.
A Deeper View
Remember that typically, an enumerative geometry problem involves finding the number of solutions that satisfy a collection of equations. Kass and Wickelgren’s insight was not to try to solve those equations directly — it rarely worked in settings other than the complex numbers. Instead, the pair rewrote a given enumerative geometry question (set in a given number system) in terms of spaces of equations and functions that described the relationship between those spaces.
With the problem reformulated in this way, they could apply motivic homotopy theory to it. This allowed them to compute a quadratic form. Now they had to figure out what information that quadratic form contained about their original problem.
When they were working in the complex numbers, they realized, all they had to do was count up the number of different variables in the quadratic form they’d computed. That number gave them the number of solutions to their enumerative geometry problem. Of course, this wasn’t particularly interesting to them: Mathematicians already had good techniques for getting this answer.
So they moved on to other number systems. For the real numbers, it got a little trickier. Once they computed the quadratic form in this setting, they had to look at its signature instead. And the signature didn’t give the precise answer: It gave a minimum for what the answer could be. That is, for any enumerative geometry problem involving real numbers, they had a way to calculate a lower bound — a good starting place.
But most exciting of all was that when they computed a quadratic form for other, stranger number systems, they could also glean important information. Take a looping system of seven numbers that operates on what’s called clock arithmetic: In such a system, 7 + 1 equals 1 instead of 8. In this system, they rewrote their quadratic form as an array of numbers called a matrix. They then calculated a quantity called the determinant and proved that while it didn’t tell them the total number of solutions, it did tell them something about what proportions of those solutions had certain geometric properties.
In 2017, Kass and Wickelgren showcased this for one of enumerative geometry’s most famous theorems: that a cubic surface can contain at most 27 lines. Using their new methods, they showed that indeed, the answer is 27 in the complex numbers. They replicated a known lower bound for the real numbers — and provided new numerical information for every finite number system. It all came in one package.
It was one of the first times mathematicians had been able to say anything significant about enumerative geometry problems for systems outside the complex and real numbers. Moreover, while the problem’s answer might change depending on the number system and the configuration of shapes within it, for the first time mathematicians had found one theory that could encompass all those potential different answers.
“It’s not just about the real numbers or the complex numbers,” Wickelgren said. “They’re just special cases of a result that holds in any number system.”
And that was only the beginning.
A New Start
In the years since, Wickelgren, Kass and others have reframed a host of other enumerative problems using motivic homotopy theory, deriving the relevant quadratic forms in various number systems.
“All the geometric constructions used to give people integer answers,” said Marc Levine, a mathematician at the University of Duisburg-Essen who has been independently exploring the same ideas. “Now you can feed [the problem] in and get something which will give you a quadratic form as an answer.”
Mathematicians have made a lot of progress since Kass and Wickelgren’s original work when it comes to understanding what information a quadratic form can give them in different number systems. Sometimes, though, they’re not sure what to look for in the quadratic form. “We’re still kind of mystified about what exactly it tells you,” Levine said. There’s a lot left to interpret.
“At this point,” said Aravind Asok of the University of Southern California, trying to glean information about enumerative geometry problems from quadratic forms “is an entire industry.” It’s also concrete and accessible, which has attracted the attention of young mathematicians, he added. “It’s exciting because students can get into something with meat sort of quickly.”
Such concreteness is unusual in today’s abstract mathematical landscape. “The math keeps going one level higher in abstraction, and then sometimes I feel like I don’t know what I’m talking about anymore,” said Sabrina Pauli, who was Wickelgren’s first graduate student and is now a professor at the Technical University of Darmstadt in Germany. But this new area of research gives her a way to bring that high level of abstraction back down to earth.
Wickelgren, Kass, Levine and others have recently used their techniques to revisit enumerative questions related to string theory — but in new number systems and settings.
In all these cases, mathematicians have found a new way to explore how points, lines, circles and far more complicated objects act differently in different numerical contexts. Kass and Wickelgren’s revived version of enumerative geometry provides an unlikely window into the very structure of numbers. “It would be hard for me not to be drawn to the question that asks how many rational curves are there on a sheet of paper,” Wickelgren said. “That’s a fundamental part of the mathematical reality of a sheet of paper.”
