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.
Sheldon Katz is intrigued by the connection between questions in enumerative geometry and string theory. Fred Zwicky
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: