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New Math Revives Geometry's Oldest Problems -- Quanta Magazine
https://www.quantamagazine.org/new-math-revives-geometrys-oldest-problems-20250926/Using a relatively young theory, a team of mathematicians has started to answer questions whose roots lie at the very beginning of mathematics.
A pretty dense (for me) explanation of some recent findings in branches of number theory that seem to have broad implications for much of mathematics.
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 (opens a new tab), 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 (opens a new tab), a mathematician at Stanford University. “If you do it again and again, it’s a theory.”
. . .
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.”
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 (opens a new tab), 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 (opens a new tab), a mathematician at Stanford University. “If you do it again and again, it’s a theory.”
. . .
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.”