Aug 29, 2022 08:34 PM
(This post was last modified: Aug 29, 2022 08:38 PM by C C.)
https://www.quantamagazine.org/old-probl...-20220825/
INTRO: A basic fact of geometry, known for millennia, is that you can draw a line through any two points in the plane. Any more points, and you’re out of luck: It’s not likely that a single line will pass through all of them. But you can pass a circle through any three points, and a conic section (an ellipse, parabola or hyperbola) through any five.
More generally, mathematicians want to know when you can draw a curve through arbitrarily many points in arbitrarily many dimensions. It’s a fundamental question — known as the interpolation problem — about algebraic curves, one of the most central objects in mathematics. “This is really about just understanding what curves are,” said Ravi Vakil, a mathematician at Stanford University.
But curves that live in higher dimensions, despite having been studied with state-of-the-art tools for hundreds of years, are tricky beasts. In two-dimensional space, a curve can be cut out by a single equation: A line might be written as y = 3x − 7, a circle by x2 + y2 = 1. In spaces of three or more dimensions, however, a curve gets much more complicated, and it is often defined by so many equations in so many variables that you cannot possibly hope to fully understand its geometry. As a result, a curve’s most basic properties can be exceedingly difficult to grasp — including the seemingly simple notion of whether it passes through some collection of points in space.
For centuries, mathematicians have been proving cases of the interpolation problem: Can you, for instance, put a curve with certain specified properties through 16 points in three-dimensional space, or a billion points in five-dimensional space? That work has not only allowed them to answer important questions in algebraic geometry, but also helped inspire developments in cryptography, digital storage and other areas well beyond pure mathematics.
Still, Vakil said, it’s not enough to understand interpolation for most curves. Mathematicians want to know it for all of them.
Now, in a proof posted online earlier this year, two young mathematicians at Brown University, Eric Larson and Isabel Vogt, have finally dealt the problem its final blow, solving it completely and systematically. The paper marks the culmination of nearly a decade of work, during which they gradually chipped away at the question, solved important related problems about what curves look like and how they behave — and also got married.
“It’s really a remarkable story,” said Sam Payne, a mathematician at the University of Texas, Austin, “for [people] that young and that early in their mathematical development to latch on to such a deep, hard problem, and then to be so persistent.” (MORE - details)
INTRO: A basic fact of geometry, known for millennia, is that you can draw a line through any two points in the plane. Any more points, and you’re out of luck: It’s not likely that a single line will pass through all of them. But you can pass a circle through any three points, and a conic section (an ellipse, parabola or hyperbola) through any five.
More generally, mathematicians want to know when you can draw a curve through arbitrarily many points in arbitrarily many dimensions. It’s a fundamental question — known as the interpolation problem — about algebraic curves, one of the most central objects in mathematics. “This is really about just understanding what curves are,” said Ravi Vakil, a mathematician at Stanford University.
But curves that live in higher dimensions, despite having been studied with state-of-the-art tools for hundreds of years, are tricky beasts. In two-dimensional space, a curve can be cut out by a single equation: A line might be written as y = 3x − 7, a circle by x2 + y2 = 1. In spaces of three or more dimensions, however, a curve gets much more complicated, and it is often defined by so many equations in so many variables that you cannot possibly hope to fully understand its geometry. As a result, a curve’s most basic properties can be exceedingly difficult to grasp — including the seemingly simple notion of whether it passes through some collection of points in space.
For centuries, mathematicians have been proving cases of the interpolation problem: Can you, for instance, put a curve with certain specified properties through 16 points in three-dimensional space, or a billion points in five-dimensional space? That work has not only allowed them to answer important questions in algebraic geometry, but also helped inspire developments in cryptography, digital storage and other areas well beyond pure mathematics.
Still, Vakil said, it’s not enough to understand interpolation for most curves. Mathematicians want to know it for all of them.
Now, in a proof posted online earlier this year, two young mathematicians at Brown University, Eric Larson and Isabel Vogt, have finally dealt the problem its final blow, solving it completely and systematically. The paper marks the culmination of nearly a decade of work, during which they gradually chipped away at the question, solved important related problems about what curves look like and how they behave — and also got married.
“It’s really a remarkable story,” said Sam Payne, a mathematician at the University of Texas, Austin, “for [people] that young and that early in their mathematical development to latch on to such a deep, hard problem, and then to be so persistent.” (MORE - details)