Last year, two young assistant professors of mathematics: Eric Larson and Isabel Vogt made mathematical history by achieving a breakthrough that had stumped other mathematicians since the 1800s. In a proof published last January, Larson and Vogt solved the so-called interpolation problem — a discovery that has been hailed aa one of 2022’s most significant mathematical developments.
They also happen to be married to each other.
Algebraic geometry is a branch of mathematics that deals with curves or surfaces, describing them as both as geometric objects and solutions of algebraic (specifically, polynomial) equations. A simple example is a circle of radius 1 (the unit circle) which is defined by the set of all points that are at a unit distance from its center, but is also the set of points (x,y) satisfying the polynomial equation, x2 + y2 = 1
There are deep connections between this branch of math and another important one – number theory that has far-reaching applications in information technology, specifically cryptography, which is crucial in building defense against cybersecurity attacks. Indeed, the deeper the connection between algebraic geometry and number theory one can find, the more options there are in expanding our cybersecurity toolkit. This is another startling example of how seemingly “useless” mathematics can turn out to be extremely useful in the real world.
The problem, of course, is that deep connections are hard to visualize and even harder to proof mathematically. That is why the work of Larson and Vogt is causing ripples in the mathematical world.
The interpolation problem asks the following question:
How many random points in r-dimensional space can a curve of genus g and degree d pass through?
With two points on the plane, one can pass a unique straight line touching both points. With three points, you can draw a unique circle touching all three points. With five points, a unique conic section passes through the five points. These are the simplest cases we can see and intuit.
Mathematicians are interested in generalizations from these simple cases to “points counting” in curves of any dimension. This is where it gets hairy. For starters, one must be sure that for a given genus g and degree d, curves do exist in any dimension, r.
In 1870, the mathematicians Alexander von Brill and Max Noether (father of the famous mathematician, Emmy Noether) came up with a prediction on the existence of curves using only three properties of a curve: the genus (g), the number of dimensions (r), and the degree (d). Very briefly, the genus of a curve is the number of holes it has (a doughnut has genus one), the dimension refers to the space in which the curve lives in (the ones we can easily visualize are up to three dimensions), and the degree is a measure of how “twisty” a curve is, or how many times it intersects with a hyperplane (example: a circle has degree 2 because if you slice it with a line, the line cuts at two points).
Brill and Noether conjectured that one can embed a curve of a given genus in a space of a given dimension provided the degree of the curve is sufficiently large. They wrote out their conjecture as a precise inequality in terms of g, r and d though they did not supply rigorous proof of their conjecture. That proof was derived only in 1980, by two mathematicians, Phillip Griffiths and Joe Harris, using new advances in algebraic geometry.
Back to the Interpolation Problem
Now that the Brill-Noether conjecture was proven true (meaning highly complex curves do exist given “the right conditions”), mathematicians returned to the interpolation problem, hoping to figure out how many random points in an r-dimensional space a curve of genus g and degree d could pass through. Nobody could solve this problem until Larson and Vogt in 2022. While they were only undergraduate students at Harvard University, Larson and Vogt got interested in the problem and set their minds on proving it. They guessed that the answer to that problem would depend not only on g, r, and d but also n, the number of points.
Then as a graduate student, Larson worked on another major problem in algebraic geometry known as the maximal rank conjecture which was seemingly unrelated to the interpolation problem. But as it turned out, the proof of the maximal rank conjecture, which required that techniques to first break a curve into small curves, studying their properties, then gluing them back together in just the right way, proved also to be the crucial tool in solving the interpolation problem. This is because to glue those simpler curves together, one must make each of them pass through the same group of points. Here is a simple example of this “curve breaking” technique.
Meanwhile, Vogt was already working on interpolation in graduate school where she had proved all cases of interpolation in three-dimensional space. To address higher dimensional spaces was a big challenge. So, she teamed up with Larson and they solved the problem in four-dimensional space as well as got married.
Since their marriage, Larson and Vogt have fused their married live into their working lives both at Brown University and at home. Their set themselves to the challenge of proving the interpolation problem in its full generality, i.e., for all dimensions, an extremely hard problem to say the least. But the young couple persisted, until they finally cracked it in January this year. “It’s sheer perseverance. It’s more than that. It’s brilliant actually, to be able to finish it,” said Gavril Farkas, a mathematician at the Humboldt University of Berlin.
While their proof might mark the end of one narrative thread, the story is far from over. Larson and Vogt are continuing to forge ahead in illuminating what different kinds of curves look like, how they behave, and what all this imply for other mathematical problems, which the scientific world awaits with bated breath.
This post is adapted from a longer story published in the 25 August 2022 issue of Quanta Magazine.