Old Problem About Algebraic Curves Falls to Young Mathematicians

Any two points in the plane can be used to draw a line. It is unlikely that a single line will pass through all of them. You can go through any three points and a conic section through any five.

When you can draw a curve through a lot of points is a question mathematicians want to know. One of the most important questions in mathematics is about the intersection of curves. The purpose of this is to understand what curves are.

Even though curves have been studied with state-of-the-art tools for hundreds of years, they are difficult. A curve can be cut out by a single equation in two-dimensional space. A curve is more complicated in spaces of three or more dimensions because it is often defined by so many equations in so many variables. A curve's most basic properties can be difficult to comprehend, including the seemingly simple notion of whether it passes through a collection of points.

If you want to put a curve with certain properties through 16 points in three-dimensional space or a billion points in five-dimensional space, you need to prove it. They have been able to answer important questions in algebraic geometry thanks to the work that has been done.

It is not enough to understand the differences between curves. It is important for mathematicians to know it.

Two young mathematicians at Brown University have solved the problem completely and systematically in a proof posted online. The paper marks the culmination of nearly a decade of work, during which they solved important problems about what curves look like and how they behave, as well as getting married.

Sam Payne, a mathematician at the University of Texas, Austin, said it was a remarkable story for people to be so persistent in their mathematical development.

Embedding Curves

Work dating back to the 19th century is the basis of the solution to the problem. What types of curves are out there?

The curve is a one-dimensional object. It is not always clear how to describe a curve using specific equations. The curve's dimensions are the first of these. The degree of the curve is the number of times it intersects with a hyperplane. When a circle is sliced with a one-dimensional line, the line generally hits it at two points. The degree of a curve in 20-dimensional space is the number of times it intersects with a 19-dimensional hyperplane. The degree is a way of measuring how twisted up the curve is.

The third number used by mathematicians to describe a curve is called the generat. A curve is a one-dimensional object that is defined in terms of complex numbers and each point can be written as a pair of real numbers. A curve can have holes if it is a two-dimensional surface. The surface of a doughnuts. The number of holes is called a curve's genus.

Before mathematicians could even think about what curves of a given degree might look like, they had to figure out when they could. That turned out to be a huge challenge.

Alexander von Brill and Max Noether formulated a prediction using only three properties, the number of holes, degree, and twistiness. If the degree of the curve is large, you can put a curve in a space of a certain number of dimensions. If the inequality held, a curve of your choosing would be possible.

Their argument didn't meet the standard of proof. That wouldn't happen for more than a century, when in 1980 Phillip Griffiths and Joe Harris used modern techniques to show that the Brill-Noether Theorem was true. Since then, mathematicians have produced around half a dozen different proof of the theorem and created a rich theory around it.

The result made it possible for mathematicians to figure out how many random points a curve of g and degree d could pass through. The curve doesn't get embedded in space in a special way because it's said to be "general." They had an educated guess as to what the answer should be. The curve's parameters needed to be satisfied by a particular inequality that was written in terms of g, d and r, but also in terms of n.

There were exceptions to the rule where the geometry of the curve restricted how many points it would have been expected to pass through. Payne said that it was an indication that this is a hard theorem and that it would take a lot of work to complete.

The problem that they were interested in was that. When they were undergraduates at Harvard University, they met Harris, who was one of their professors. When they were graduate students at the Massachusetts Institute of Technology, Harris and Vogt became co-advisers of one another.

Breaking the Problem

While he was working on the maximal rank conjecture, he began to work on the interpolation problem. When he was a graduate student, he set his sights on this conjecture, which had been open for more than a century. He was chasing something that people much older than him had failed at.

He presented a proof that he was a rising star in the field.

The proof needed to be worked out in different cases of the problem. The approach to the maximal rank conjecture was to break a curve of interest into multiple curves, study their properties, and glue them back together in the right way. He had to make each curve pass through a group of points in order to glue them together. Interpolation can be used to build curves that are more complex.

He was working on it. In the first paper she wrote in graduate school, she proved all cases of interpolation in three-dimensional space, as well as in four-dimensional space, with the help of her partner. This is how the couple began working together. They got married in the same year that he posted his proof. They work through problems on the chalkboards they have in their home after dinner.

If a curve can pass through a set of random points, the problem is solved. They had to show that the curve could wiggle in space. Three points on a line is considered. If you move one point away from the line but keep the other two points fixed, you can't shift the line in any way that would allow it to pass through the new configuration of points Trying to hit all three at the same time would cause the line to bend. A line can go through two points, but not three.

The mathematicians wanted to figure out a way to shift the curves in higher-dimensional spaces to study how they move.

The normal bundle of the curve controls how the curve can wiggle around. The normal bundle of a curve could be used as a problem to solve the interpolation question.

These are difficult to study because they are more complicated than the curves that the two men were worried about. The same strategy was used in the proof of the maximal rank. They broke it into pieces. They broke it in the right way so they could see what was happening.

An example can be taken. A hyperbola is a single curve that looks like a pair of mirror- image arcs facing away from each other. You can form this curve until it splits into two simpler curves, in this case a pair of lines crossing each other in an X shape. The geometry of those lines reflect some aspects of the hyperbola. Since the lines are simpler, they are easier to work with.

It is not possible to simply look at the individual lines and understand the normal bundle of the hyperbola. The normal bundle behaves differently at the point where the two lines meet. The normal bundle needs to be studied by mathematicians.

They weren't looking at lines but at more intricate situations. They would split a curve into two parts: a line and a simpler curve that met the line at either a single point or multiple points. They would break the more complicated curve into two, and repeat the process again, and again, and again, until they reduced everything to simpler curves, which is the sort of thing that you can work out with your bare hands. To prove what they needed to prove, they had to keep track of the normal bundles of the pieces and the modifications to them.

These methods weren't enough to break the curves up. They didn't work for all of the curves.

The method for breaking up their curves had to be a different one. It was a challenge to figure that out, not only because it might not do what they wanted it to at a given step in their argument, but also because they had to watch out for the exceptions where the statement didn't hold true. You can't ever have an exception as your base case, so your argument has to be complex. It would be terrible.

They were able to find a way to do this. It is very hard. It is a very demanding construction argument. I believe it requires someone with exceptional skills to carry it out.

The methods for dealing with the modifications to the normal bundle were developed at the same time. Gavril Farkas, a mathematician at the University of Berlin, said it was an amazing feat to keep track of all the data.

Eric is good at this. Izzet Coskun is a mathematician at the University of Illinois who frequently works with the two men. He said that Eric is a bit frightening. We give up and our eyes glaze over but he doesn't give up. There isn't anything complicated for him.

They proved that curves will always interpolate through the expected number of points. Four types of curves interpolate through an unexpected number of points. They had finished the problem once and for all.

They make the arguments easy to understand. Dave Jensen is a mathematician at the University ofKentucky. This is a result that other people couldn't prove.

It's sheer persistence. More than that is what it is. Farkas said it was brilliant to be able to complete it. It is something to behold.

A Family Legacy

The story is far from over, even if this proof marks the end of a narrative thread.

You can ask a lot of questions about curves. There is a recipe of sorts for getting a hold of these mathematical objects. A lot of the classical problems are easier to understand. You can ask about things that we would have thought was impossible.

The younger sister of Larson is a mathematician, and she is currently a Clay fellow. Her adviser said that she was a machine. She can do whatever she wants.

She came up with a new proof of the Brill-Noether Theorem. She and her brother have been working together to prove an analogue of the Brill-Noether theorem for certain curves. Jensen said that the family was an impressive one.

Hannah said it was fun to work with her brother and sister-in-law.

Hannah was inspired to study the material after taking a class. She attributes some of her interest in the subject to Eric and Isabel. She said she wanted to try it because she saw how much fun someone was having doing math.

They get along very well. People should not get along with the other two people.

They are continuing to illuminate what different kinds of curves look like, how they behave, and what that means for other mathematical problems. Hannah said that the story was not finished in any way.