Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics.

Ye and Xi, a researcher at Zhejiang University and alumni of University of Science and Technology of China Mathematics Department in 2003, together with two scholars from Cambridge University and Harvard University, applied dynamical systems to number theory, solving a problem that has perplexed mathematicians for decades.

The research results were published in Annals of Mathematics, a bimonthly academic journal, which has published only more than 30 academic papers in the past two years.

This is also the first time for Zhejiang University to publish results in this journal in more than 40 years.

Ye and Xi, combined with the dynamical system method, prove a very important problem in number theory.

Dynamical systems, which study how all points in space change with time. The subject is best known as the Lorenz attractor in the "butterfly effect".

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals
△ Lorenz attractor

Number theory, on the other hand, deals with the properties of integers.

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals

These two seemingly disparate fields have been cleverly combined by mathematicians.

So how do they relate to each other, we have to start with two equations.

Two equations

1, y? =x? +ax+b

2, f (z) = z? +c

The first equation represents an elliptic curve, which is shaped differently as A and B keep changing, like squeezing a "bubble" out of the curve.

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals

Elliptic curves are an important tool in number theory and are used by mathematicians to prove Fermat's Great Theorem.

On an elliptic curve, you can even add two points.

If we have two points, P and Q, then the mirror image of the third point where PQ intersects the curve, R, on the X-axis, is P plus Q. The mirror image of R is minus R, so minus R is equal to P plus Q.

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals

Because an elliptic curve is symmetric up and down, so P plus Q must also be on the elliptic curve.

So what happens when P is added to itself (P+P)?

So let's imagine that Q gets closer and closer to P, and then PQ becomes tangent to P, so P plus P is the mirror image of where this tangent intersects the elliptic curve.

If P is repeatedly added to itself, after a finite number of additions (P+P+... +P) goes back to P, and P is called the Torsion Point.

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals

Let's look at the second equation, the mathematical formula: f(z)=z^2+c, which is not a quadratic, but has to do with another branch of mathematics, dynamical systems.

Z is not a real number here, it's a real plus an imaginary number. If we draw a plane coordinate, where the x-coordinate is the real part of it, and the y-coordinate is the imaginary part of it, z is a point.

Let's draw two points z and f(z) on this plane, and then put f(z) into the equation to get f(f(z)), and then f(f(f(z)))...

This "infinite doll" operation, draw all the points, you can get the following figure.

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals

Some of you might have noticed, well, isn't that what fractals are, how do they relate to elliptic curves?

The graph above has a limited range, indicating that some Z values are still finite after the infinite dolls.

Assuming c=-1 and z starts at 2, the resulting combination of numbers is 2, 3, 8, 63... , this group will keep growing; If the initial value of z is 0, then the following numbers are -1, 0, -1, 0... , will keep going.

In the second case, each point after an infinite number of iterations is within a finite range, and the set of points within a finite range is called a "Julia set".

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals

In the dynamical system, things like -1, 0, -1, 0... In this way, not only is the range limited, but a group of points that can return to the starting point is called the Finite Orbit Point.

In this way, elliptic curves are associated with dynamical systems, and finite orbital points are the simulation of torsion points on elliptic curves.

"A torsional point on an elliptic curve is the same as a finite orbital point on a dynamical system, and that's what we used over and over again in our paper," said DeMarco, Ye's supervisor.

Prove a mathematical guess

But the problem the three mathematicians were working on -- the Manin-Mumford conjecture -- was much more complicated than that.

The Manin-Mumford conjecture is something more complicated than an elliptic curve, such as y^2 = x^6 + x^4 + x^2, right? 1. Each curve with a different parameter is associated with a geometry.

The Manin-Mumford conjecture was proved by Raynaud in 1983, that is, any smooth algebraic curve with genus greater than 1 has at most a finite number of torsion points.

For a closed directed surface, the genus is the number of holes in the surface.

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals

The corresponding geometry of an elliptic curve is a "donut" with genus 1.

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals

Ye and Xi took the Manin-Mumford conjecture a step further. He, together with Holly Krieger and Laura DeMarco, combined with the dynamical system, proved that the number of tortuous points of smooth algebraic curves is not only limited, but also has a uniform upper bound under the condition that the credit is 2.

Unlike elliptic curves, the complex curves in the Manin-Mumford conjecture do not have structures that allow addition.

But their corresponding geometries can be added, and they have torsion points like elliptic curves.

They give the shape of the solution to a particular cluster of curves to be solved: something like the surface of two doughnuts (genus 2).

Each doughnut represents an elliptic curve.

To prove the upper limit of the number of torches, we need to calculate the number of intersecting points between torches on an elliptic curve.

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals

However, it is not possible to directly compare the torsion points on the two elliptic curves because they do not necessarily overlap.

Several scholars have come up with a way to compare whether they are in the same relative position on the doughnut.

They compared the positions of torsion points by plotting the solutions of the two elliptic curves on a plane.

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals

Then, just count the number of times the points overlap to give the Manin-Mumford conjecture a definite upper bound.

This is where the dynamical system comes into play.

Using dynamical systems, they showed that these points can overlap only a certain number of times, and that this number does exist -- that the upper bound on the Manin-Mumford conjecture does exist.

Regarding their proof, Patrick Ingram, an assistant professor at York University in Canada, said:

They succeeded in proving a particular problem. Until now, the problem had been relegated to number theory, and no one thought it had anything to do with dynamical systems. It really caused quite a stir.

Solve important conjectures with former mentors

In fact, the three scholars behind the conjecture's proof are mentors and old friends.

Ye and Xi, along with the paper's author, Holly Krieger, were former students of Laura Demarco.

In 2013, they received their doctorates from the University of Illinois at Chicago under the latter's supervision.

Since then, Laura DeMarco is now a professor at Harvard University, and Holly Krieger is a mathematics lecturer at Cambridge University.

Ye and Xi chose to return to China and become a mathematics researcher at Zhejiang University.

But in the meantime, they did not stop working together.

In 2017, Ye and Xi, together with Laura DeMarco and Holly Krieger, studied the bounded height problem in dynamical systems. The results were published in 2019.

In 2019, after proving the Manin-Mumford conjecture, they also explored another problem in dynamical systems in depth and adopted a similar approach.

Harvard and Cambridge scholars from Zhejiang University have joined hands to solve a decades-old puzzle in mathematics. The results were published in top math journals

This article is currently available as a preprint.

Their proof of the Manin-Mumford Consensus was published in the Annals of Mathematics on April 15, 2020.