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Part – 2 Continued (from https://www.freemathhelp.com/forum/...ion-to-unlock-their-secrets-i-quantas.128429/)
www.quantamagazine.org
But elliptic curves come packaged with a special internal structure that creates a different type of arithmetic. This structure is called a group, and the result of adding points together using its self-contained arithmetic rules is quite different.
If you add two points on an elliptic curve according to the group structure, the sum is always a third point on the curve. And if you continue this process by, for example, adding a point to itself over and over, the result is an infinite sequence of points that all lie along the elliptic curve.
Different starting points will result in different sequences. The “home base” points are starting points with a very unique property. If you repeatedly add one of these points to itself, it does not generate an infinite sequence of new points. Instead, it creates a loop that comes back to the point you started with.
These special starting values that create loops are called torsion points. They are of immediate interest to number theorists. They also have a striking correspondence to a specific type of point on dynamical systems — and it was this correspondence that really set arithmetic dynamics in motion.
“That’s truly the basis of why this field has become a field,” said Krieger.
Repeating Patterns
Dynamical systems are often used to describe real-world phenomena that move forward in time according to a repeated rule, like the ricocheting of a billiard ball in accordance with Newton’s laws. You begin with a value, plug it into a function, and get an output that becomes your new input.
Some of the most interesting dynamical systems are driven by functions like f(x) = x2 − 1, which are associated with intricate fractal pictures known as Julia sets. If you use complex numbers (numbers with a real part and an imaginary part) and apply the function over and over — feeding each output back into the function as the next input — you generate a sequence of points in the complex plane.
This is just one example of what’s called a quadratic polynomial, in which the variable is raised to the second power. Quadratic polynomials are the foundation of research in dynamical systems, just as elliptic curves are the focus of a lot of basic inquiry in number theory.
“Quadratic polynomials [in dynamical systems] play a similar role as elliptic curves in number theory,” said Baker. “They’re the ground that we always seem to return to to try to actually prove something.”
Dynamical systems generate sequences of numbers as they evolve. Take for example that quadratic function f(x) = x2 − 1. If you start with the value x = 2, you generate the infinite sequence 2, 3, 8, 63, and so on.
But not all starting values trigger a series that grows larger forever. If you begin with x = 0, that same function generates a very different type of sequence: 0, −1, 0, −1, 0, and so on. Instead of an infinite string of distinct numbers, you end up in a small, closed loop.
In the world of dynamical systems, starting points whose sequences eventually repeat are called finite orbit points. They are a direct analog of torsion points on elliptic curves. In both cases, you start with a value, apply the rules of the system or curve, and end up in a cycle. This is the analogy that the three mathematicians exploit in their new proof.
“This simple observation — that torsion points on the elliptic curve are the same as finite orbit points for a certain dynamical system — is what we use in our paper over and over and over again,” said DeMarco.
Setting a Ceiling
Both Krieger and Ye received their doctorates from the University of Illinois, Chicago in 2013 under DeMarco’s supervision. The trio reconvened in August 2017 at the American Institute of Mathematics in San Jose, California, which hosts intensive, short-term research programs.
“We stayed in a room for five days. We needed to work through some questions,” said Ye.
During this period, they began to envision a way to extend the crucial analogy between torsion points of elliptic curves and finite orbit points of dynamical systems. They knew that they could transform a seemingly unrelated problem into one where the analogy was directly applicable. That problem arises out of something called the Manin-Mumford conjecture.
The Manin-Mumford conjecture is about curves that are more complicated than elliptic curves, such as y2 = x6 + x4 + x2 − 1. Each of these curves comes with an associated larger geometric object called a Jacobian, which mimics certain properties of the curve and is often easier for mathematicians to study than the curve itself. A curve sits inside its Jacobian the way a piece sits inside a jigsaw puzzle.
Unlike elliptic curves, these more complicated curves don’t have a group structure that enables adding points on a curve to get other points on the curve. But the associated Jacobians do. The Jacobians also have torsion points, just like elliptic curves, which circle back on themselves under repeated internal addition.
Mathematicians Set Numbers in Motion to Unlock Their Secrets
A new proof demonstrates the power of arithmetic dynamics, an emerging discipline that combines insights from number theory and dynamical systems.
www.quantamagazine.org
But elliptic curves come packaged with a special internal structure that creates a different type of arithmetic. This structure is called a group, and the result of adding points together using its self-contained arithmetic rules is quite different.
If you add two points on an elliptic curve according to the group structure, the sum is always a third point on the curve. And if you continue this process by, for example, adding a point to itself over and over, the result is an infinite sequence of points that all lie along the elliptic curve.
Different starting points will result in different sequences. The “home base” points are starting points with a very unique property. If you repeatedly add one of these points to itself, it does not generate an infinite sequence of new points. Instead, it creates a loop that comes back to the point you started with.
These special starting values that create loops are called torsion points. They are of immediate interest to number theorists. They also have a striking correspondence to a specific type of point on dynamical systems — and it was this correspondence that really set arithmetic dynamics in motion.
“That’s truly the basis of why this field has become a field,” said Krieger.
Repeating Patterns
Dynamical systems are often used to describe real-world phenomena that move forward in time according to a repeated rule, like the ricocheting of a billiard ball in accordance with Newton’s laws. You begin with a value, plug it into a function, and get an output that becomes your new input.
Some of the most interesting dynamical systems are driven by functions like f(x) = x2 − 1, which are associated with intricate fractal pictures known as Julia sets. If you use complex numbers (numbers with a real part and an imaginary part) and apply the function over and over — feeding each output back into the function as the next input — you generate a sequence of points in the complex plane.
This is just one example of what’s called a quadratic polynomial, in which the variable is raised to the second power. Quadratic polynomials are the foundation of research in dynamical systems, just as elliptic curves are the focus of a lot of basic inquiry in number theory.
“Quadratic polynomials [in dynamical systems] play a similar role as elliptic curves in number theory,” said Baker. “They’re the ground that we always seem to return to to try to actually prove something.”
Dynamical systems generate sequences of numbers as they evolve. Take for example that quadratic function f(x) = x2 − 1. If you start with the value x = 2, you generate the infinite sequence 2, 3, 8, 63, and so on.
But not all starting values trigger a series that grows larger forever. If you begin with x = 0, that same function generates a very different type of sequence: 0, −1, 0, −1, 0, and so on. Instead of an infinite string of distinct numbers, you end up in a small, closed loop.
In the world of dynamical systems, starting points whose sequences eventually repeat are called finite orbit points. They are a direct analog of torsion points on elliptic curves. In both cases, you start with a value, apply the rules of the system or curve, and end up in a cycle. This is the analogy that the three mathematicians exploit in their new proof.
“This simple observation — that torsion points on the elliptic curve are the same as finite orbit points for a certain dynamical system — is what we use in our paper over and over and over again,” said DeMarco.
Setting a Ceiling
Both Krieger and Ye received their doctorates from the University of Illinois, Chicago in 2013 under DeMarco’s supervision. The trio reconvened in August 2017 at the American Institute of Mathematics in San Jose, California, which hosts intensive, short-term research programs.
“We stayed in a room for five days. We needed to work through some questions,” said Ye.
During this period, they began to envision a way to extend the crucial analogy between torsion points of elliptic curves and finite orbit points of dynamical systems. They knew that they could transform a seemingly unrelated problem into one where the analogy was directly applicable. That problem arises out of something called the Manin-Mumford conjecture.
The Manin-Mumford conjecture is about curves that are more complicated than elliptic curves, such as y2 = x6 + x4 + x2 − 1. Each of these curves comes with an associated larger geometric object called a Jacobian, which mimics certain properties of the curve and is often easier for mathematicians to study than the curve itself. A curve sits inside its Jacobian the way a piece sits inside a jigsaw puzzle.
Unlike elliptic curves, these more complicated curves don’t have a group structure that enables adding points on a curve to get other points on the curve. But the associated Jacobians do. The Jacobians also have torsion points, just like elliptic curves, which circle back on themselves under repeated internal addition.
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