D
Deleted member 4993
Guest
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
Joseph Silverman remembers when he began connecting the dots that would ultimately lead to a new branch of mathematics: April 25, 1992, at a conference at Union College in Schenectady, New York.
It happened by accident while he was at a talk by the decorated mathematician John Milnor. Milnor’s subject was a field called complex dynamics, which Silverman knew little about. But as Milnor introduced some basic ideas, Silverman started to see a striking resemblance to the field of number theory where he was an expert.
“If you just change a couple of the words, there’s an analogous sort of problem,” he remembers thinking to himself.
Silverman, a mathematician at Brown University, left the room inspired. He asked Milnor some follow-up questions over breakfast the next day and then set to work pursuing the analogy. His goal was to create a dictionary that would translate between dynamical systems and number theory.
At first glance, the two look like unrelated branches of mathematics. But Silverman recognized that they complement each other in a particular way. While number theory looks for patterns in sequences of numbers, dynamical systems actually produce sequences of numbers — like the sequence that defines a planet’s position in space at regular intervals of time. The two merge when mathematicians look for number-theoretic patterns hidden in those sequences.
In the decades since Silverman attended Milnor’s talk, mathematicians have dramatically expanded the connections between the two branches of math and built the foundations of an entirely new field: arithmetic dynamics.
The field’s reach continues to grow. In a paper published in Annals of Mathematics last year, a trio of mathematicians extended the analogy to one of the most ambitious and unexpected places yet. In doing so, they resolved part of a decades-old problem in number theory that didn’t previously seem to have any clear connection to dynamical systems at all.
The new proof quantifies the number of times that a type of curve can intersect special points in a surrounding space. Number theorists previously wondered if there is a cap on just how many intersections there can be. The authors of the proof used arithmetic dynamics to prove there is an upper limit for a particular collection of curves.
“We wanted to understand the number theory. We didn’t care if there was a dynamical system, but since there was one, we were able to use it as a tool,” said Laura DeMarco, a mathematician at Harvard University and co-author of the paper along with Holly Krieger of the University of Cambridge and Hexi Ye of Zhejiang University.
Moving on a Curve
In May 2010, a group of mathematicians gathered at a small research institute in Barbados where they spent sunny days discussing math just a few dozen feet from the beach. Even the lecture facilities — with no walls and simple wooden benches — left them as close to nature as possible.
“One evening when it was raining you couldn’t even hear people, because of the rain on the metal roof,” said Silverman.
The conference was a pivotal moment in the development of arithmetic dynamics. It brought together experts from number theory, like Silverman, and dynamical systems, like DeMarco and Krieger. Their goal was to expand the types of problems that could be addressed by combining the two perspectives.
Their starting point was one of the central objects in number theory: elliptic curves. Just like circles and lines, elliptic curves are both numbers and shapes. They are pairs of numbers, x and y, that serve as solutions to an algebraic equation like y2 = x3 − 2x. The graph of those solutions creates a geometric shape that looks vaguely like a vertical line extruding a bubble.
Mathematicians have long been interested in quantifying and classifying various properties of these curves. The most prominent result to date is Andrew Wiles’ famed 1994 proof of Fermat’s Last Theorem, a question about which equations have solutions that are whole numbers. The proof relied heavily on the study of elliptic curves. In general, mathematicians focus on elliptic curves because they occupy the sweet spot of inquiry: They’re not easy enough to be trivial and not so hard that they’re impossible to study.
“Elliptic curves are still mysterious enough that they’re generating new math all the time,” said Matt Baker, a mathematician at the Georgia Institute of Technology.
Mathematicians are particularly interested in points on elliptic curves that act like a home base for a special way of moving around on the curves. On an elliptic curve, you can add points to each other using standard addition, but this approach is not very useful: the sum is unlikely to be another point on the curve.