John Doyle
Department of Mathematics
Dr. John Doyle joined the Oklahoma State University faculty in fall 2020 as an assistant professor of mathematics.
"I received my Ph.D. in mathematics from the University of Georgia in 2014, then held
a postdoctoral position at the University of Rochester and a tenure-track position
at Louisiana Tech University before arriving at OSU. My research is generally in the
field of number theory, and my research in this area has led to roughly 20 publications
and funding from the National Science Foundation.
Number theory is the study of properties of number systems like the whole numbers,
integers, or rational numbers. Things like prime numbers and perfect squares fall
into the category of number theory. I was drawn to study number theory for a couple
of reasons. Number theory is one of the oldest disciplines in all of mathematics.
People were studying it more than 2,000 years ago; for example, the ancient Greek
mathematician Euclid was the first person to prove that there are infinitely many
prime numbers. Many problems in number theory, though challenging to solve, are relatively
simple to state. Are there infinitely many pairs of "twin primes" -- prime numbers
that differ by two, like 3 and 5, 5 and 7, 11 and 13, 17 and 19, etc.? It's a simple
enough question to ask, but we still don’t know the answer.
My specific research area is arithmetic dynamics, which is a mix of number theory
and dynamical systems. "Dynamical systems" can mean a few different things, but in
this context it essentially refers to "iterative processes". As an example, take a
standard five-function calculator, enter any positive number, then press the square
root button. Press it again. And again. The repeated application of the square root
function is an example of a dynamical system. (If you do this long enough, the calculator
will eventually show 1; in dynamical terminology, 1 is an attracting fixed point for the square root function.)
My research in arithmetic dynamics involves understanding the sorts of patterns that
can arise from iterating polynomial functions. For example, what if instead of iterating
the square root function, you iterate the function x2 – 1? If you start with 0, your output is –1; if you then input –1, you end up back
at 0. This gives us a simple pattern: The values cycle between the two integers 0
and –1. It turns out there are also quadratic polynomials that cycle threeintegers, but so far no one has found a quadratic polynomial that cycles more than
three. We think three is probably the limit, but this is yet to be proven; it’s part
of one of the major open problems in arithmetic dynamics—the Dynamical Uniform Boundedness
Conjecture, which is the problem I tend to spend most of my time thinking about. Over
the last several years, we've been able to chip away at this conjecture, but there's
still plenty of work to do.
What initially drew me to arithmetic dynamics was the simplicity of some of the important
problems in the area and the fact that it was a relatively young field to work in.
Now I'm interested for other reasons, one of which is that I get to work with some
of the most beautiful pictures in mathematics. If you do a Google search for "Mandelbrot
set" or "Julia sets", you'll see some fascinating fractal images, and I've had the
good fortune of being able to contribute some small amount to our collective understanding
of these objects."