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John Doyle

Department of Mathematics

John Doyle


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."

 

 

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