Math professor Mark Hughes specializes in topology.
But before you get out your elevation map and GPS, make sure you know what Hughes studies.
“Whenever you say topology, people always hear topography, which involves maps and land forms and is not the same thing,” said Hughes.
Derived from the Greek word topos, meaning place, topography and topology both study the spatial relationships and characteristics of surfaces. But that’s where the similarities end; while practical topography belongs in the geography department, abstract topology stands firmly in mathematics.
“A topologist views everything as being made of a [hypothetical] flexible rubber that can be deformed,” said Hughes. “If you are deforming a surface, then you’re not able to preserve properties like curvature or length because everything is stretchy, but you are able to study other properties like connectedness or whether there are holes in the surface.”
Imagine a clown’s balloon. If you draw a line between any two points on the balloon, it doesn’t matter what shape you contort the balloon into—those two points will always be connected by that line. Stretch it, twist it, bend it, inflate it, deflate it—it doesn’t matter. Certain properties (like the connectedness between those two points) will never change.
“One of the areas I study is called knot theory,” said Hughes “Think of a Boy Scout tying a knot: you take two strings and you tie them together so that they are tight. Now instead of tying two separate pieces of string together, a knot in mathematics is a loop—a closed loop. Imagine that you take a piece of rubber or a string and instead of leaving the two ends open you attached the two ends together. You get a closed loop, but you potentially could have something knotted along the loop.”
Hughes demonstrates the property of knottedness to his students by tugging on a long rubber tube that has been looped over itself, knotted, and connected at the ends to form a closed circle.
“It doesn’t matter if I twist it around or something, there is still this knottedness along here that is inherent in it,” said Hughes. “I can never take this and end up with just a perfect circle [like] a hula hoop without cutting it in some way.”
But what good is a theoretical knot that can’t even keep a Boy Scout’s food out of the reach of bears?
“Physicists use knots in string theory,” said Hughes. “The reason I am interested in knots primarily is because of their relation to higher dimensional shapes. If you are interested in studying shapes that are 3- or 4-dimensional, it turns out that a lot of their structure can be encoded in knottedness. If you can understand knots, then it allows you to describe and understand 3- and 4-dimensional shapes in a way that is difficult otherwise.”
Without the benefit of hyperdimensional rope (which, unfortunately, doesn’t exist), studying these topological knots can be tricky. Because of this, Hughes is developing new computational processes to pick apart the knot complexity.
“One of the things that I am trying to do is to see if I can use methods from machine learning and artificial intelligence to teach a computer to recognize certain properties and to solve certain problems,” Hughes said.
Hughes hopes to teach knot theory and develop new research techniques while at BYU. Hughes recently took up a permanent position with the BYU Mathematics Department after working for three years as a visiting assistant professor. For Hughes, one of the greatest benefits of teaching is to show students how rewarding the field can be.
“Sometimes it’s engineering students or other students who aren’t math majors, but when you help them and see something click with them, that’s exciting,” said Hughes. “Or sometimes . . . you have a math major working on a research problem and you are able to discover something cool together.”
Hughes’s students will realize—with the help of rubber tubes, artificial intelligence, and a sturdy clove hitch—that math is much more than tedious checklists of predetermined functions.
“For me, part of the reason I didn’t think about math [in high school] was because in high school, even into my first year of college, I didn’t know what a math person did,” said Hughes. “I didn’t realize that mathematics is such a powerful area. In some senses, it’s just learning a language. It’s a language by which much of science and technology is really described.”