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Feb. 25, 2015

Matthew Hedden: Solving Knotty Problems

Feb. 25, 2015

Matthew Hedden is an assistant professor of mathematics in the College of Natural Science. His research area is low-dimensional topology, with a focus on knot theory. He received his bachelor’s degree in mathematics and physics, summa cum laude, from the University of Notre Dame in 2001, and his doctorate in mathematics from Columbia University in 2005. He joined the MSU faculty in 2009.

If you’d asked me what I wanted to be when I was a kid, I would have said I wanted to be an artist. If you’d asked me during my freshman or sophomore year in high school, I would have said I wanted to be a writer. Math was something fun for me, but it was not always my passion.

I can pinpoint the defining moment in my math career. I was in the third year of grad school at Columbia University. It was during a snowstorm that had canceled classes and shut down the city of New York.

My thesis problem involved the computation of some new knot invariants. I was doing some computations and I was realizing that something was about to happen. I recall my advisor saying, “Don’t sleep! Just do it!”

So I sat in the department lounge—there was no one else around—and computed and computed nonstop for a couple of days. I could see the patterns emerging. Around 3 a.m. on the third night, I realized I knew what the answer was. I knew exactly how to prove it. Within 45 minutes, I had written down the proof (which I still have in one of my notebooks in my office). This was my first theorem. My thesis was solved at that moment. The whole experience was just amazing. I was hooked.

illustration of a know

Today, a significant part of my research focuses on knots. Knot theory, which has gained a lot of notoriety during the past 35 years, sits within a branch of mathematics called topology—the study of shapes, from a very loose perspective. To a mathematician, a knot is a closed piece of string with no loose ends or, in mathematical parlance, the embedding of a circle in three-dimensional Euclidean space. It is quite unlike the definition of a knot in everyday life, such as in a shoelace or a rope, which has two loose ends.

One of the things I find fascinating about knot theory is that it connects with a lot of other areas of mathematics. Knots are the building blocks for all three- and four- dimensional shapes and are relevant to the problem of finding meaningful models of our universe. In fact, some of the most powerful tools for studying knots come out of theoretical physics. Physics aside, the classification of four-dimensional shapes is one of the most notoriously difficult and important problems within geometry and topology.

On a more concrete front, there are emerging applications for knot theory in biology. For example, the DNA of various bacteria can become “knotted.” In order to replicate, the DNA must “unknot” itself. Specific enzymes, known as topoisomerases, perform the act of “cutting” the string (or circular strand of DNA) to accomplish the unknotting. The mathematical complexity of the knots becomes highly relevant for understanding this process.

The emergence of “big data” also calls for topological insight. We want to understand—in a huge zoo of data—what the salient features are. What are the reasons that we’re seeing data points in this way? Maybe what looks like just a big cloud of data points has some shape, or some underlying structural features. We can attempt to measure those features using tools from topology. This quickly emerging field is called topological data analysis.

What are the immediate, practical applications that might result from my research? As a pure mathematician, I’m not necessarily motivated by practical applications. Pure math is conducted for the sake of pure math. It’s not always clear—as the math is being developed—what is going to be most useful for the “real world.” It is often the case, however, that years later, mathematical tools that were developed end up being quite relevant for people in science and industry.

Photo by Harley Seeley
Illustration by Andrew Ward