Coffee cups, doughnuts and geometry: UTM mathematician wins prestigious Sloan Fellowship

A topologist is someone who can’t tell the difference between a doughnut and a mug of coffee.

That’s a math joke, but it begs for an explanation. Topology is a sub-field of geometry that studies how shapes can be deformed, stretched and twisted. The shape of a coffee mug and a doughnut share the same basic form; they are both centered around a hole, and were they not constrained by the properties of their materials, one could be reshaped into the other – a little like play-dough, but for the mathematical mind.

Get it? Michael Groechenig does. The assistant professor of mathematics at UTM has been awarded a 2022 Sloan Research Fellowship, a two-year $75,000 award from the Alfred P. Sloan Foundation that recognizes the distinguished performance of early career researchers with a unique potential to make substantial contributions to their field.

To be a pure mathematician can be a little bit like being a philosopher. Mathematicians use logic, analysis and contemplation to identify patterns and discover relationships between numbers and space. In the course of their work, they conduct thought experiments that uncover anomalies, unexpected symmetries – and just about anything else that captures the mathematical imagination.  From this, they create conjectures, which are plausible hypotheses that they then try to prove. This is how our knowledge of mathematics has continued to grow.

Topology allows Groechenig to use number theory and geometric techniques to solve problems, and it is the combination of different branches of mathematics that drive his work.

“One of my specialties is to approach geometric and topological questions from a number theoretic point of view. My research combines methods from many different areas of mathematics, and physics provide some of the questions,” says Groechenig.

“Sometimes seemingly unrelated problems can have the same solution, which reflects an underlying symmetry that we cannot yet explain. Such as describing the basic shape of an object that doesn't change when it is deformed – like the coffee cup and the doughnut.”

By uncovering new mathematical relationships, topology could have applications in fields that include quantum computing and materials research, but it will likely be many years before the exact applications of current research are known.

“Math is very slow in comparison to many other fields. It can take years to actually see the bigger picture. The research I will do over the next two years will be one piece of a larger piece of the puzzle I am trying to solve. When you are a mathematician, you often don’t know the applications of your work until long after you are dead.”

For pure mathematicians like Groechenig, the discipline harkens back to something more fundamental in the human experience. Just as Plato once posited that there exists a timeless and eternal mathematical world that awaits discovery through human reasoning, today’s mathematicians keep this tradition alive through their work.

“One aspect of being a mathematician is trying to preserve a part of human culture. In order to preserve math, you have to do it. In a way, it is a kind of living thing. Mathematics is an essential part of human knowledge that is about problem solving,” says Groechenig.  

“It is really exhilarating to solve problems. When you can answer a question, it does not matter if it is a difficult question or an easy one. It's all the same, it's always exciting. This is really something that I would like to share with everyone, the ability to approach questions by just thinking independently and analytically.”