**the ‘fun’ in commutative algebra**

**The word “fun” is a subjective one, but that is how Anne Fayolle describes mathematics as a discipline.**

A graduate student in mathematics at the University of Utah, Fayolle is a recent recipient of a multi-year scholarship from The Natural Sciences and Engineering Research Council (NSERC) which is Canada’s equivalent of the National Science Foundation’s Graduate Student Fellowship in the U.S.

##### Math credentials

As a Ph.D. student working with Professor Karl Schwede, Fayolle, who was born in France and grew up in Canada, clearly has her mathematics credentials. Before coming to the U, she studied first at McGill University, one of Canada's best-known institutions of higher learning and one of the leading universities in the world. There, she quickly developed an affinity for the independent learning model in which one is paired with a professor in a one-on-one setting and in which both determine together what textbooks and papers they will study together. This conversational model of learning proved to be better than the lecture-and-classroom-style model for Fayolle and helped solidify her desire to go to graduate school.

Following her bachelor’s degree, Fayolle returned to Europe at École Polytechnique Fédérale de Lausanne (EPFL), one of the most celebrated public universities in Europe. As a master’s student, Fayolle studied with Dr. Carvajal-Rojas in commutative algebra. This involves working in modular arithmetic, a type of arithmetic in which addition works similar to time on a clock: 5 + 10 = 3. “The numbers working differently,” she says, “means one cannot use the traditional tools of calculus to study polynomial equations and the shapes they define. "For instance, since the numbers work differently, we can't draw graphs exactly like we would over the real numbers. So our intuition derived from those graphs doesn't always work here. One has to rely on the underlying algebraic structures."

This algebraic abstraction and the understanding it brings is one of Fayolle’s favorite parts of doing math, and it may speak to what Fayolle identifies as the “weird” aspect in commutative algebra, followed closely by the feeling of it being “fun.” The appeal is also philosophical “You can get to the heart of why something works the way it does. I really enjoy the abstraction that comes with [commutative algebra]––trying to find the structure in abstract things.”

Part of her enjoyment in doing math is that singular moment when someone (or “some ones”) in the math sector solves a persistent problem. “It’s [only] ‘hard’ until someone comes along and finds the right object or point of view of how things are working,” she says of breakthrough findings. “It’s suddenly less ‘weird’ because it makes more sense.” She explains that she’s been working in this positive characteristic realm for the past few years and is now used to it. Fayolle is especially interested in studying singularity theory, she says, in positive and mixed characteristics and, fortunately, in Schwede has found a principal investigator/mentor at the U who “does cool math.”

**Everyone can do math**

The multi-year NSERC fellowship will free up more of Fayolle’s time for research. Her ambition is to continue in academics as a post-doctoral researcher and then as a faculty, if possible. “I like having stuff that has more world impact independent from [just the study of] math. I think that pure math is intrinsically valuable, hard to justify by linking it to real world applications, but still necessary. ” This includes teaching.

“I think math is very scary to a lot of people. I personally think that everyone can do math. Everyone struggles, and I think that’s very important to emphasize when you’re teaching. I struggle in math. I don’t think struggling in math should be a barrier to doing math.”

In the meantime, Anne Fayolle continues in graduate school, sharing mathematics by organizing BIKES, the student commutative algebra seminar here at the U as well as co-organizing an Association for Women in Mathematics conference later this year. She also skis on the weekends. “I was skeptical,” the Montreal native says, “when I first saw the [Utah “Greatest Snow on Earth”] license plates. But after I went skiing, I agreed.” It helps, she says, that it doesn’t get too cold and is not too icy.

“I think the license plates might be right.”

by David Pace