### “There’s this misconception that some people are good at math, and some people aren’t. But the truth is, it’s a continuum. There are many, many skill sets required to be a really good mathematician; I just happen to have some of them.”

Dan McQuillan, PhD patiently walks a visitor through his area of expertise— topological graph theory—with a collegial tone signaling his confidence that even an abecedarian can grasp its complexities.

McQuillan’s work utilizes sophisticated mathematics to calculate the crossing numbers of complete graphs embedded on a surface—but McQuillan doesn’t explain it this way. Instead, he invokes common geometrical shapes to illustrate the precise area of his inquiry.

“Imagine the vertices of a square as people, and the lines between those vertices as relationships,” he begins, conjuring a simplified version of a graph Facebook might use to represent its subscriber base. “Now, imagine drawing the two diagonal lines between opposite corners; this symbolizes the full complement of possible relationships among the people.”

He continues, “It’s my job to figure out, in any given graph, how many edge crossings there are—the point where those diagonal lines meet in this example— and whether there’s a way to reduce them. Why? Because there are concrete instances where such crossings are undesirable: when you’re designing an electrical circuit, for example.”

Aha: his visitor understood. This pleases McQuillan.

“If you can view math as a language, and you know what the words and sentences mean, then it becomes much more fun and meaningful!” he says.

Indeed. In conversing with McQuillan in English, his knack for teaching math as a second language becomes immediately clear, and for good reason: he has been speaking mathematics fluently since 1989. That year, as an undergraduate student at Carleton University in Ottawa, Canada, he co-authored his first paper with mathematics professor R. Bruce Richter titled On the Crossing Numbers of Certain Generalized Petersen Graphs. This seminal event launched a career of inquiry into a topic McQuillan remains immersed in today.

**A pure Mathematician**

McQuillan is a pure mathematician: a scholar involved in basic research for which there may be no immediate, or even future, application. As such, he feels fortunate to be at Norwich.

“A lot of the best research has no grander intent than to ask questions simply for the sake of asking them,” he explains. “Yet many decision-makers don’t understand this. At Norwich, I don’t have to justify everything that I do. The leadership here supports the value of investigation, regardless of outcome.”

One of McQuillan’s current queries is, “How many edge crossings are possible for a complete graph with 15 points?”

“We literally do not know the answer,” McQuillan says. “It becomes mathematically impossible to avoid edge crossings in complete graphs with as few as five points; with 15, no one can conceptualize what’s going on. Even a computer can’t. That’s what I’m working on right now.”

McQuillan can’t say whether the results of this work will be useful in his lifetime. But the fact that it might—even a century from now—is precisely the reason he perseveres.

“The way most difficult problems are solved, in any field, are by using things that people didn’t originally think would be part of the answer,” he says. “Yet the ‘publish or perish’ culture of academia often pressures researchers to seek straightforward projects. I am extremely grateful to Norwich for encouraging me to take significant risks. That’s when truly special work emerges.” A genuine collaborator

“Since Dan first arrived at Norwich 15 years ago, he has raised the collective level of scholarship within the mathematics department,” says Dave Westerman, Charles A. Dana Professor of Geology and Vice President for Research at Norwich.

McQuillan, as humble as he is brilliant, demurs.

“There’s this misconception that some people are good at math, and some aren’t. But it’s a continuum. There are many, many skill sets required to be a really good mathematician; I just happen to have some of them.”

If McQuillan is missing a particular skill set, it’s hard to identify. A prolific author, he has published several impactful papers in topological graph theory, as well as in the diverse areas of discrete mathematics, linear algebra, and calculus. His article On the Crossing Number of K13, also co-authored with B. Richter and S. Pan, has ranked number one on Science Direct’s list of “hottest papers.”

Adept at identifying his colleagues’ complementary competencies—and regularly seeking their counsel on particularly difficult questions—McQuillan has co-authored several articles with Norwich faculty as well, including department chair Rob Poodiak, Darlene Olsen, and Jeremy Hansen. He also has a paper in progress with Joe Latulippe.

“When you see mathematics everywhere, then it’s easy to engage with your colleagues in creative ways. Sometimes, those conversations lead to a publication. But it’s important to note that this is not particular to me. Our entire department is very receptive to starting interesting discussions to see where they lead.”

**An inspiration in the classroom**

A passionate teacher and mentor, McQuillan regularly brings his research into the classroom—using it to catalyze students’ understanding of complex concepts, and to spark investigations of their own.

Recalling his own experience as an undergraduate, when he often didn’t fully grasp the material until years later, he strives “to be aware of what sorts of things take time to sink in, and what will be helpful to students a decade from now.”

By equipping students to solve problems through real-world examples, as opposed to memorizing equations just to get through the course, McQuillan hopes his approach to teaching will have a lasting impact.

This is another area in which he believes Norwich shines.

“Small class sizes offer the opportunity to experiment,” he says. “When I use my work to illustrate the application of a particular topic, I know right away whether or not it ‘clicks’ with the students,” he says. “That level of interaction doesn’t exist in 500-person lecture halls.”

Nor, perhaps, would the opportunity to publish as an undergraduate—something McQuillan actively encourages. To date, he has co-authored four peer-reviewed publications with Norwich students.

“I challenge students to reach beyond their comfort zones,” he says. “Because truthfully, there are unsolved questions in mathematics where undergraduates know enough to contribute to their solutions,” he says.

McQuillan also believes that Norwich undergraduates are fully capable of considering problems at the highest level, and signals this through his oversight of the university’s participation in the annual Putnam Competition—considered one of the most prestigious mathematics tests in the world.

**Looking ahead**

Raised in Ottawa, McQuillan earned his master’s and PhD degrees in mathematics from the University of Western Ontario. He joined the Norwich faculty in 2002 after serving as a visiting assistant professor at Lakehead University in Ontario, and a lecturer at Southern Illinois University at Carbondale.

Since his arrival, McQuillan has taught 17 different mathematics courses, often carrying a full load. He has mentored 20 students in their summer research and independent study projects.

Noting the importance of long-term research goals, he is excited to be developing the concept for a textbook on the process of mathematical problem-solving with Norwich colleague (and former student) Addie Armstrong. In the more immediate future, he anticipates (within two years) the publication of “by far the best” paper he’s ever done with international collaborators B. Richter, Alan Arroyo and G. Salazar.

In recognition for his outstanding research, scholarship and teaching, President Richard M. Schneider named McQuillan as a Charles A. Dana Professor—the first in Mathematics since the award’s inception in 1974—during Commencement on May 14, 2016.

“My best work has occurred since joining the Norwich University community,” McQuillan says. “And I believe that—thanks to the support I receive here— even better work lies ahead.”