Math is creative. Math is fun. It’s shapes and patterns and play.
And it’s a team activity.
If that’s not the math you remember from school, that’s no surprise. But that’s the math they practice and preach through the Illinois Geometry Lab, a new math department initiative now in its fourth semester.
“We want to show students that mathematics is something that’s alive and constantly changing,” says math professor Jayadev Athreya, the director and co-founder of the lab, and in many ways its lead evangelist.
The lab provides a “gateway of visualization” for undergraduate students – not just math majors – who want to learn what math research is all about, and what mathematicians really do, Athreya said. One of the things mathematicians do is work together on research, he said.
“The purpose of the lab is to really get them into this idea that they’re going to be exploring and figuring out genuine new things, and that that’s just going to be absolutely an enormous amount of fun,” he said.
Visualization is important because it gives you another way to look at math problems, says graduate student Grace Work, the lab manager. Focus only on the numbers, and you might miss things, she said. “It’s better if you can say ‘Here’s a picture of what I’m trying to prove,’ or ‘Here’s a model of how this thing works.’ ”
“We help translate problems even in many areas of mathematics where you wouldn’t expect visuals, or geometry, to play a role,” Athreya said.
The lab space itself is a large conference-size room on the basement floor of Altgeld Hall. Two walls are lined with computers, and they have a couple nifty tools: a paper cutter that spits out paper cut and marked for folding into specific shapes (polyhedrons); and a 3-D printer that can produce solid shapes from spools of colored plastic. You don’t walk in, however, feeling you’ve entered high-tech space.
But then, that’s not the point.
The lab is really centered on its community of faculty members and students – working together on research, taking seminars on tools and software such as Mathematica, and doing outreach with K-12 students.
Yet another aspect of the IGL community: weekly lunch. Every Wednesday, faculty members and students are invited to meet at the lab for a lunch excursion somewhere in Campustown.
At the center of the IGL are the research teams, most of them advised by a faculty mentor, led by one or more graduate students, and composed of a small group of undergraduates. The spring semester had 12 teams with a total of 37 participating undergraduates, each expected to devote about 10 hours a week to the project.
The project topics can range from the purely theoretical (“Apollonian circle packing density” and “number-theoretic random walks”) to those with more immediate practical application, such as studying different structures for lithium batteries and their effect on efficiency.
Brian Freidin, a senior in math from Northbrook, Ill., said he joined the lab in the spring semester of last year, after taking a course with Athreya.
Freidin already was interested in the geometric side of things, so that was motivation to get involved, he said. He joined a team mentored by professor Stephanie Alexander, studying Minkowski space, the geometric setting for Einstein’s theory of special relativity. That project is now in its third semester.
They’ve progressed through several phases of research, he said, and they’re working on a paper.
That research involvement with a professor has been a big benefit to the experience, Freidin said. “It’s different seeing a professor teaching versus seeing a professor doing research math. It’s just a whole new side that’s opened up.”
Ananya Uppal, a sophomore from New Delhi, is now a math major but started college with plans to study engineering. “I never actually thought of it (math) as a major,” she said.
A couple of classes convinced her that was where she should be, however, and involvement in the IGL appears to have reinforced that. Her project with professor A.J. Hildebrand, on n-dimensional integrals, started with one question, but has evolved, she said. “It’s very dynamic, it keeps changing.”
Which is the point Athreya hopes students learn well through the IGL – that there’s plenty left to discover.
“Saying that what students learned previously in school is math is like saying that learning grammar is learning literature,” he said. Or, as Freidin put it: “The math you learn in high school and before is math from 200 or 2,000 years ago.”
Today’s engineers and scientists need better tools, Freidin said, and mathematicians are developing them. “We study math because it’s cool, it’s interesting and it’s neat,” he said, “and then someone else thinks it’s neat for a different reason and they figure out how it can be used to solve newer and more-challenging problems.”
Number theory, for instance, was an area of math developed decades ago with no practical application, but is now used extensively in computers and cybersecurity, Uppal said. “It’s something that you would think that, oh yeah, we’re just playing with numbers, but now it’s an integral part of our life.”
Although the research teams are a central focus of the IGL, there are other ways for undergraduates to be involved, and a key one of those is doing outreach with K-12 schools and the community. (One outing involved setting up a demonstration at the local farmers’ market.)
“Our belief is that math isn’t something only for the top 1 percent,” says graduate student Noel DeJarnette, the IGL’s outreach manager. Even in the lab’s outreach, they’re often thinking research, he said.
The idea is that professors develop an idea, which is distilled for graduate students, then for undergraduates, and then for K-12 students. “At every level, each person is doing the part that they’re capable of,” he said.
In one type of program, K-12 students come to the lab for an interactive activity, where they’re split up into “research groups,” with IGL undergraduates acting as facilitators, DeJarnette said. They present the younger students with leading questions designed to get them understanding a big idea.
It’s designed to be exploratory and hands-on, and they might use the 3-D printer or the paper cutter to produce shapes, or use interactive demonstrations on the computers, DeJarnette said. Part of the thinking is “if we can get people to have a physical connection with math, then they’re much more likely to have a mental connection with math,” he said.
In other programs, IGL members visit area classrooms and after-school programs, DeJarnette said. Even when they don’t have the same resources at hand, the plan is still the same: make it hands-on, ask thought-provoking questions and, ideally, bring them to a new level of understanding. And maybe tell them a little about their experiences with math in college, he said.
Without the IGL, Uppal said she likely would have stuck to studying and working only on her own, since that’s the way she’d always done it and that’s what she was used to. “I think that the lab just gets you to work with people, to actually get to know that you can work with people,” she said.
Modern science and modern industry involve a lot of teamwork and collaboration, DeJarnette said, which makes those skills important.
And besides, math is a team activity.