It’s not easy to make confusing mathematics topics understandable, let alone interesting, to non-mathematicians, but University of Pennsylvania professor Robert Ghrist has figured out the formula.

Ghrist, a Penn Integrates Knowledge Professor with appointments in the School of Arts & Sciences and the School of Engineering and Applied Science, studies a branch of math called algebraic topology.

“Topology is based on the notion of proximity,” he says. “It answers questions like what’s its shape, how many holes does it have, what type of holes does it have? It’s intrinsically qualitative in nature. It’s not asking for distances between points.”

Despite how abstract it sounds, it’s a field of pure math with many applications, and Ghrist is at the forefront of several.

One he points out is the intersection between neuroscience and math. An area neuroscience researchers care about, he says, is data generated from neurons in the brain, how rapidly they fire or how their connections work. The problem is the dataset is massive. “Trying to extract meaningful information about how those neurons are wired, it’s difficult for classical mathematical techniques, calculus, linear algebra,” Ghrist says.

Topology, which specializes in understanding spaces and holes between items, offers a different approach.

“New problems demand new methods,” he says. “We’ve been working to develop these methods, then find new ways to compute topological variants to access brain network structures.”

The “we” in this equation is Ghrist and postdoc Chad Giusti, a fellow at Penn’s Warren Center for Network and Data Sciences who recently published a paper in the Proceedings of the National Academy of Sciences about these brain networks. Giusti’s work details a machine that uses complex math to find spatial patterns underlying neural activity.

“The machine takes neural activity, neurons, as its input, and as its output it gives you these signatures of brain models,” Giusti says. “It gives you a hint about what to look for when you don’t have anything to begin with.” Put another way, it offers a new method to see how the networks in the brain are organized conceptually rather than physically.

The field is so new, its applications so complex that much of the work on the brain networks has been “a lot of false starts and careful idea probing before finding the right path,” he says, a good deal of throwing ideas at the wall to see what sticks. Giusti says Ghrist is particularly good at pushing forward even when every idea seems to bounce.

“He has a very different view on how math interacts with the world than many mathematicians and many scientists,” Giusti says. “He’s got a unique perspective and a whole lot of faith in the tools.”

That positive mentality helps for projects like the one Ghrist is doing with Greg Henselman, a fifth-year graduate student and part of the Penn Applied Algebraic Topology Research Network. When Henselman pares it to down to layman’s terms, he describes the work as a computer program that can measure difficult-to-measure shapes.

“Robert is interested in looking at spaces you construct out of building blocks, like Legos. For some of these spaces you need a lot of Legos to build them,” Henselman says. With his program, “computers can understand millions and billions of complex parts.”

Anyone working in topology deals with such complex spaces, but there aren’t great ways to process them right now. Henselman, with guidance from Ghrist, is aiming to change that, with Ghrist’s legwork making it possible.

“He’s done the hard work of convincing people there are problems we can solve,” Henselman says. “Not only can we solve them in theory, but we can actually solve them with this software.”

All three mathematicians express passion about the field but Ghrist most of all.

“The beautiful thing about the subject is it’s inherently very visual,” he says. “It’s very appealing to those who like pictures.”

Behind much of what Ghrist does sits this notion of reaching a broad audience. It’s evident when he discusses his role as an educator on the Penn campus, focusing on the tools that have pull with his students, i.e., anything technology-related. It’s also clear when he brings up a calculus class he authored and developed through Penn’s Online Learning Initiative.

The free online course has now run seven times and recently transitioned to a work-at-your-own-pace mode. More than 100,000 people have signed up, and viewers have watched more than half a million hours of the video content. Ghrist says he frequently gets requests for higher-level classes. It’s not for lack of personal interest that he hasn’t yet done it; it just takes time.

That’s something Ghrist may now have more of, actually, given a five-year, $2 million grant from the Department of Defense he earned in 2015. It’s an opportunity for him to take some risks in his research that don’t require an immediate payoff, he says.

“That is how science has to happen,” he says.

The project he proposed through the lengthy DOD application process extracts data at the local level to understand its global structure. Consider for a minute how cell phones work.

“They communicate with each other wirelessly. If they don’t have a GPS or if they’re indoors or underground, they don’t have exact latitude and longitude coordinates,” Ghrist explains. “Nevertheless a collection of cell phones can determine how everyone is arranged: what the hallways look like, how many classrooms there are, how many floors in the building there are. These are all qualitative properties.”

Topology, he says, can bring order to a group of seemingly distinct items. But why, exactly, do we need this order?

Ghrist offers another example, the pathway of malaria. Research conducted at Stanford University used topology to show that the parasite causing malaria doesn’t move along a straight line from infection to full-blown disease, then back the way it came during recovery, resulting in a healthy individual. Rather, it travels a loop.

The first half, the top of the oval or starting point, is getting the disease, he says. “The recovery phase is actually different; it’s not running in reverse but making a loop back to a state of health.”

Using the framework of his mathematics passion, Ghrist, the Andrea Mitchell University Professor at Penn, breaks down these complex topics in language anyone can understand. “Eventually I want to explain applied algebraic topology to the masses,” he says, not just to mathematicians but to scientists and researchers and maybe even retirees looking to expand their knowledge.

“You never know,” he says. “That’s the future I want to see happen.”