Some mathematical and statistical challenges are so fundamental that the solutions can be applied to an array of real-world applications—which we all benefit from. But it’s not necessarily the applications that excite researchers. They’re on the hunt to develop an elegant set of equations or a more accurate model because of the work’s inherent beauty and satisfaction, and, simply, because it’s fun.

This past fall, members of the UMBC Department of Mathematics and Statistics received at least three major grants to pursue work with significant real-world potential, from self-driving cars to treatments for Alzheimer’s. And while the mathematical work underpinning these potential uses can seem opaque, abstract, and just plain hard to many people, these mathematicians and statisticians see the process of thinking about the challenges as exciting and stimulating in and of themselves.   

Optimizing optimization

Andrei Draganescu (courtesy of Draganescu)

Andrei Draganescu, associate professor of mathematics, has received a grant from the National Science Foundation to pursue a creative new approach for solving optimization problems. Anytime a system performs a basic functionality, but you would like to make it better or more efficient, that’s an optimization problem, Draganescu explains. 

Optimization means maximizing or minimizing something, such as cost, time, or power output. Think about today’s large language models, such as ChatGPT: Before it can respond effectively to any prompts, the model must rely on an algorithm to “learn” an optimal set of parameters from a vast amount of training data. Only once those parameters are fixed can it use them as a basis to determine the best response to your prompt. The time, electricity, and human work hours involved in training can cost billions of dollars, not including the capital investment in the hardware itself—so even a very small increase in efficiency makes a huge difference.  

All of the possible combinations of parameters can be represented as points along a curved surface with hills and valleys. An algorithm developed by a mathematician can search for the lowest or highest point on that surface to solve the optimization problem, depending if you are looking for a minimum or a maximum. One risk with these algorithms is that they may end up at a local minimum or maximum—the peak of a single hill or the base of a single valley—that isn’t actually the global maximum or minimum across the entire surface.

In his grant application, Draganescu proposed a novel way of solving this kind of problem. “Because this type of problem is so old, it’s hard to find something new. But I think I did,” Draganescu says. Rather than searching for the minimum or maximum along a series of straight lines that approximate its curved surface, Draganescu has proposed a way to search directly along a set of well defined curves. The idea for the new approach “is a completely new angle,” he says, and in correspondence with colleagues to date, “nobody has contradicted me so far.”

All of the possible combinations of parameters for a system can be represented as points along a curved surface with hills and valleys. (Fiona Suherman/UMBC)

Sticking with the hard problems

In math, there are many problems that are easy to express but extremely difficult to solve. Some have puzzled mathematicians for centuries. Draganescu’s Ph.D. advisor encouraged his students to think about these problems from time to time, in case they found a solution, of course, but also to keep their creative juices flowing. Draganescu continues the practice today.

“Sometimes I’ve spent days in a row on some of these very hard problems that nobody knows how to solve,” he says. “And of course I would love the glory, but honestly, that’s not where the fun is. I’m old enough to know that the fun is actually thinking about it.”

“And of course I would love the glory, but honestly, that’s not where the fun is. I’m old enough to know that the fun is actually thinking about it.”

Andrei Draganescu

Draganescu’s sticktoitiveness sometimes pays off in his everyday work. In the case of the idea that underpins his recent proposal, “I was surprised that it actually worked out. But ‘working out’ means many many months of work,” he says. “From having the initial idea to trusting that it could go somewhere, there were a lot of mistakes in the calculations and all that.” So why did he persist, knowing there was a good chance it wouldn’t “work out”? “I stuck with it because it was fun. I enjoy thinking about these problems.” 

Draganescu is continuing to develop his new technique for solving optimization problems, which could improve efforts to optimize all kinds of systems, from AI to airline schedules to agriculture—all because he found joy from sitting with a sticky problem.

Replicating the real world

Ansu Chatterjee (courtesy of Chatterjee)

Ansu Chatterjee, professor of statistics, is another UMBC math mind whose work could have a major impact on a wide range of fields. He and UMBC colleagues Animikh Biswas, professor of mathematics, and Karuna Joshi, professor of information systems, have received an NSF grant to expand the use of “digital twin” technology. The grant proposal highlighted the potential to advance diagnosis and treatment for neurodegenerative diseases such as Alzheimer’s or Parkinson’s, but digital twin technology has a huge range of potential applications.

A digital twin is “essentially a clone of something in the physical world replicated inside a computer,” Chatterjee says. Additional applications could include forest management or weather forecasting. These twins can offer up predictions about which direction a hurricane will turn, which treatment protocol might be most effective for a particular patient, or whether it’s the right time for a prescribed burn in a forest—but there will always be uncertainties in the models. 

“Measuring those uncertainties is a huge task,” Chatterjee says. How to deal with those uncertainties is one of the things the team wants to focus on with the new grant, which is very statistics-heavy, Chatterjee notes. Understanding and reducing the uncertainties requires understanding the physical processes at play in the real-world version of the system, and then working them into the model—and that’s hard. 

Beyond the black box

Typical AI models are frequently “black box models,” which means even the people who created them don’t always understand exactly what they’re doing. For one thing, they are not constrained by objective truth in the physical world, Chatterjee explains. “It’s not even attempting to get the biology or the chemistry correct,” he says, and that’s not acceptable for a digital twin that’s modeling the human brain and may be used to determine a real human’s treatment plan. 

“You don’t get the model right unless you get the underlying math of the molecules that constitute it right. So mathematics and statistics have to be the underpinning of anything related to digital twins.”

Ansu Chatterjee

“You don’t get the model right unless you get the underlying math of the molecules that constitute it right,” Chatterjee says. “So mathematics and statistics have to be the underpinning of anything related to digital twins.”

The digital twins Chatterjee and colleagues hope to further with their new research need not only replicate the outcomes of a system, but also the system’s mechanisms and internal processes. Again, that’s hard—especially when humans don’t yet fully understand many of these systems, such as the human brain, how cancer develops, or how various environmental factors influence carbon sequestration, for example.

“We are at the foundation of digital twins,” Chaterjee says. Once they are developed to a point where they are usable, digital twins and AI more generally would “open up a vast amount of opportunities for making fairly rapid progress with the actual science. This is where AI can help in scientific progress. It can suggest opportunities and possibilities which then can be verified,” Chatterjee says. “It would rule out certain possibilities. There would of course be false positives and false negatives, but as digital twins keep improving, it would be more and more convenient to use those first to find out a lot more interesting things.”

A digital twin is “essentially a clone of something in the physical world replicated inside a computer,” Ansu Chatterjee says. (Illustration by Sadaf Rehman ’19/UMBC)

Fundamental building blocks

Chatterjee and colleagues are aware of the risks associated with digital twins. “There’s quite a lot of ethical issues that are related to digital twins in general, and it goes up several notches when it’s digital twins related to humans.” For example, there are regulatory concerns around who owns the data put into a digital twin (such as someone’s genomic data) and who is compensated when the twin is used for profit. “It’s exciting, and it’s also kind of a nightmare,” Chatterjee says. 

That’s why he and colleagues will be working directly with federal regulatory agencies to make recommendations about how to handle these AI models. “This grant is sort of laying down the first fundamental steps—saying this is what should be done, this is how the first steps should be done,” Chatterjee says. 

While that work is critical, it’s the math that truly excites Chatterjee. “For me, it’s about building the fundamental building blocks, whether it’s cancer or neurodegenerative diseases or forests” he says. “The essential ingredients for the digital component of it rely on the same math, the same statistics.”

Defining possible

Matthew Kvalheim (courtesy of Kvalheim)

Matthew Kvalheim, assistant professor of mathematics, is also working on developing tools that can help people in a wide range of situations. His ultimate goal? To be able to tell researchers whether whatever they are trying to do with their system is possible or impossible. Sound vague? That’s the point. 

“I’m never thinking about a specific application,” Kvalheim says. “I’m always working at the fundamental level, with a class of mathematical models that can describe all of them.” For Kvalheim, it’s the underlying math that is exciting—which is fortunate for all the people working on applications who will eventually use his multi-purpose tools. 

“For any real world system, you might want it to do some behavior, or not do some behavior,” Kvalheim says. This might be making sure two self-driving cars don’t collide, preventing a humanoid robot from falling on its face, or keeping an electronic device at a safe temperature. 

“Lots of people work on the set of models that I study,” he says, and the vast majority of them are investigating a particular application. These other researchers are very good at coming up with solutions that result in stability, or maintaining a certain state in a system, and safety, or preventing dangerous states in the system, “but there’s one thing they can’t do,” Kvalheim says. “They can’t determine if the thing they are trying to do, to make their system stable and safe in the way they want, is just fundamentally impossible. There could be a fundamental law of nature that says too bad, you can’t do it. You could try forever, but the heat death of the universe will occur first.”

A tool Kvalheim has already published to help prove impossibility relies on math that is closely related to something called the “hairy ball theorem.” If you imagine certain 3D shapes covered by directional vectors—or, to make it simpler, hairs—”Try as you might, you can’t comb that hair flat everywhere without creating a cowlick somewhere,” Kvalheim says. (Illustration by Sadaf Rehman ’19/UMBC)

A license to look

Kvalheim would like to discover new fundamental limitations that hold true for any application you could come up with, and also new fundamental capabilities. The U.S. Air Force recently awarded him a grant to further this work. He’s trying to generate a sort of mathematical litmus test that any practitioner can use to find out whether the thing they are trying to do with their system (whatever it is!) is fundamentally possible or impossible. If it’s impossible, it saves them a lot of time, energy, and money. And if it’s possible? “It’s like giving people a license to go look for it,” Kvalheim says.

The best part is that the end user of one of Kvalheim’s tools wouldn’t need to understand all the pure math behind it. “The idea is to produce useful tools that they can use even if they don’t have the time or interest in diving into all the details,” he says.

Kvalheim’s stability research is applicable even to the Lorenz system, which is a set of ordinary differential equations. Solutions to this system can vary chaotically depending on the initial parameters, but converge toward and stay within a bounded space. This space appears butterfly-shaped when represented visually.
(Illustration by Fiona Suherman/UMBC)

Math undergirds just about everything, Kvalhiem says, so it’s important to study the general case, rather than focus on specific applications. He wants people to see that it’s important “to know when there are fundamental laws of nature telling them that it is impossible or possible to do things that they want, and that math allows you to discover such laws of nature—and not just for one real-world system, but for many systems all at once.”

Somewhere a math or stat researcher is helping prevent your car from crashing, providing life-saving medical tools for yourself or a loved one, reducing wildfire risk where you live, or improving your experience with an AI tutor—and you might never even know it. There are humans behind each of these innovations who find joy and value in the math puzzles and problems that are all around us. And some of them work at UMBC.