Spotlight Scholar

Mathematical Mind

Expert mathematician explores phenomena of how things work
  • UMass Amherst Professor Andrea Nahmod stands in front of a chalkboard filled with equations.

“I like approaching a problem in new ways, to move the problem forward in a different fashion. That is a strength at UMass Amherst.” –Andrea Nahmod

Professor Andrea Nahmod’s interest in mathematics began in early childhood. Her fondest memories include sitting in her mother’s kitchen as an elementary school child doing her math homework. “I found it very peaceful,” says Nahmod.

Growing up in Argentina, Nahmod found the logic of mathematics a safe harbor in the political upheaval of the 1976-1983 military coup and dictatorship that threw the country into turmoil during her high-school and first university years. “Things were so oppressive during that period in our history," says Nahmod. "Math was a place where things were logical. It made sense to me at a time when what was happening around me made no sense."

Having come from a family of medical researchers and physicians, Nahmod decided early on to study medicine. But her initial focus on how mathematics applied to genetics, as well as a key conversation with a former neighbor who worked for NASA and the Smithsonian Astrophysical Observatory, led her to reconsider her choice. “He showed me how math was the basis of many other disciplines,” says Nahmod, “and encouraged me to do math as a career. Somehow he opened my head for math and I was convinced.”

It turned out to be good advice. An expert in the two separate but interrelated classical areas of mathematics called harmonic analysis and partial differential equations, Nahmod has risen to national prominence in her field. Since receiving her PhD from Yale University in 1991 she has received many prestigious honors: a Radcliffe-Sargent Faull Fellowship in 2009 took Nahmod to the Radcliffe Institute at Harvard University for a year to expand her research program; she was selected for a highly-competitive Simons Fellowship in 2013 allowing her a full-year sabbatical at MIT; and she’s been twice selected a Member of the Institute for Advanced Study at Princeton. In 2014 Nahmod was one of a select few mathematical scientists from around the world to be named a Fellow of the American Mathematical Society in recognition of her contributions to nonlinear Fourier analysis, harmonic analysis, and partial differential equations, as well as service to the mathematical community.

Most recently, Nahmod received a Simons Professorship at the Mathematical Science Research Institute in Berkeley, California, where she also co-directed an international effort in which more than 200 researchers worked in concert on a set of problems in partial differential equations. Additionally the National Science Foundation recently awarded Nahmod and collaborators a $1.4 million Focused Research Grant to fund their research for the next three years. “This continues her sterling record of NSF funding in every year that she has been at UMass Amherst,” says Farshid Hajir, professor and head of UMass Amherst’s Department of Mathematics and Statistics.

When asked about her research, Nahmod says “I’m an analyst. I study how to decompose objects in forms we can understand and that give us information about their most relevant features, their structure and patterns.” Using techniques called harmonic and nonlinear Fourier analysis, Nahmod and her colleagues apply these decomposition techniques to problems in the material world in order to find solutions and to understand their behavior.

“We can break down images and signals such as speech, radar or wave propagation in optics into modulated wave-forms, the signal’s basic building blocks which capture their main features and are easy to compute,” says Nahmod. “At the same time this gives us a way of putting them back together using only some parts without losing the signal’s basic qualities. It’s an important process that has revolutionized digital technology.” This ability to compress signals into wave packets has been helpful when reconstructing digital fingerprints or developing face recognition software, two technologies important to public safety and national security.

Nahmod’s related area of expertise is in nonlinear partial differential equations (PDEs) modeling different wave propagation phenomena in nature. Nonlinear wave models arise in quantum mechanics, fiber optics, ferromagnetism, water waves, Bose-Einstein condensate, and many other phenomena. One of the most ubiquitous of these PDEs is the nonlinear Schrödinger equation (NLS).

“Because waves in nature interact in a nonlinear fashion as they propagate and have different properties such as amplitude, length, oscillation, speed, and position over time, it’s important to understand how they may behave under certain conditions or when introduced to certain media,” says Nahmod. Understanding the most efficient way to send a signal through a fiber optic cable or being able to anticipate the properties of a gas when the temperature approaches absolute zero are two very different phenomena in nature but are both aspects of solutions to the same equation, she adds.

“Being able to understand and describe the dynamics and behavior of solutions to NLS is fundamental to accurately predict wave phenomena,” says Nahmod. That’s an important tool to have in your toolkit when studying the natural world.”

What’s next for her research? Nahmod has always been interested in bringing new tools to bear on current mathematical methods in order to open up new research directions or define new paradigms. “I like being able to approach a problem in new ways,” she says, “to move the problem forward in a different fashion. That is a strength for me and of the faculty at UMass Amherst.” To this end, her current research investigates the role of data randomization in nonlinear wave phenomena and how probability can be applied to shed light on behavior and dynamics of ‘generic’ solutions.

“Roughly,” Nahmod explains, “the idea is that you don't need to look at the dynamics of equations for every single initial profile of a class to predict an outcome. Probability introduces the notion that you can look at it generically: pick one at random that has certain prescribed properties, and understand the long-term dynamics of it by approaching it from a non-deterministic viewpoint.”

Two attractions drew Nahmod to UMass Amherst: it gave her the opportunity to teach at a public institution and it fulfilled her desire to be at the forefront of something new. “When I came to campus in 1998 the department was in transition,” she recalls. “It soon lost 15 faculty members to early retirement and began hiring a lot of young faculty. There was this energy—a group of young people who were somehow going to take the lead in shaping the future of the department. UMass was a place where I saw something bright, a place where nobody else was really doing what I was doing and in which I could have a role in building something. I liked that.”

As a product of the public school system in Argentina, Nahmod appreciates the role public universities such as UMass Amherst play in citizens’ lives. “There is something that I like that is a little bit of a challenge at a public school,” she says. “The body is larger and more diverse and includes an enormous number of bright kids. To find them and tap their potential, to reach out to them is rewarding. I was helped in that way when I was their age. It's time to pay it forward."

Karen J. Hayes '85