Growing up in Italy, Sara Maloni was interested in lots of things: art, cinema, literature, and math. When she thought about what she might want to do for a living, she was a bit torn. “I was struggling about whether to go into math or humanities,” she said. “But I decided that if I leave mathematics it will be much more difficult to keep as a personal interest. You can go to the cinema without studying it.”
Not only that, she was also afraid that studying the humanities would take away the magic it had for her. As she explains it, “You start to lose the poetry behind it.” But the more the new Tamarkin Assistant Professor learns about math, the more poetry she finds.
“To me,” she said, “mathematics is the notion of beauty, the notion of symmetry, the notion of truth. Mathematicians are curious people. We like to understand things.”
Chief among Maloni’s research interests is hyperbolic geometry, a non-Euclidian geometry. It’s non-Euclidian because it discards one of the five postulates that govern Euclidian or “flat” geometry. The postulate — known as the parallel postulate — states that if you have a line and a point outside the line, there is exactly one line through the point that is parallel to the first line. That holds true on flat surfaces, but breaks down in curved space. In spherical geometry, the study of positively curved space, there are no lines that satisfy the postulate, while in hyperbolic geometry, or “negatively curved” geometry, there are infinitely many.
“In some sense hyperbolic geometry is the most mysterious and also the most interesting because many things are still unknown,” Maloni said. “It’s also the geometry that is most common in some sense.” Hyperbolic planes can be found on lettuce leaves, sea slugs and coral reefs, and physicists use hyperbolic geometry in representing space-time.
Maloni got interested in the subject in college at the University of Genoa when one of her professors handed her a book called Indra’s Pearls. Coauthored by David Mumford, professor emeritus of applied mathematics at Brown, the book describe the work of the 19th-century mathematician Felix Klein. It starts from some basic mathematical topics and describes simple algorithms that create delicate fractal figures. Klein's vision of infinitely repeated reflections is only recently possible to generate thanks to computers. Hyperbolic geometry provides the underpinnings of Klein’s work.
“The book manages to explain pieces of very difficult mathematics to a general audience without any specific background,” Maloni said. “They start from the basics, and using lots of pictures and examples, they are able to introduce topics that I use in my research. I think that is a perfect example of the poetry behind mathematics.”
Maloni has bachelor’s and master’s degrees from the University of Genoa. She earned her Ph.D. from the Mathematics Institute at the University of Warwick in the United Kingdom. Before coming to Providence, she was a postdoctoral researcher at Université de Toulouse and Université Paris-Sud.
At Warwick, she enjoyed a close and productive relationship with her adviser. It’s the kind of relationship she hopes to have with her students at Brown.
“In Italy, you are a student and your professor is far above you,” she said. “Usually there is a feeling that you can’t really be a peer. In the U.K. and U.S., there’s a feeling that both professor and student are going in the same direction because we are both trying to prove new things together. I really like this teamwork.”