What does it mean to do research in math? According to Tara S. Holm, Mathematics, it’s not as complicated as some might think. “It’s always surprising—the simple things that we still don’t know,” she says.
Contrary to popular assumptions, Holm adds, research in mathematics doesn’t take talent or genius so much as hard work and obsession. “You get obsessed with certain problems, and you’re not able to stop thinking about them,” she says. “It’s curiosity-driven. Mathematicians study mathematics because they have to know.”
The Math of Shapes
Holm’s field is geometry, the math of shapes. She studies the properties of symplectic structures, shapes with a certain degree of floppiness.
She explains that there’s a spectrum of rigidity in geometry. On the one end is differential geometry, which considers the most rigid shapes. In differential geometry, a small circle is different from a big one because the smaller one is more tightly curved. At the other end of the spectrum is topology, which is concerned with the essential features of the shape and not how big it is. “In topology, you can think of shapes as floppy, like rubber sheets you can stretch,” Holm says.
Holm studies just how floppy symplectic structures can be. She is researching the conditions under which one symplectic shape fits into another. While theoretical, this research is motivated by physics—the effort to understand how particles are positioned and how they move in a physical system.
Squeezable versus Rigid
Holm and collaborators—Dan Cristofaro-Gardiner (Harvard University), Alessia Mandini (Pontifical Catholic University of Rio de Janeiro), and Ana Rita Pires (Fordham University)—are working on a project that derives from the Non-Squeezing Theorem, first proved in 1985 by Mikhail Gromov (New York University).
This theorem proved that symplectic structures were not entirely floppy; they had some rigidity. Fellow mathematician Ian Stewart (University of Warwick) explained, using an analogy of a camel fitting through the eye of a needle. “To a topologist, a camel should be as floppy as you like, so you could squeeze it through the tiniest of holes,” Holm says. “Gromov’s work shows that symplectic geometry is a bit more rigid because the symplectic camel doesn’t squeeze through too small of an eye of a needle.”
Gromov showed, in this theorem, that you can’t simply squeeze a ball into a cylinder if you want to maintain the two-dimensional structure of the ball. “What he found is that you can only preserve the two-dimensional structure of the ball when the radius is less than or equal to the cylinder’s radius,” Holm says. “We’ve been working on refining this and trying to understand things other than balls mapping onto things other than cylinders.”
Holm describes these other things, or shapes, as four-dimensional symplectic gadgets. She represents the problem on a blackboard with a triangle and a polygon, pointing from one to the other. Holm asks, “When can we fit this symplectic gadget represented by the triangle into this other one?”
Twelve Shapes and the Infinite Staircase
The work has yielded surprising and mysterious connections to number theory. Holm and her team plugged coordinates into a computer to graph the relationship between the size and shape of various gadgets and the required volume of the shape into which it fits. With some configurations, the resulting graph has the pattern of an infinite staircase. “This is where the weird number theory seems to come in,” Holm says.
“Doing research in mathematics is like being in a dark room with a book of matches, and you’re discovering more and more bits of it, but the room is like the universe.”
The team plugged the heights of the stairs into the Online Encyclopedia of Integer Sequences—a database of formulas and number sequences. For one pair of shapes, the numbers correspond to a famous sequence called the Fibonacci numbers. “The Fibonacci numbers show up mysteriously in math a lot. Why? We’re not sure,” Holm says.
These particular shapes also corroborated observations from other fields of mathematics. “It turns out that there are 12 shapes where an infinite staircase shows up and in all others there’s not,” Holm says. “And those 12 shapes were already known as interesting to other fields of mathematics.”
Holm and her collaborators are currently working to prove that these 12 shapes are the only ones with the particular properties that lead to the infinite staircase pattern. The team is also in contact with specialists in number theory to explore the connection between the shapes and the number theory-related sequences that arise.
While the 12 shapes are proving to be the main fruit of their labor, Holm had some funny mishaps along the way to identifying them. “We had this pair of shapes we were working with, and we graphed it and found an infinite staircase. Then we plugged the numbers into this online encyclopedia, hoping that something interesting would come up. Up pops exactly one sequence—that’s the one, right? Well, it was the numbered stops on a SEPTA railway line—not a mathematical sequence at all.” Holm laughs. “We decided that maybe we can now predict where the next stop is going to be.”
Fueling the Quest for Learning
Despite entering a field of numbers and shapes, Holm was most motivated to pursue mathematics because of the people. “I never really thought I would be a mathematician. I went to college not really planning to major in math,” Holm says. “But there was a math professor who was really kind and encouraging and patient, nurturing, and he got me involved in research projects. I owe him a huge debt of gratitude.”
In graduate school, Holm similarly gravitated to her specialization because of her adviser and the group of graduate students studying symplectic geometry. “They were fun to talk to,” she says. “And my adviser loved math, and it rubbed off.”
Holm says she likes Cornell for the same reasons—she describes the math department as incredibly friendly. “One of the things I’ve experienced at other universities is that people aren’t around very much or their doors are closed. At Cornell, people are always happy to talk to you about your math questions, and the students are amazing,” she says. “Even first semester calculus students, many of whom won’t go on to major in math, work really hard at it. It’s an inspiring place to be, where there’s so many amazing things going on and so much learning.”
This energy fuels the quest to know, the mathematician’s curiosity for curiosity’s sake. “One of the important things about mathematics is that we just never know when it’s going to be important. The parts of mathematics that are currently used to encrypt data when you make a credit card transaction—that comes from one of the more theoretical areas of mathematics,” Holm says. “Doing research in mathematics is like being in a dark room with a book of matches, and you’re discovering more and more bits of it, but the room is like the universe, it just goes on forever. The ultimate goal is making sense of all the flashes of insight, eventually understanding how it all fits together.”