In the early 1980s, Yair Minsky was at a crossroads. After earning his bachelor’s degree from Princeton—where he was a student of the legendary geometer William Paul Thurston—he had enrolled in a graduate program in computer science at Columbia. Certain that computing would pave the way to career stability, he was nevertheless “mourning the loss of mathematics” in his life.
“I had always liked graphical things—liked pictures, liked to draw…,” Minsky, the Einar Hille Professor of Mathematics and an expert on three-dimensional topology and hyperbolic geometry (a non-Euclidean geometry), explained in his Dunham Laboratory office last week. And as an undergraduate studying with one of the pioneers of modern topology, he had been given a glimpse of the big questions to which Thurston and others in his field were seeking answers. The rigidity theorem of George Daniel Mostow—Yale’s Henry Ford II Professor Emeritus of Mathematics and the 2013 recipient of the prestigious Wolf Prize in mathematics, who died on April 4 at age 94—had revolutionized geometry in the late 1960s, showing that a hyperbolic manifold of three dimensions or greater, and of finite volume, is unique in its shape. And Thurston’s geometrization conjecture sought to prove that any compact 3-manifold (a space that locally appears as Euclidean three-dimensional space) is made up of components that can be categorized into eight specific geometric structures.
So, with his master’s degree in computer science in hand, Minsky returned to Princeton to pursue his Ph.D. in mathematics, and to work alongside his mentor on a program of questions and methods Thurston had developed to explore the structure of shapes.
Three decades later, the project Thurston started in the 1970s is nearing its conclusion, and Minsky—a member of the FAS faculty since 2003—has been a driving force behind its progress. The recipient of numerous National Science Foundation (NSF) research grants stretching back to 1996, he teaches undergraduate and graduate courses in geometry, group theory, and differential topology, and has published extensively in the Annals of Mathematics, the Journal of the American Mathematical Society, Geometric and Functional Analysis, and elsewhere. He was director of undergraduate studies of the Department of Mathematics from 2007 to 2009, chaired the department from 2010 to 2014, is a member of the FAS Senate, and has served on the Physical Sciences and Engineering Tenure and Appointments Committee.
In the interview in his office, Minsky emphasized his interest in the connections between geometry, topology, and symmetry. On his desktop computer, he used images of symmetric tilings to demonstrate how they can be “rolled up” into a smaller manifold by identifying points in the tiling that are indistinguishable by virtue of the symmetries. (Imagine a room walled by perfect mirrors: there is no way to tell, by naïve observation, whether you are in a single room with mirrors or in an infinite array of rooms with repeated identical copies of yourself.) “This is an old classical idea, but it has a lot of very subtle and useful consequences,” Minsky said. “The work on hyperbolic geometry and its symmetries has led to a lot of insights into the structure of symmetry groups in general.”
Over the past 10 years, many of the principal questions originally posed by William Thurston have been solved; the task now is “a kind of refinement” focused on the lingering mysteries of the field. For example, Thurston’s geometrization conjecture (now proven through the work of Russian mathematician Grigori Perelman) shows that each topological 3-manifold admits a geometric structure, but does not always provide a way of explicitly “translating” from the topological description to a quantitative description of the object’s geometry—how big it is, its diameter, etc. For certain families of objects, this “recipe” is known. But, Minsky said, “there is no overarching, complete picture”—no comprehensive theorem that has enabled mathematicians to create, so to speak, the entire cookbook. And thus the work of “trying to map out this universe of objects” continues.
Outside of his research, Minsky—following the model of his mentor at Princeton—has dedicated himself to making mathematics accessible to students and general audiences. Five years ago, inspired by his children’s participation in Yale’s “Science Saturdays,” he began his department’s NSF-funded Math Mornings series. Aimed at middle- and high-school students, each installment in the series includes math demonstrations and games (facilitated by undergraduate volunteers) and an interactive lecture by a faculty member from Yale or a peer institution. The program draws a loyal following of local students and their families and, reliably, enthusiastic engagement by the participants. “Kids—especially the young ones—tend to ask good questions,” Minsky said.
So, all these years later, does Yair Minsky regret his brief hiatus from mathematics? Not at all: in fact, his foray into computer science proved invaluable to his ultimate path as a researcher and scholar. Although Minsky’s work is not in computational geometry, the scholarly questions he pursued at Princeton under William Thurston—and the field of topology in general—intersect with computational issues in many ways. And, he added, the “computational point of view” lends helpful perspective to the work he is undertaking today—cataloguing the structures that give shape to our world, one step at a time.
-Reported and written by Alison Coleman for the FAS Dean’s Office