After 47 years of teaching and research at Columbia, Jonathan Gross retired last semester, following a highly active career that allowed him to indulge his lifelong love of mathematics while doing pioneering work in graph theory, three-dimensional topology, shape modeling, and sociological modeling.
Professor Gross’s main specialty is topological graph theory, a math subdiscipline straddling combinatorics and geometry and marked by a strong visual component. In several of his 17 books and in over 100 papers and journal articles, Gross expanded topological graph theory by initiating new programs of investigation and by developing new methods for them, often collaborating with Thomas W. Tucker. Together Gross and Tucker authored the influential and comprehensive Topological Graph Theory, which at its release in 1987 represented the state-of-the-art in graph theory. Their objective in writing that book was to create a single source that would provide someone new to topological graph theory with sufficient background to move as quickly as possible into frontier research. It remains a standard reference today.
Gross invented the voltage graph construction in 1973, which is the basis for a concise algebraic specification of infinite families of large graphs and also of placements of such graphs on increasingly complicated surfaces. Gross’s joint work with Tucker on its generalization, published in 1977, includes some of the most frequently cited publications in topological graph theory. The name voltage graph plays on the fact that one of the key properties that sometimes occurs in the specification of placements in surfaces is an algebraic generalization of the Kirchhoff voltage law, which is a property of electrical circuits well known to electrical engineers and physicists. Another paper by Gross and Tucker explains how the voltage graph construction unifies dozens of special cases that occur in the solution of the Heawood map-coloring problem.
Topological graph theory has connections to many other areas of mathematics, including combinatorial and probabilistic models, as well as to knot theory. Since 2009, Gross has been working with Jianer Chen, one of his former Columbia PhD students, to apply topological graph theory to the computer graphics area called shape modeling. Another area that Gross tackled and examined for several years is behavioral and cultural rule systems, for which he developed information-theoretic models and measurement techniques. Working with the eminent British anthropologist Dame Mary Douglas, Gross demonstrated how such high-powered tools can be harnessed to better understand human social behavior. In his book, Measuring Culture, Gross and his co-author Steve Rayner describe how to measure information content in societal patterns, making it possible to obtain objective comparisons of different target populations.
For his research, Gross has earned multiple honors and awards: an Alfred P. Sloan Fellowship, an IBM Postdoctoral Fellowship, and numerous research grants from the Office of Naval Research, the National Science Foundation, the Russell Sage Foundation, and, most recently, from the Simons Foundation.
Gross began his formal mathematics education as an undergraduate at MIT, graduating in 1964. From MIT, he went to Dartmouth College where his PhD thesis on three-dimensional topology (1968) solved a published problem of Fields Medalist John Milnor. After graduate school, he joined the Mathematics Department at Princeton University, working with Ralph Fox, renowned for his work on knot theory and three-dimensional topology.
Though primarily a mathematician, Gross had an early interest in computers, and it was in computer science that he felt that his teaching would have greater impact. He has believed since his high school days that computing was for everybody, and his earliest books are concerned with computer programming. It was to set up a computer science curriculum for arts and science students that he was invited in 1969 to join the Statistics Department at Columbia. His first class in introductory computer programming at Columbia had eight students. Within a few years, 300 students in that same course filled the seats in the large lecture room in Havemeyer. The university expanded the computer science contingent that he headed within Statistics one by one, to five faculty members.
In the late 1960s and the 1970s, computer science was also taught by a small nucleus of professors of Electrical Engineering. In 1978-79, while Gross was Acting Chair of Statistics, Dean Peter Likins of SEAS committed funds from a substantial gift to SEAS to found a separate Computer Science Department, which both contingents agreed to join. Merging the computer science course offerings from Statistics and from Electrical Engineering was among the first initiatives that Gross orchestrated for the new department. He strongly encouraged faculty to balance their teaching assignments between undergraduate and graduate levels. His role in starting Columbia’s computer science department was fundamental; as the department grew over the years—it now numbers 44 professors and 5 lecturers—Gross was the organizer of department-wide efforts to keep the academic curriculum at the educational forefront. Over the years, he became the keeper of institutional memory.
“Not only did we have no cell-phones or personal computers when I was young, most families did not have a television before 1950. We would start being nice to the rich kid around Thursday, in the hope that he would invite us to watch television at his house over the weekend.”
Mathematician, researcher, author, and computer scientist, Gross was also an instructor to thousands of Columbia students. He taught discrete mathematics, graph theory, and combinatorial theory, lecturing with humor and with what he called “enhancement,” short historic anecdotes from science and mathematics as well as from his own mathematical career and personal history. “Enhancements” were as integral to his courses as his meticulously put-together notes, often giving students insight into a different time and place.
He proved popular with students, who variously described him as devoted to his work, brilliant, idiosyncratic, and highly quotable.
- When I say a baby-level proof, that’s just how mathematicians talk.
- I don’t actually know any babies who can do algebraic topology.
You have to say it very quickly, or you’ll get it wrong.
- I have no idea what liquid soap will make your dishes sparkle,
but I recommend liquid Joy for making high-quality knotted soap bubbles with interesting mathematical properties.
– From a collection of quotes compiled by students
For his excellent teaching, Gross received two SEAS awards; in 1994 he received as well the career Great Teacher Award from the Society of Columbia Graduates.
In late career and retirement, Gross continues his research work with his co-authors around the world. Each year he produces numerous journal papers in topological graph theory, and he continues to travel to national and international mathematics meeting to give talks about his research and to chair sessions in his specialty. One math friend has joked, “Jonathan, you are in danger of flunking retirement.” To this, Gross responds that math is too much fun to stop and that he intends to flunk retirement for years to come.
His conclusion of active service at Columbia was marked in December with a dinner amidst remembrances by colleagues and family. Among those who shared their personal stories of Professor Gross, it was perhaps his daughter Rena who most closely articulated how much mathematics infused her father’s life when she recounted how, as a child and misbehaving, her father would threaten “Stop, or I’ll map you into the complex plane.”