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Emily Riehl is a Professor of Mathematics at Johns Hopkins University, working on higher category theory, homotopy type theory, and computer formalization. She is one of the world’s leading mathematicians who has written advanced mathematical textbooks and serves on the editorial boards of three journals. She is invited to speak at leading conferences such as the International Congress of Mathematicians, the world’s biggest gathering of mathematicians that happens every four years. She builds the structural frameworks that bind algebra, topology, and logic.
But when you meet her, what you may not realize is that she is perhaps the most unorthodox mathematician out there. She is an international footballer who has captained the United States national team in Australian Rules Football. She is a classical violist who has spent years performing in orchestras. She is the bass guitarist in the rock band Unstraight. In the field of frontier mathematics, which demands absolute devotion, how do you build an aggressively multidimensional career? Let’s meet the mathematician who managed to do it.
Emily Riehl has crafted a career that defies easy labels. She is a research mathematician at the top end of abstraction, working in fields like higher category theory and homotopy theory. These areas are so theoretical that they deal with mathematical relationships between relationships. Category theory is the mathematics of mathematics itself because it provides a unifying framework in which disparate mathematical ideas can be treated as instances of one general concept. Riehl has established herself at the forefront of this field. She ranks amongst the top two or three mathematicians to arise in category theory in the past 15 years.
Riehl is also known as a framework builder. Much of her recent work has focused on clarifying and strengthening the foundations of infinity categories. This is a modern language for describing relationships between relationships across all of math. This language lets mathematicians capture structure at every level, from objects to processes to processes of processes.
She comes from Thousand Oaks, California, but her family moved frequently before settling in the Midwest. She grew up in Bloomington-Normal, Illinois, which is a college town that provided fertile ground for a young math nerd. “I mean, as long as I can remember, I have been interested in math and patterns and numbers and sort of the whole thing," she says. As an elementary student, she pored over puzzle books with titles like The I Hate Mathematics Book and Math for Smarty Pants. She delighted in brainteasers and logic problems. As a child, she was fascinated with calendars. She loved spotting patterns in how dates fell on certain days of the week.
“I had great math teachers all the way through,” she recalled in an interview with Johns Hopkins magazine. “They had a vision of this world of advanced math that I did not know yet, and they encouraged me to explore it, which was wonderful. I mean, absolutely wonderful. These teachers went beyond their class syllabi. They chatted with her after class. They recommended extra problems. They even guided her to enrichment opportunities. By junior year of high school, Riehl was advanced enough that she enrolled in mathematics courses at Illinois State University down the road. In her senior year, when one of her math teachers took a short leave, Riehl actually stepped in as the substitute teacher for the Advanced Placement calculus class.
In 2002, as a high school student, Riehl entered the Intel Science Talent Search. This is one of the premier research competitions for U.S. high schoolers. She won third place in the nation for a math project titled "On the Properties of Tits Graphs". Alongside nerding on math, she began studying the viola as a child and became serious enough to perform in orchestras during her teens. She learned to keep time. She was also physically active and competitive. She ran cross country in high school.
Riehl got into Harvard University as an undergraduate. Stepping into the math department at Harvard, Riehl found herself in an environment that was both exhilarating and intense. In her first semester as a freshman, she enrolled in Math 55. This is Harvard’s infamously challenging introductory course for wunderkinds covering nearly four years of math in two semesters. Surviving Math 55 is a badge of honor. Riehl did more than survive. She thrived. By the end of her undergraduate years, she had completed all the core undergraduate and many graduate-level math courses. Being suddenly surrounded by peers who were just as talented, and having to tackle proofs that needed real creative thinking, was intense. But it was also super exciting! For the first time, she felt she was entering a world where proofs were a craft.
Riehl later said that when she discovered category theory as a student, the proofs just felt like the right way to think about math. She also enjoyed teaching and wanted to communicate mathematics. She served as a course assistant for seven classes during her undergraduate tenure. This is an unusually high number. By senior year, she was assisting Dean Benedict Gross, a renowned mathematician. Gross took Riehl under his wing. He invited her to co-teach problem sessions and even to present material. He also mentored her research. At his suggestion, Riehl tackled a senior thesis on Lubin-Tate formal groups in local class field theory. “This is a very abstract number theory. It is hard to explain," she laughs, noting that her thesis largely involved carefully understanding proofs from 1960s research papers and then rewriting them in her own words.
For all four years, Riehl played viola in the Harvard-Radcliffe Orchestra, the university’s principal orchestra. Rehearsals took up many hours each week. Performances required focus and cooperation. At the same time, she became a standout in women’s rugby. On a whim during freshman orientation, the former cross country runner decided. By sophomore year, she was elected team president.
By 2006, Riehl had an undergraduate degree magna cum laude from Harvard. She had been accepted to a top PhD program at the University of Chicago, but she decided to take a detour first: a formative gap year at the University of Cambridge in England. There she enrolled in Cambridge’s famous Part III of the Mathematical Tripos. Part III is sometimes described as a math boot camp or a tasting menu of modern mathematics. Students attend courses on a wide array of specialties.
At Cambridge, Riehl made what she later called a fortuitous decision. She remembered that the University of Chicago, where she was headed next, had historic ties to something called category theory. One of its founders, Saunders Mac Lane, had been a professor there decades ago. So out of curiosity, she signed up for the category theory course offered in Part III. “I liked it instantly. I fell in love," she says of category theory. “I feel like it chose me as much as I chose it." The subject clicked with her in a way nothing else had. After years of exploring different fields like number theory, geometry, and analysis, she had found a viewpoint that unified them. “The proof felt like the right way of thinking about mathematics," she recalls. It was love at first sight. “Once you understand the statement of a theorem in category theory, you can probably supply the proof yourself," she notes.
Riehl entered her PhD at the University of Chicago in 2007, where she joined the research group of J. Peter May, a towering figure in algebraic topology and homotopy theory. Under his mentorship, Riehl plunged into problems at the intersection of homotopy theory and category theory. Homotopy theory, broadly speaking, is the field that asks "When are two shapes the same?" in a very flexible sense. Riehl’s PhD thesis, Algebraic Model Structures from 2011, was devoted to this kind of foundational work. In essence, she was building general architectures within which mathematicians can prove that one object is equivalent to another, even when direct comparison is hard. One way to describe it is that she studied category theoretic foundations for homotopy theory, making sure that the intuitive idea that these two shapes can deform into each other can be captured by precise algebraic conditions and that those conditions behave well in a category. This was deep abstract stuff, but incredibly important for pure mathematics. If you think of algebraic topology as a house built on somewhat shaky stilts—the myriad ad hoc definitions of the same shape that existed—Riehl and others were replacing those stilts with a solid concrete foundation by using category theory. Her thesis work contributed to the theory of model categories, which are a tool to turn homotopy type questions into something like a well-behaved category where equivalences can be systematically handled.
On returning to the U.S., even as she was writing her thesis, she sought out any opportunity to play footy, which in practice meant traveling to join club teams in different cities since the sport was and is quite obscure in America. This led to her playing with teams in Milwaukee and New York and going to the 2011 Aussie Rules International Cup in Australia as part of the USA women’s squad. At the 2011 USAFL National Championships, she was named Most Valuable Player.
With her doctorate in hand, Riehl moved back to Harvard in 2011, this time not as a student but as a Benjamin Peirce Postdoctoral Fellow. At Harvard, Riehl certainly made those years count. She published prolifically. By 2017, a scant two years after her postdoc, she had 21 research articles to her name. She also secured major grants, including an NSF grant and the prestigious NSF CAREER award, to support her research. In 2014, she published her first book, Categorical Homotopy Theory with Cambridge University Press, essentially turning her thesis and related research into a reference volume. She continued to experiment with teaching. She taught a Harvard online course called "Fat Chance: Probability from the Ground Up," bringing together math and real-world data to teach probability. She also became active in broader outreach, appearing in popular math YouTube videos like Numberphile to explain topics such as the Stable Marriage Problem.
“I have always focused on working well rather than working long," she says. So how does she do it all? How does she manage to excel at everything, every time, all at once? She has a few tips to share. “My main time management strategy is to start work on the thing that is due the soonest, when I will be the most focused," she told the American Mathematical Society. For instance, she would prepare her lecture notes just an hour before class, racing the clock so that she would not over-prepare and could save mental energy. This approach, she said, is risky on occasion. She does not strive for perfect daily balance; instead, she operates in seasons of focus. There are times when math is front and center, for example, when a research deadline looms or she is deep in writing a book. At those times, rugby practice or gigs might take a backseat or be scaled down. Then there are periods, perhaps summer or sabbatical, when she can give more time to training for an international footy tournament or touring with a music ensemble. She has explicitly mentioned that one key to her productivity is procrastination—purposeful procrastination. She lets long-term projects like research or book writing occupy her regular schedule, and only just before, say, a teaching duty or a referee report is due, does she shift focus to knock those out.
Secondly, Riehl protects deep work time fiercely. She learned from Hardy the value of those morning hours for math. She likely carves out similar protected time for other pursuits. For instance, she probably has her designated rugby practice times that she treats as inviolable unless an absolute emergency. She maintains focus on each area by dividing her schedule and adhering to the dedicated time blocks.
Thirdly and perhaps most importantly, Riehl does not view her non-academic identities as distractions; she believes they make her academic work better. The mental refreshment from a rugby practice can lead to insight the next morning in math as the subconscious had time to work while the body was active. She has said that some of her best ideas have come while running or stepping away from the desk. Also, these outside activities provide emotional sustenance. After all, doing mathematics at a high level can be isolating and frustrating. She mentioned the isolation in research as a downside. But if you then go to rugby practice and laugh with teammates or jam with bandmates, it balances that solitude.
As Riehl looks to the future, she sees a mathematical landscape that is more interconnected than ever. The barriers between fields are breaking down. Algebraists are talking to topologists. Logicians are talking to computer scientists. And category theory is often the language they use to communicate. Her work on infinity categories is helping to build the syntax for the next century of mathematics. But she is also looking at how technology will change the field. She is interested in formal proof verification. This is the idea of using computers to check mathematical proofs for errors. As math becomes more complex, humans are struggling to verify the longest proofs. Computers offer a solution, but they need a language they can understand. Homotopy type theory, which is closely related to Riehl’s work, offers a promising path. It provides a foundation for math that is both rigorous enough for a computer and intuitive enough for a human. Riehl has begun to explore these connections. She has given talks on the prospects for formalizing infinity categories.
Emily Riehl envisions a future where the partnership between mathematicians and machines significantly accelerates discovery. In this collaboration, the mathematician contributes the high-level strategy and creative insight, while the machine manages the tedious verification and low-level logic. One day, it could allow mathematicians to tackle problems that are currently too complex for the human mind alone.