Emily Riehl thinks hard about objects that don't exist in the material world yet mysteriously seem to underlie many things that do. These objects have no concrete existence of their own, but they do have connections that once discovered can be used to solve knotty mathematical problems. This means employing ever-more abstract thinking about things that were abstract to begin with, and Riehl is fine with that. She has spent the last several years probing esoteric forms of mathematics that rethink the most basic of terms, like "equals" and "same." As a result of work by her and others, "the same" will never be the same again.
Riehl is an associate professor in the Johns Hopkins Krieger School of Arts and Sciences (and surely the only mathematician in Hopkins history to play several stringed instruments and to also play on the U.S. national team in Australian rules football). She has established herself at the forefront of theories that are galvanizing higher mathematics: category theory and infinity category theory and homotopy type theory. Which means … what? Travel writer Paul Theroux once said that listening to Afghans speaking Pashto made him feel like a dog listening to music. If you're not a mathematician, delving into Riehl's work can induce the same feeling. Try to describe it without recourse to equations or graphs or mathematical symbols or esoteric technical jargon—that is, with everyday words—and you will ever be only approximately right because those words are abstractions.
But then, so are numbers, for all their illusion of concrete specificity and precision. This is counterintuitive for most of us. Five is five, 10 is 10, 200 is twice 100. The square root of 36 is neither 5.999 nor 6.111 but exactly 6, always and forever. But all that is not as concrete as it seems. "Mathematical things don't exist in the real world," Riehl says. "There's not a perfect circle stored in a vault somewhere that is, like, the standard perfect circle. Even something relatively familiar like the natural numbers—0, 1, 2, 3, 4—what is a number, really? That's actually a very deep and complicated question."
"Two" doesn't seem to pose a deep and complicated question: two kids, two shoes, two peas in a pod. What's so hard? But "two" itself is not a thing, it's a mathematical object. It's an idea that helps us think about other things. You can't pick up a two, throw a two, or eat a two because it doesn't exist. Yet it inheres in everything, everywhere that pairs exist. In an article she wrote recently for Scientific American, Riehl quoted John Horton Conway, an esteemed English mathematician: "What's the ontology of mathematical things? There's no doubt that they do exist, but you can't poke and prod them except by thinking about them. It's quite astonishing and I still don't understand it, despite having been a mathematician all my life. How can things be there without actually being there?"
Riehl lives in this liminal space where the most abstract thinking creates new knowledge of immaterial things that are the blueprint of all that is real. She says, "One of the most personally difficult things about being a mathematician is it's hard to share the stuff that you're so excited about with the people in your life. There's just a lot of background necessary to understand the research questions that I'm interested in. It takes a long time to explain."
But it's not every day that people turn math on its head. So, let's give it a try.
First, a few concrete things about Emily Riehl. She is in her mid-30s. She was born in Thousand Oaks, California, and moved several times with her family, ending up in Bloomington-Normal, Illinois. She has degrees from Harvard, Cambridge, and the University of Chicago. She joined the Johns Hopkins faculty in 2015. She plays the viola and guitar, as well as bass in a Boston-based band named Unstraight, which describes itself as "all women, all queer, and all you ever wanted." She is skinny and not very tall, about 5-foot-4, but for seven years played Australian rules football, a violent contact sport that resembles rugby with 36 players running around and colliding on an oval field. She is a co-founder of Spectra, an association for LGBT mathematicians. She has already written three books. Important papers frequently come from mathematicians in their 30s, but not important books.
Her most recent, co-written with Dominic Verity of Macquarie University in Australia, is Elements of ∞-Category Theory, scheduled for publication by Cambridge University Press in February 2022. The authors spent years building on their own research and the work of others to write the first comprehensive explication of infinity category theory (we'll get to what that means), aimed at students and mathematicians alike, that is "model-independent" (we'll get to what that means, too). Clark Barwick, a professor at the University of Edinburgh and one of the mathematicians whose work informs the book, has praised it for "bursting at the seams with gorgeous insights."
Last January, Riehl became the sixth recipient of the $250,000 President's Frontier Award, which supports Johns Hopkins researchers poised to become leaders in their field. To several of her colleagues, Emily Riehl isn't poised to become a leader—she's already there. Her collaborator, Verity, says, "I would certainly rank Emily amongst the top two or three mathematicians to arise in category theory in the past 15 years. Her influence extends well beyond the consummate technical sophistication that she brings to her research. She is a brilliant mathematical expositor with three major books under her belt and an enviably strong record as an invited speaker." In his letter of support of her nomination for the Frontier Award, David Savitt, chair of Hopkins' Department of Mathematics, called Riehl "a force of nature for her ability to attract visitors and postdocs, offer sought-after seminars and graduate classes, and pursue a formidable research program all at once."
One more concrete thing about Riehl: She's more comfortable dealing with abstractions. As an undergraduate, she began encountering things that seemed intuitively obvious but somehow were hard to prove, and this made her uneasy. "So, in a geometry or topology course, let's say you're making some argument about a geometric space that you can visualize if it's in sufficiently low dimensions. You sort of use your visual intuition." To illustrate an example, Riehl draws a simple circle and a trapezoid on a chalkboard. "But there's a very famous theorem in topology called the Jordan curve theorem. You have a plane and on it a simple curve that doesn't intersect and closes—in other words, a loop. There's an inside and an outside to the loop." As Riehl draws this, it seems obvious enough, but here's the problem: No matter how much your intuition tells you that there must be an inside and an outside, it's very hard to prove mathematically that this holds true for any loop that can be drawn. She says, "I found that very uncomfortable—that there is something that feels so obvious but is somehow very deep. In algebra, by contrast, the things that you are proving aren't things that you encounter in the real world in any sense." No loops drawn on a blackboard or cubes or spheres or parallel lines. "They are things called groups and rings and fields, and those are words from the English language, but in math they don't mean anything concrete. It's an abstract number system. The abstract stuff can seem kind of dry, but for me, I dunno, I find it really compelling."
Riehl's mathematical path from Harvard to the present could be graphed as a curve toward ever-greater abstraction. After earning her bachelor's degree summa cum laude in 2006, she committed to a doctoral program at the University of Chicago. But she decided first to take a gap year and go to the University of Cambridge in England for a famous one-year master's program that's simply called Part III. While at Cambridge she made a fortuitous decision. "I had heard that the University of Chicago had a history with this thing called category theory because one of its two founders had been on faculty there. So I thought while I was in England I would take a category theory course."
What she sampled for the first time in England was a new form of metamathematics launched in the 1940s by two guys who had no idea that they were starting a revolution. In 1945, Samuel Eilenberg and Saunders Mac Lane (the latter was the one who ended up in Chicago) published a paper titled "General Theory of Natural Equivalences." In that paper, they introduced a new fundamental mathematical object that they called a category. A category was like a container of other mathematical objects. They also introduced functors, which are comparisons between different categories, and natural transformations, which are more abstract comparisons between different functors.
Understanding why all that matters requires a brief digression into formal systems. To write poems, you must work within the system we call poetry, a system that formalizes meter, rhyme scheme, the structure of poetic stanzas, etc. In similar fashion, to do math you have to work within a formal system of mathematical concepts and logical rules. Since the 1870s, the fundamental formal system has been set theory. In set theory, all mathematical objects are built from sets, and the elements of sets are other sets; all of this is accompanied by logical rules for proving theorems about sets. For 150 years, the theorems of modern mathematics have been derived from the framework of set theory.
Eilenberg and Mac Lane were studying algebraic topology, which turns shapes and surfaces into algebra to study them more rigorously. Says Verity, "They realized that in order to keep a clear head while doing this, they needed a more powerful and expressive type of algebra. And so, category theory was born." Categories, as defined by Eilenberg and Mac Lane, are collections of a higher order—all the sets that mathematicians meet in their daily lives can be considered objects in a single category, for example. And categories do more than just group things into new sorts of metasets. They also reveal new data about the objects in the category, such as how they relate to each other, including how they are equivalent.
That last word, equivalent, is key. The equals sign (the familiar =) first shows up in mathematical literature in 1557. But the concept of equality goes back at least to the ancient Egyptians and Babylonians, who understood basic addition (albeit with words, not numerals or signs). That means equality has been at the heart of mathematics for at least four millennia. Eilenberg and Mac Lane steered the discussion away from equality toward the concept of equivalence—mathematical objects that are not equal, per se, yet are the same in profound ways that revealed whole new avenues of exploration.
Explains Riehl, "The set of symmetries of an equilateral triangle and the set of ways three people can order themselves in a line are not the same, but they are mathematically equivalent. It's like the surface of a doughnut, this thing called a torus. There are different ways of constructing it." For example, picture a doughnut, not as the delicious breakfast food but as a stretchable abstract object shaped like a doughnut. In math, this shape is called a torus. Now, imagine pulling on the left side of the doughnut and stretching it thin until it looks like a handle. Then use your thumb to depress the top of the larger right-hand portion to form a bowl. Your doughnut is now shaped like a coffee cup. "Are the two equal?" Riehl asks. "No. Yet they are somehow the same object." Doughnut or cup, each is still a torus, regardless of their different uses at the breakfast table. They are mathematically equivalent in their torus-ness.
Eilenberg and Mac Lane did not mean to spark a radical new way of thinking about math, Riehl says. "They introduced some new terms and then made some sort of basic observations about those new terms. They thought what they were doing was providing this new technical language to describe things that you need technical language for. There were a lot of aspects of category theory that they didn't anticipate."
In the ensuing decades, mathematicians began working with this new thing, the category, and this new idea, equivalence. In so doing they created a revolutionary new approach to mathematics, category theory, that many see as supplanting set theory. Imagine if writers had spent 150 years representing the world only through basic description: This is a red ball. That is a 60-foot tree. This is a dog. Then one day someone discovers metaphor. Suddenly, our ability to find new ways to represent the world explodes, as does our knowledge of writing as a discipline. Category theory has the potential to do the same thing for mathematics. "Theorems proved 2,000 years ago are still true. But perspectives change," Riehl says. "The notion of equality in mathematics is way too restrictive. It's just very uncommon that you're going to find things that are actually equal, even when things are, in some very strong sense, the same. Category theory gives us a rigorous language for expressing the sameness between things that aren't literally equal."
Category theory pulls back from the granular details of sets or groups or geometric spaces to consider the relationships among them and how one maps to another (in mathematics that's called morphism). It facilitates the transfer of ideas from one area of study to another, and it gains new perspective by climbing to higher levels of abstraction. Working in the set theory framework allowed mathematicians to produce important detailed descriptions of trunks, ears, feet, and tails. Working in category theory let them see the elephants.
Riehl has said that she first realized she could have a career in math when she was 9 and watched Jeff Goldblum portray a mathematician in the film Jurassic Park. "I've been a math nerd my whole life," she says. "I mean, as long as I can remember I've been interested in math and patterns and numbers and sort of the whole thing. I remember being interested in patterns in the calendar—when certain dates would fall and whether that was a Monday or a Tuesday or a Wednesday. My elementary school library had these books called the Brown Paper School book series. There was The I Hate Mathematics! Book and Math for Smarty Pants [both by Marilyn Burns]. And then Sideways Arithmetic from Wayside School [by Louis Sachar]. I loved all of those and all sorts of logic puzzles."
Her schooling was ideal for a girl like her. "I had great math teachers all the way through. They would talk to me outside the classroom or over the summer and say, 'You could do this, and you could do this.' They just had a vision of this world that I didn't know yet, and they encouraged me to explore it, which was wonderful. I mean, absolutely wonderful." By her junior year of high school, she was attending classes at nearby Illinois State University; in her senior year, one of her math teachers went to Japan for a week and Riehl took her place, substitute teaching AP calculus to her classmates. "My teachers really gave me a glimpse of what math was like at the college level, the creative side of mathematics as opposed to the calculational side of mathematics. By the middle of high school, I was doing proof-based math courses and at that point I knew for sure what I wanted to do."
Always athletic, Riehl ran cross-country and played soccer in high school. Her first semester at Harvard, she attended an activities fair and pondered either playing ultimate frisbee or joining the rugby club. The rugby girls seemed more fun, so she became a rugby player. She continued playing through grad school in Chicago, breaking an arm along the way. Starting in November 2009, she spent six months on a fellowship in Sydney, which had become a center for work on category theory. While there, she got to try Australian rules football, what the Aussies call footy. Smitten with the game, she looked for chances to play once she returned to the States, eventually joining teams in Boston and Baltimore. In 2010, she flew to New York for the U.S. national team tryouts. "They were kind of desperate, so I made the team," she says. Every three years, Australia hosts an international tournament for national squads, and Riehl played for the U.S. in 2011, 2014, and 2017. During a second fellowship in Australia in 2017, she played intensely, working on her game five days a week. (Her collaborator Dominic Verity says, "I regard Emily as being an honorary Australian.") After that, she moved on from the game—the quality of play and coaching in Australia had spoiled her for playing in America. Now she mostly bikes and runs. A few days before one of our conversations, she ran a 20-mile race as training for an upcoming 50-mile ultramarathon.
Riehl took up the viola in second grade in Minneapolis and has played a stringed instrument ever since, these days mostly bass and some guitar. She has likened the viola to category theory: one creates a richer orchestral sound, the other a deeper mathematics. And like music, math has an aesthetic dimension for her. She gravitated to category theory in part because she found the proofs aesthetically pleasing. "Like a slick proof of a cool problem—I find that really beautiful. There are these classic analogies between math and poetry or math and music, and I think mathematicians do math in part because we think it's beautiful."
For the last several years, what has drawn Riehl's attention are theories that in significant ways go beyond category theory. If category theory is the study of equivalences, what happens when you explore a higher abstraction—equivalences of equivalences? One thing that happens is infinity category theory, the subject of Riehl's forthcoming book. As she has written, "If mathematics is the science of analogy, the study of patterns, then category theory is the study of patterns of mathematical thought."
Infinity category theory could be thought of as the study of the patterns of the patterns, however far out you want to go—a tool for pulling back through infinite levels of abstraction to see what more can be learned. To find even more connections and with mathematical rigor prove their validity. Cornell mathematician Inna Zakharevich, who has known Riehl since they were Harvard undergraduates, says, "In mathematics, there is always a thread of meta- mathematics: How do we do math? What does it mean to prove something? Understanding something at the root is a lot of what mathematicians want, and if you're going to chase down the roots of something, why not have it be math itself?"
Infinity category theory has developed in only the last 20 years, and much of the foundational work has been done by Jacob Lurie of the Institute for Advanced Study in Princeton, New Jersey, following pioneering work by André Joyal at the University of Quebec in Montreal. Lurie put it all in two books, Higher Topos Theory and Higher Algebra. Those are dense volumes of 944 and 1,553 pages, respectively, which poses a practical problem: To master and begin using the theory, mathematicians would need to invest years in the study of Lurie's work, and not many have open years on their calendars. Lurie's ideas could transform mathematics again, but for mathematicians excited to explore this new approach, there is no easy way in. And if you are a referee evaluating a paper that used the theory, how can you know whether it is valid without first slogging through those 2,500 pages? Do you just sign off on it, hoping for the best because you don't really understand its theoretical foundation?
Starting in 2012, Riehl began working with Dominic Verity on a new approach to Lurie's work that tried to make it more accessible by creating simpler proofs of the theorem. They collaborated on a series of papers that became the basis of the new book. A cornerstone of their work has been to make the theory independent of models. As Verity explains, "Unfortunately, it is not the case that there is one universally accepted definition of what an infinity category is. Instead, researchers share only a common intuition implemented rigorously in the form of a number of competing models. Our work seeks to unify these disparate models by providing a common framework in which to develop arguments that apply simultaneously to all models."
Riehl told Quanta Magazine: "Part of what we're trying to do is give a user-friendly rewrite of Lurie. The theorems are the same, the proofs are very different. There are some new insights. Mathematicians feel that they haven't really understood something until it seems simple. So, part of the dream for my field is that someday the community will understand it well enough that we can explain it to undergraduates."
With Elements of ∞-Category Theory completed, Riehl has branched out into a new area again. "That's one of the exciting things about being an academic; there's room to grow," she says. "My new area is something called homotopy type theory. The subject did not exist 20 years ago. It's not just new to me, it's kind of new to everybody." It works with mathematical types, which are not exactly sets and not exactly categories, to create, potentially, yet another foundation system. Steve Awodey, a mathematician at Carnegie Mellon who also works on the theory, says, "Very roughly speaking, one can say that set theory describes the world of mathematics as seen from a static, atomic, or discrete point of view. Homotopy type theory better describes the world of continuous variation. It is a rigorous logical system that can be used to construct error-free proofs which can be checked on a computer."
"We're not trying to unify all mathematics," Riehl says. "Math is just too broad to be captured in categorical language. There are lots of different explanations for why things turn out the way they do. I guess what we're looking for is the most distilled explanation—what we hope gets to the core of the matter."
The core of the matter. Sounds concrete and material, doesn't it? Then you remember that even something as familiar as the number 10 is actually an abstraction, and just like that you're back in the mind-spinning space where the real world and the Riehl world coincide.
Posted in Science+Technology
Tagged mathematics, category theory, infinity category theory