Connecting the Family Tree of Mathematics: A Profile of Andrei Okounkov
To his colleagues, award-winning mathematician Andrei Okounkov’s is a “family man,” a moniker that can refer to both his parenting and his research, which brings together many branches of mathematics. As a graduate student, Okounkov contemplated his thesis problem while pushing his first daughter in a stroller on cold Russian winter nights.
His second daughter was born while he was a postdoc at the University of Chicago. “We had a desk with my computer on it and I had this little bassinet, and so I would move it with one hand and type with the other,” recalls Okounkov, who is a Simons Investigator in Mathematics and a Senior Fellow with the Simons Society of Fellows. “Now I think about it, how is it possible?”
Then again, Okounkov is used to ‘impossible’. Connecting disparate people and ideas is his specialty; he won a Fields Medal (often called the ‘Nobel Prize for mathematics’) for “his contributions bridging probability, representation theory and algebraic geometry,” according to the award statement. His oeuvre reflects an age-old ideal of mathematics, rarely realized since the Renaissance: that all mathematical fields are part of one big family tree and contain connections waiting to be discovered.
Okounkov’s thesis and several of his papers from the 1990s solved problems in a subfield called representation theory. Some of his technical ideas have since been used in fields ranging from differential geometry to number theory. In the 2000s, he proved famous conjectures in probability theory and mathematical physics; some of his work with two algebraic geometers and a theoretical physicist has become part of the foundation of enumerative geometry.
“What Andrei is really gifted at is drawing out those connections,” says his collaborator Davesh Maulik, who was a graduate student when he first met Okounkov in 2002. “He embodies that spirit in a way that is extremely rare — it’s a sort of thing that was common like 100 years ago, but as math becomes more specialized, it’s just a much harder thing to execute.”
Building the roots
Okounkov took a relatively circuitous route to pure mathematics, having started college as an economics major. At the time, Russian economics included political philosophy, and the young Okounkov thought he could apply economic ideas to bettering society. But after a peacetime stint getting drafted into the army, he changed his mind and switched to math, graduating in 1993.
“Certainly, it’s much easier to solve problems in math than to solve problems in society,” Okounkov says, the corners of his eyes crinkled by his ever-present smile. “I cherish the fact that math is, by and large, apolitical. The same theorem stays true, regardless of whether I agree or not with the person who proved them on any kind of political ground.”
During the economic chaos following the dissolution of the USSR in 1991, Okounkov found a job at the Institute for Information Transmission Problems in Russia while pursuing his doctorate. He worked in representation theory, which studies mathematical objects by analyzing their symmetries. For example, one could study the hydrogen atom by considering the movements of its single electron and the symmetries of those movements.
After finishing his thesis, Okounkov headed to the U.S. Serendipitously, his academic career lined up with his wife’s career in finance: when Okounkov began his postdoc at the University of Chicago in 1996, his wife, Inna Okounkova, enrolled in the MBA program there; his 1999 tenure-track position at the University of California, Berkeley coincided with her work in Silicon Valley during its dot-com boom; and moving from his tenured position at Berkeley to a tenured one at Princeton University in 2003 meant their young daughters could grow up in a quieter town while Inna Okounkova commuted to nearby New York City. When his eldest daughter started college at Princeton, the rest of the family moved in 2010 to the bustle of the big city, as Okounkov began his current position at Columbia University, where he is now the Samuel Eilenberg Professor of Mathematics.
In another happy accident, the mathematician Rahul Pandharipande (who would later become Maulik’s adviser) had an office next door to Okounkov’s in Chicago. When Pandharipande, an algebraic geometer, published a problem referring to a particular numerical series called Hurwitz numbers, Okounkov reached out. He had realized he could solve it using representation theory. Okounkov was especially keen to learn more algebraic geometry, the study of curves given as solutions to polynomial equations (for example, the parabola y = x2) and apply his representation theory know-how. Today, Okounkov and Pandharipande are each other’s most frequent collaborators, having published more than a dozen papers together over the past two decades.
In fact, Okounkov has published so many papers in algebraic geometry that one could easily assume he did his thesis in the field, says Maulik, now a professor of mathematics at the Massachusetts Institute of Technology. “The fact that there are multiple subjects that you might guess that that was actually what his background is in is a reflection of his kind of insane breadth.”
Growing different branches
While serendipity factored into his successful two-career family and fruitful collaborations, Okounkov would likely still have found success had things played out differently. Or, as he puts it, “as somebody who works in mathematical physics, I understand our world is kind of random. It’s kind of silly to try to maximize things.”
Mathematical physics, as its name implies, is a field of pure mathematics that deals with problems in physics. These problems are derived from the ‘real,’ or physical, world rather than the theoretical realms of much abstract mathematical research.
Mina Aganagic, another collaborator of Okounkov’s, calls mathematical physics “a very, very happy interaction of two fields.” Aganagic is a Berkeley physicist who works in string theory, a subfield of theoretical physics that replaces particles with ‘strings’ and seeks to explain relationships between quantum physics and regular mechanics. She says that “what string theory provides to math is a context, a sort of a road map.”
“If you have some abstract problem (and there are plenty of those in math), very often you either have no intuition about it, or you don’t have any way of gauging the importance and connections of that problem,” Okounkov explains. “Mathematical physics, amazingly and remarkably, has all kinds of context.”
Physics isn’t the only field touched by Okounkov’s work. One classical chemistry problem concerns the combinatorics of molecules, which involves counting the number of ways the molecules can glue together. Okounkov recognized that he could apply algebraic geometry to the problem with his collaborator Rick Kenyon.
“That was a big surprise, as people who have classically done algebraic geometry are used to working in some high dimensions and some mysterious domains,” says Kenyon, who works in combinatorics and probability theory. “They really are far from physics and far from the real world.”
Again, Okounkov’s tendency to forge his own path led to an unexpected connection within the mathematical family tree. In fact, the techniques he and Kenyon developed in their solutions led to exciting results in both probability theory and algebraic geometry, adding more blossoms to disparate branches.
The MNOP papers: growing a new branch
The physicist Nikita Nekrasov remembers meeting Okounkov at Princeton, where Nekrasov was a graduate student, when Okounkov was spending the spring of 1996 at the Institute for Advanced Study. Years later, in 2002, Nekrasov “just completely by chance” ran into Okounkov on a train platform outside of Paris, and the two struck up a conversation. Nekrasov was stuck on the final step of the proof of one of his conjectures, which he thought had to do with summing over Young diagrams, a technique used in representation theory.
“His eyes got kind of misty and dreamy, and he said, ‘Oh, how interesting,’” recalls Nekrasov, now a physicist at the Simons Center for Geometry and Physics in New York. “The next morning, he wrote me an email: ‘It’s very lucky that we met because summing over Young diagrams is what I do from dusk till dawn and from dawn to dusk. And so I might be able to help you with your conjecture.’”
That combinatorics help was key to wrapping up the proof of Nekrasov’s conjecture, which relates several parts of physics, the way Okounkov’s work relates branches of mathematics. The proof, which is directly related to the work by Kenyon and Okounkov, hinges on a phenomenon that pops up in different areas of math and physics: We can predict how random processes at a small scale can end up at a large scale. For instance, crystals grow when molecules glue together in a particular format.
“That conjecture was the stone which was covering the entry to a cave full of treasures,” Nekrasov says. “Once we dislodged that stone, we actually found that was just the beginning of a very beautiful story.”
Inspired by crystal growth and Okounkov’s work with several other mathematicians and physicists, Nekrasov and Okounkov started investigating three-dimensional Young diagrams. From this work they derived some well-known results, published jointly with Maulik and Pandharipande in 2006. These foundational ‘MNOP’ papers (named after the authors’ combined initials) set the stage for enumerative geometry, which studies the number of curves that satisfy various geometric conditions.
Those papers represent the next chapter in a saga that began in the 1970s and ’80s with separate publications by the mathematician Mikhail Gromov and the physicist Edward Witten. In string theory, the way each ‘string’ sits in space-time can be viewed as a complex curve and hence analyzed with the tools of algebraic geometry. In the 1990s, researchers combined Gromov’s and Witten’s results to form a theoretical framework to count those curves in different geometries. The MNOP papers conjectured that for a particular class of shapes, Gromov-Witten counts precisely correspond to what they call Donaldson-Thomas theory, which uses a completely different understanding of curves. That theory builds on a 1998 algebraic geometry paper by the mathematicians Simon Donaldson and Richard Thomas.
That surprising connection, in line with Okounkov’s motif of connecting disparate fields, has paved the way for an entire field seeking similar correspondences. The four co-authors won the prestigious inaugural Compositio Prize in 2009 for their paper, described in the award statement as “an outstanding piece of mathematical research.”
Teaching the future
In addition to his award-winning research, Okounkov also infuses his unique viewpoints into the teaching and service portions of his job. Henry Liu, one of his current graduate students at Columbia, chose Okounkov as an adviser after taking his introductory graduate course on Lie groups and representations.
“I was just really blown away at first by how confused I was,” Liu says, laughing. “Usually, for Lie groups, there’s a fairly canonical way to approach it pedagogically. Andrei did not take that approach. Eventually, I came to realize his perspective on all this stuff is very beautiful.”
During the COVID-19 pandemic, Okounkov put together a weekly Zoom summer school on enumerative geometry and geometric representation theory. He ended up continuing the series for a total of 33 lectures lasting until December 29, with videos and notes for the course posted online. This feat exemplifies Okounkov’s commitment to education. Melissa Liu, a colleague of his at Columbia who shares a grant for a research cluster with him, says, “He has done a great job educating the young generation, not only his own students and postdocs at Columbia but young researchers all over the world.”
One of Okounkov’s proudest achievements was co-organizing a semester-long mathematical physics program in 2018 with Aganagic and others at the Mathematical Sciences Research Institute in Berkeley. He calls this “the happiest time of my professional life,” when he was surrounded by colleagues from around the world, all focused on enumerative geometry problems.
Lately, Okounkov has kept busy co-organizing the International Congress of Mathematicians, which meets every four years and will take place in St. Petersburg in July 2022. His yearly visits to Russia over the summers have resulted in his supervising a lab focused on mathematical physics and representation theory, shared between two of the largest research universities in Russia.
Nearly all the world’s mathematicians have an entry in the mathematical genealogy project, a massive online family tree depicting advisers and advisees for the past several centuries. Okounkov, who has 10 descendants on the tree, is now seeing his own children join the world’s scientists. His younger daughter is readying herself for graduate school in physics, while his older daughter works at the Simons Foundation’s Flatiron Institute as a computational astrophysics research fellow. Years ago, Okounkov gave mathematical lectures at their middle school, reflecting his belief in the apolitical universality of the field.
“There are many, many deficiencies still, but in principle, math can be made absolutely free and absolutely universal,” Okounkov says. “Math can be absolutely democratic and very freely available to everybody. Everybody needs it.”
