In the year 2000, the Clay Institute of Mathematics selected seven yet-unsolved “Millennium Prize” problems as the most important in mathematics, offering a million-dollar cash prize for each correct solution. The first verified solution, which was for the Poincaré conjecture in 2006, shocked the academic world and created one of the most infamous New Yorker cartoons of the past century. None of the remaining problems have been solved––or even credibly attempted––until now.
On February 14th, 2023, at the age of eighty-eight, American mathematician James Glimm released what he believes is a solution to a second.
For the past seven years, Glimm has dedicated himself to the Navier-Stokes existence and smoothness problem, one of the first steps to cracking the mystery of turbulence. The Navier-Stokes equations are a series of partial differential equations that have been used to describe the motion of fluid since the mid-1800s, but no one has ever proven that they always have smooth solutions.
By coincidence, I spent a significant amount of time with Glimm over the same period as he attempted to crack the Navier-Stokes. He and his wife have an apartment in New York on the Upper East Side, and they often host me in their spare room. During these trips, I liked to watch Glimm at his large mahogany desk in the morning, writing on his yellow legal pad or staring into space. Often, he wakes up in the middle of the night to keep working.
When Glimm and his wife, Adele, got married over sixty years ago, she did not realize how much time he would dedicate to his work. And yet, she is not surprised that he has continued pushing so hard and for so long. She says that Glimm is driven by a desire to impact the world, to add to the sum of humanity’s knowledge.
When I asked Jim what motivates him, he had a simpler response.
“I want to get the answer.” And then a pause.
“I don’t have the answer.”
What Makes a Genius
James Gilbert Glimm is one of the most accomplished mathematicians of his generation. Glimm is the winner of the 2002 National Medal of Science, considered the highest honor possible for an American mathematician. The sum of an academic career is hard to measure, but he has an h-index—a rough measure of research output—on par with most Nobel Prize winners. At academic conferences, other scientists used to pull his young daughter aside to ask her variations on the same question. Is he real? They wanted proof, from his own flesh and blood, that such an incredible mind woke up and put on clothes in the morning like everyone else.
The label of genius has been, for lack of a better word, democratized in recent years. Every child can be a genius, for a fee. Every fast-talking twenty-something with a software product is publicly declared a genius––at least until a newer, younger genius comes along. At the same time, an increasing number of people would deny that innate ability exists at all. When I was younger, my dad used to try to prove to me that all variations in human achievement could be explained by hard work. He bought me every pop psychology book on expertise––like Malcolm Gladwell’s beach read Outliers, which famously declared that anyone can become an expert at anything in ten thousand hours. I never believed him about talent or genius––if only because a certain class of minds had always seemed so different from my own.
Glimm has one of those minds. He never forgets anything. Once, when he discovered that his granddaughter was writing a dissertation on Irish poetry, he began quoting entire stanzas of the poet William Yeats back at her. And he learns quickly. Before presenting his work at international conferences, he used to read foreign newspapers—in Russian, German, Italian, or French—and then turn around and give his talks in the native language. As an undergraduate, Glimm was constantly accused of cheating because his mathematics tests never showed any work—save for all of the correct answers, neatly circled.
The word genius comes from the Latin genere, which means to beget or give birth to. Certain definitions associate genius with creativity, or output over time, but the phrase has always implied the existence of individuals whose exceptional capacity is determined at birth. The fixed nature of genius, or any form of giftedness, is the source of much contemporary unease. Major newspapers and tabloids alike are eager to celebrate the outputs of genius, but many deny the insinuation––innate in the term––that some people are simply born more able than others.
Before I knew Glimm as a mathematician, he was a family friend. I met him when I was nineteen; my own grandfather had died a few months prior. I don’t think he thought much of me at the time. Over the next few years, however, we developed a quiet rapport. It is strange to connect with someone who is both so grounded and yet exists on another plane. One night, when I was jet-lagged, I woke up early and found him at the kitchen table, notepad in hand, sitting exactly as he had been five hours before when I had gone to bed.
Occasionally, as a game for myself, I would try to slip a half-truth by him. He always won. Once, when I tried to pitch him a start-up I was working for, Glimm tore apart the business model until I admitted that I had no faith in the company either, and had only taken the job for cash while I was traveling. “Oh,” he said. “That makes sense.”
Glimm himself never sold out for money or prestige. His family says that he never cared about academic prizes, but he won them all anyway. It is a bit unfair to hold him up as a standard of a meaningful life, as I often do, because he has succeeded, always, by simply being more intelligent than other people. I learned much from that formative period with him. He gave me a strong sense of my place in the world––whatever I did with my own career, it probably was not worth worrying about too much, because someone like James was out there, pushing the world forward. And yet, watching him at his desk, day in and day out, was an important reminder that it is possible to produce good work, and to live a good life, and that you should always do the best you can with what you have.
He takes his obligation as a genius seriously. Whenever Glimm has had the opportunity to take the easy way out or rest on his considerable laurels, he always eschewed it to continue pushing the limits of his own ability.
A Normal Life
Glimm’s early life was unusual only in that it was so usual. He was born in 1934 in Peoria, Illinois, and grew up as the beloved youngest of three to an engineer father and stay-at-home mother. He was popular with the local girls because he was handsome; in the summers, he worked as a lifeguard. Even today, Glimm is quite dapper, and not just by the standard of mathematicians—he wears loafers and pocket squares, and holds Adele’s arm when they go out to dinner.
He lacks much of the anguish typically associated with extreme intelligence. When the time came to leave for college, Glimm made the simple decision and chose Columbia University because his father had gone there.
While at Columbia, Glimm was far from the typical nerd. He lived in the Alpha Delta Chi fraternity house on 114th Street, where he used his considerable engineering talent to make various repairs. Those who know him from college recall that he had friends that included Jews, Catholics, Protestants, slackers, intellectuals, and jocks alike.
As he was finishing his fourth year of undergraduate studies, Glimm had been hopelessly torn between engineering, physics, and mathematics. He was an engineer at heart but found the field insufficiently rigorous for his taste. For a while, he seriously considered pursuing physics. But the more he looked at physics, the more he realized he wanted to do mathematics––it was the most exact.
He decided to pursue a Ph.D. in mathematics, but remained at Columbia “without much thought” because he liked the university and was already there. He completed his doctorate in 1959 with a forty-page thesis titled On a certain class of operator algebras, which dealt with the classification and description of a novel category of C*-algebra, a space with operations (like addition and multiplication) that obeys certain properties.
In his personal life, Glimm similarly preferred to listen to his intuition. As a first-year graduate student, he found that most of the women––in keeping with the era––were more interested in finding a husband than in intellectual life. Once, he complained to a female friend that he was having a hard time meeting smart women, and she offered to introduce him to her friend, the bookish Adele.
Adele Strauss was an eighteen-year-old Barnard sophomore from Philadelphia. She was sweet and a bit shy—then, as today, she loved reading and has published both fiction and nonfiction. The two got coffee and immediately hit it off. Jim had found his match.
And as always, he was decisive. They were married within a year. Notably, neither family approved of the marriage, which was an unusual Protestant and Jewish mix. Still, the wedding proceeded on June 30, 1957, and proved a resounding success. In the sixty years that followed, they never left each other’s side, and remain each other’s best friend and closest confidant.
When Glimm was twenty-seven, he had his first major academic accomplishment: two theorems that were an outgrowth of his thesis work. His exceptional talent had already been noted in graduate school, but after those two publications, he became incredibly well-known in academic circles. In 1960, he published his first paper, A Stone-Weierstrass theorem for C*-algebras, which provided a partial answer to a major conjecture in functional analysis. And in 1961, he published a second that built on the same topic. He recalled that like any major breakthrough, there was “a certain controversy” associated with the publications (“people get jealous”). But the results were swiftly accepted.
Today, the uniformly hyperfinite algebras he explored in his early work are still known as “Glimm Algebras” in his honor. The young mathematician was off.
Depth Before Breadth
The great wisdom of his career, as Glimm likes to tell it, is depth first, then breadth. Do not waste time on a million directions at once—do one thing well, and then master another. Over time, you can have deep knowledge across a wide range of subjects. He is unusual, even among elite mathematicians, in that his research actually spans multiple academic disciplines. By contrast, most academics remain in the same narrow specialty for their entire career, particularly after a major accomplishment in their early years. Glimm could have supported a long trajectory in academia by riffing on his early discoveries with C*-algebras for decades.
But Glimm had a different idea. To the shock of his colleagues, he abandoned pure math altogether and pivoted to another field. He wanted to do physics.
After a stint at the prestigious Institute for Advanced Studies at Princeton, Glimm joined MIT as an associate professor of mathematics. He wasted no time in switching fields. By 1964, he had begun publishing on topics related to mathematical physics—specifically quantum field theory—which would become his focus for the next twenty years. However, he saw gaps in his knowledge and visited Princeton to learn more.
During his time at Princeton, Glimm encountered a young physics Ph.D. student named Arthur Jaffe––a protegé of the legendary mathematical physicist Arthur Wightman. The two spoke briefly after Glimm delivered a lecture, and then met again at a workshop a few months later. The next year, Jaffe invited Glimm to a workshop at Stanford over the summer.
That chance encounter would lead to one of the most productive partnerships in modern mathematics history.
The stereotypical genius works alone, either because they are antisocial or believe they have nothing to learn from other people. But Glimm has always been quite practical: he will engage with anyone who can make a worthwhile contribution. Over that summer in California, Glimm and Jaffe spent hours talking about mathematical physics. They published their first joint paper in 1968. After a few initial successes, they would go on to work side by side for over two decades—an incredible feat in the typically cutthroat world of academia, where the most prolific scholars tend to keep to themselves.
When I asked Glimm why he and Jaffe worked so well together, he had no particularly profound response. They simply had complementary skill sets––and both saw that they could achieve more together than alone. They co-authored dozens of papers together, which have received over six thousand citations.
Together, Glimm and Jaffe pioneered a new field called Constructive Quantum Field Theory, which sought to show that quantum field theory is compatible with special relativity. Since the 1940s, physicists had observed that fields, not particles, appeared to be the basis of reality. The hypothesis was predictive in high-quality experiments. There were, however, some holes—the fields could not be modeled using conventional mathematics. Instead, physicists were using a convoluted process called “renormalization” to erase inconvenient infinities in their calculations––a tortured procedure that called the legitimacy of quantum field theory into question. Many mathematicians who held experimental data in low regard were skeptical that the new fields could ever be defined.
Glimm had an intuition that the physicists, not the mathematicians, were right about quantum field theory. They just needed more mathematical support.
Working together, Glimm and Jaffe developed new mathematics to show that quantum fields could be defined exactly—one write-up described their task as analogous to Newton developing calculus to explain planetary motion and gravity.
When they first started working together, Glimm was still stationed at MIT, while Jaffe was a young professor at nearby Harvard. As Jaffe struggled to adjust to the harsh Boston winters, Jim and Adele would often invite him over for a warm dinner—Jaffe would later return the favor when he could. The professional association blossomed into a personal friendship that continues to this day.
When Glimm moved back to New York University at the end of 1968, they vowed to continue their partnership despite the distance, working “remotely” as we would call it today. Except there was no video calling or even fax machines by which to send complicated equations. Instead, the partners corresponded via photocopies and long letters––for extra postage, a letter sent from Cambridge could be delivered to New York by the next morning. Whenever Jaffe had the opportunity, would spend a few nights on the couch of the Glimms’ Brooklyn Heights apartment. Later, when they got more space, Jaffe was upgraded to a guest room.
One of the most important findings of their joint work was that the dimension in spacetime, d, is crucial when defining a field. This proved both a valuable insight and an unexpected impediment. Glimm and Jaffe’s work was confined to three dimensions; today, establishing the same results for d = 4 remains the most significant unsolved problem in mathematical physics.
They continued publishing together until the mid-1980s, before parting ways after two decades of collaboration. In an unusual admission of sentimentality, Glimm acknowledged that he stayed in the working relationship for a little longer than he should have after they stopped seeing results.
After their collaboration ended, both partners went on to become influential members of the mathematics community. By the early eighties, Glimm was solving major problems in physics by himself. And although Glimm is quite humble about his achievements, his family did recall that some of his colleagues—particularly those who had dedicated their lives to physics—were a bit miffed when he eventually proved to be the superior practitioner. As his academic star rose, Glimm retained an extremely conservative public presence. He only has one full lecture available online, an obscure talk at a Brazilian mathematics institute. To this day, he has no personal website and a matter-of-fact Wikipedia page.
Jaffe never left Cambridge, and rose to become the chair of Harvard’s Mathematics Department. And the partnership may soon complete a full circle. In 1998, Jaffe co-founded the Clay Institute, which today is best known for its seven Millennium Prize problems.
Genius is often not revealed in its own time. In 1834, the physicist Sadi Carnot, now recognized as the father of modern thermodynamics, died with a single citation and no professional recognition. Franz Kafka lived and died as a failed insurance officer in Prague, leaving the vast majority of his writing unpublished in a desk drawer. It is a common fallacy to compare mainstream contemporary culture with the unearthed excellence of the past and find it lacking. But while Kafka was alive, no one––even in his direct vicinity––knew they were in the presence of a literary genius. For every generation, there is a formula for how to look like a genius. But the contributions that are remembered are not always the ones that make themselves so obvious.
The world of academic ladder-climbing never interested Jim. He wanted to continue solving important problems. In 1989, he left NYU for the last time to join Stony Brook University as a distinguished professor in Applied Mathematics and Statistics.
Glimm’s new department was unranked and had produced few notable alumni—an unorthodox move for one of the most desirable, and hireable, researchers on the market. However, Stony Brook offered him what no other institution could, which was the opportunity to build and lead a research group. By the time Glimm stepped down as department chair, the same applied mathematics department had improved rapidly and was ranked seventh in the country by the National Research Council. Today, they are considered one of the premier research groups in the nation.
In 2002, Glimm received the National Medal of Science—an illustrious honor earned over decades of work and “a mixture of merit and politics.” He waited all day at the White House to receive his award. Then-president Bush was busy with some crisis or other, but he eventually returned to his office to place the medal around Glimm’s neck himself. Photos from the day show both men beaming with joy. And yet, future researchers are unlikely to have the same experience. Somewhat inexplicably, no National Medals of Science have been awarded since 2014. President Trump was the first not to give out the award since its inception in 1957, and President Biden has yet to bestow his.
Today, Glimm keeps his medal in a cabinet above his kitchen counter, safely in its original box. He handles the velvet with care when he opens it to show me. Of all of his prizes, that one––an honor from his country––is his favorite.
The award was never formally discontinued. Recent administrations seem to have just forgotten about it.
What is lost when we, as a society, do not support our geniuses? The most oft-cited answer is technology. Much of Glimm’s work actually does have immediate applications––for petroleum engineering, shockwave calculations, and the modeling of earthquakes. Practicality aside, however, there is something distinctly human about the search for knowledge without reason, about trying to comprehend a universe that simultaneously does not need us and yet feels built to be discovered. When I was a teenager, I used to think that science had an endgame––maybe if we learned all of the math, then we could build a big key that unlocks the universe and escape. Now, I think that if intelligent life has a purpose, it is simply to witness the world as it is, for as long as possible.
It would be awfully sad for so much of reality to go on, for billions of years, without a single chronicler. We all live here. It feels wrong not to try.
A Final Push
When I went to interview Glimm last November, I found him sitting at his desk, writing on his yellow notepad with his back facing the street. The paper with the Navier-Stokes proof had been near-complete for months. But there was a significant gap between almost done and fully done. His team discovered a new issue every week. Still, even at the time, he did not seem particularly concerned. He loves his life. Glimm has eight grandchildren now and has greatly enjoyed watching them grow up––although none have shown his aptitude for mathematics.
We rely on so few people to push the limits of what is possible. It is a meaningful pursuit. And yet, there is a role for all of us to play by keeping the rest of society on track while they work––as a parent, spouse, collaborator, or even a friend.
At the end of our conversation, I asked Glimm if there is room, in our conception of intellectual achievement, for an amorphous concept like genius. Perhaps mathematical excellence is all hard work, the cumulation of years of exceptional effort. I expected him to demur or, in typical engineering fashion, to block the admittedly imprecise question with another: what does “genius” even mean anyway?
But to my surprise, he responded without hesitation. “No, I think you need to be a genius. There is room for genius.”
Then he thought a little more. “But you still need to work hard.”
“A hard-working genius is a good idea.”