Professor Hillel Furstenberg: In His Prime.
The gifted Hebrew University mathematician has accrued the field’s most coveted prizes while finding God’s Hand in the randomness of the universe.
Professor Hillel Furstenberg has spent a lifetime describing order in a seemingly random universe.
Which is why the manner in which the award-winning mathematician was introduced to his wife was so fitting.
His roommate from Princeton University, where Furstenberg was then enrolled in the mathematics doctoral program, had seen a young woman on a subway train reading a philosophy book – not the most common sight in 1957.
Rochelle Cohen had come from her hometown of Chicago to spend the year in New York City, where she was renting a room in Boro Park. A few months later, when Simchat Torah arrived, she made her way to a local shul to watch the men dance. As it happens, Furstenberg’s roommate was celebrating Yom Tov at that same shul, and recognizing the girl on the subway and thinking of his bachelor roommate, he approached Rochelle and said: “I know just the guy for you.”
True, a mathematician and a philosophy aficionada don’t necessarily seem like a match made in Heaven. “But I often say that one of the reasons she married me,” relates Furstenberg, the 2020 winner of the Abel Prize, the equivalent of the Nobel Prize of mathematics, “is because I convinced her that there’s beauty in mathematics. Beauty comes from the hidden, not the revealed.”
Decades later, a further indication of Furstenberg’s pursuit of order amid chaos came on the day we met. After scheduling and rescheduling our interview several times because of the coronavirus pandemic – Furstenberg is 84 and was not accepting many visitors – we met on the day the professor should have been in Norway accepting the most coveted prize in mathematics. If not for the coronavirus, he would have been in faraway Oslo, delivering a lecture on mathematics to an audience of his peers.
The Norwegian Academy of Science and Letters awarded Furstenberg, along with Yale University mathematician Gregory Margulis, the Abel Prize for “pioneering the use of methods from probability and dynamics in group theory, number theory, and combinatorics.”
If you don’t know what that means, join the club. The last time I took a math class was approximately 35 years ago (Algebra II) – and I believe I received a B-minus.
But Furstenberg, a multiple-time prizewinner in his field (including winning the Israel Prize in 1993 and the Wolf Prize in mathematics – Israel’s version of the Nobel Prize – in 2007), was patient with my numerophobia and over the course of two hours patiently guided me through the disjointedness of ergodic systems, prediction theory, and the Szemerédi Theorem. There is indeed order in a seemingly random universe, and Hillel Furstenberg’s life proves it.
In the Nick of Time
He was born in Berlin in 1935 to a religious, middle-class family, the son of a furniture store owner and a homemaker. Though both his parents were born in Germany, their parents emigrated from Poland and Russia. That fact proved to be critical to their survival. At the time, the Nazi government had targeted all Jews without generational ties to Germany for expulsion, and Furstenberg’s family was ordered to leave in November 1938, the month of Kristallnacht.
“One of my few memories from Germany is Kristallnacht,” says Furstenberg, who was three years old at the time. “I remember looking at the broken windows in our apartment. We lived right next to a shul and I remember standing there staring at the broken glass.”
Fortunately, his mother’s sister, who had immigrated to Israel years earlier, had enough money to sponsor the family’s move to England. Their intention was to stay there for a year, until they could immigrate to the United States, where his mother’s brother had established a poultry farm in New Jersey. During that time, Furstenberg’s father also sought treatment for a critical medical issue. Fearing that he would not be able to enter the United States with his condition, he underwent an experimental surgery in London – but the operation failed. “He’s buried in London,” Furstenberg says. “So my mother was a widow with two children on the run during World War II.”
They sailed to the United States in 1941, just as America was entering the war. His mother helped out on the poultry farm while young Hillel attended a public school in East Brunswick. In an indication of things to come, he skipped the first grade. Though his uncle was not strictly observant, there was a Shabbos atmosphere on the farm and Furstenberg vividly remembers the songs they sang at the Shabbos table, each sung in an orderly, yekkish fashion.
Life on the farm was just a temporary stop, however, and the family soon moved to Washington Heights, where there was already a large German Jewish community centered around Rav Joseph Breurer’s community, K’hal Adath Jeshurun. But Furstenberg’s family lived in the poorer, multiethnic section of Washington Heights, near Yeshiva University. He soon became a student at Yeshiva Rabbi Moses Soloveichik on 185th Street, which he attended through the eighth grade. During those elementary school years, he had already shown early promise in mathematics, and his sister, three years his senior, became his tutor.
“She was teaching me multiplication when my class was learning addition. I was always ahead of my class,” he says. “When you’re doing well in school, you tend to become interested in things. I’m not sure whether I had any kind of ambitions in mathematics at that time or had any plans about what I wanted to be.”
During this time, things were tough at home. His mother had found a job working as a sewing machine operator in a sweatshop but at a certain point it became clear that Hillel would have to learn a trade to support the family. He took a test and was admitted to Brooklyn Technical High School – the Harvard of New York’s public schools – to study electrical engineering. But after one day at the school, the rabbi of his local shul, Rabbi Irving Weinberg, convinced his mother that Furstenberg would have a brighter future in the rabbinate. She reluctantly agreed, and he was admitted to the Talmudical Academy, Yeshiva University’s high school. “It was clear to me that I was going to be a rabbi,” says Furstenberg, looking back instead on five decades of teaching mathematics at Hebrew University, along with stints at Princeton and the University of Minnesota.
In addition to his Torah learning, he took advantage of Yeshiva University’s large library to teach himself advanced mathematics.
In high school, in addition to his Torah learning, he took advantage of Yeshiva University’s large library to teach himself advanced mathematics. Working with a friend, he constructed a theorem in geometry and presented it to a professor at the university, Jekuthiel Ginsberg. Impressed, and recognizing Furstenberg’s difficult circumstances at home, the kindly professor provided Furstenberg with translation work to earn extra income – and gave him more math problems to solve on the side. He also connected him with Max Stern, the wealthy builder of Stern College, who saw young Hillel’s potential and provided him with a monthly stipend. Stern’s office was not far from Barnes and Noble bookstore on 14th Street, Furstenberg recalls, “and when I got the money I went directly to the bookstore and bought more math books.”
Almost a Rabbi
Hillel Furstenberg was on his way to becoming a rabbi at Yeshiva University when he first met the legendary Rav Yosef Dov Soloveitchik. Though a first-year student would normally not be admitted to the Rav’s Talmud class, Rav Soloveitchik’s brother-in-law, Professor Henry Lisman, who taught mathematics at YU, recommended Furstenberg to the Rav.
In reality, Furstenberg says, he was performing far better in mathematics than in Talmud, but nonetheless counts the two years he sat in the Rav’s class as formative.
That’s not to say it was easy. The class was in Yiddish, and although Furstenberg spoke German fluently, there were nuances that he missed. And then there was the Rav himself, who Furstenberg describes as quite intimidating.
“I was very much afraid of him, which was one of the reasons that I didn’t really learn as well as I should have,” he recalls. “He didn’t suffer fools easily, and if you made a comment that was out of place, he would let you know.”
At the time, rabbinical students could not pursue other fields of study – such as law or the sciences – while in the midst of their rabbinic studies. So Furstenberg had a decision to make: mathematics or rabbinical ordination?
“I chose mathematics and applied to MIT, Harvard, Princeton, and Yale,” he says. He was accepted at Princeton and was invited to an interview.
The year was 1955 and it wasn’t the most common sight to see a student at an Ivy League university wearing a yarmulke.
The year was 1955 and it wasn’t the most common sight to see a student at an Ivy League university wearing a yarmulke. So Furstenberg thought long and hard about how to present himself at the interview. “I thought, ‘I’m going out in the world, maybe I should take off my yarmulke,’ ” he remembers.
His mother had already suggested that he remove his yarmulke while walking around Manhattan. But in the end, he decided on a more contrarian path.
“I decided I’m not taking it off,” he recalls. “And one of the professors I interviewed with at Princeton was named Solomon Bochner. His father was a Torah scholar and he immediately took notice of me.”
The interview went well, but Furstenberg was worried about how he might maintain a Jewish life on campus. There was no kosher food, no minyanim, and no synagogue. He turned to Bochner with his concerns. “And he said, ‘Young man, someone like yourself should be able to handle this.’ And because of that, he sort of made it a challenge.”
What Happens Next?
The title of Furstenberg’s doctoral thesis, published in 1960, was “Stationary Processes and Prediction Theory.” In layman’s terms, it is a study of what we can surmise about future events based on past events. In other words, it is the search for order in seeming chaos. Furstenberg explains: “Assuming you knew everything that happened in the past and the past is infinite – for instance, that the world is created so many years ago and you have an infinite amount of information from the past – what can you say about what’s going to happen next? And it’s not just whether you can say exactly what’s going to happen; you can say with such and such probability that it’s going to happen.”
He gives an example of a person who’s been tossing a coin forever. “And you see heads, heads, tails, tails, sort of a random sequence. And then if you apply the procedure and the ideas that I was working on, you could say – assuming the coin is a true [balanced] coin – well, probability is half that it’s next going to be heads; or the probability is half that it’s going to be tails. Or if the coin is unbalanced you could say, it’s going to be more likely to be heads than tails,” he explains. “You can’t exactly know what’s going to happen but you can talk about probabilities. That becomes interesting – seeing patterns in what has been happening.”
Furstenberg’s first job was at Princeton, to which he commuted from his home in Highland Park, New Jersey. But then he was hired to teach at the University of Minnesota, where he spent four years. During one of those years, he returned to Princeton as a visiting professor. One day at a gathering, he met a young Orthodox professor of economics from Israel who was then a resident at one of Princeton’s many think tanks. His name was Yisrael Aumann and his specialty was game theory.
“Because of my yarmulke, we got to meet,” says Furstenberg of the future Nobel Prize winner Aumann, who was born in Austria, grew up in New York, and then moved to Israel to teach at Hebrew University. “And we decided we’re going to learn together. So we learned a chapter of Talmud in Tractate Shabbos. We decided we were going to do this seriously and give each other an examination after we’d finish the chapter. I still have a copy of the exam.”
Aumann was among the first to try to entice Furstenberg to move to Israel. His older sister, who was very active in Bnei Akiva, had already moved years before. Furstenberg was excited by the idea but didn’t want to leave his mother, who was by this time middle-aged and had established a life for herself in New York. She was hesitant, but sport that she was, agreed to go on a pilot trip to Israel in 1965.
The trip was successful and they decided to stay – a granddaughter, born to Furstenberg’s sister, helped – and Furstenberg and his wife moved to an apartment on Alfasi Street in Jerusalem’s Rechavia neighborhood, where they live to this day.
On the Way to Infinity
Professor Furstenberg is famous for several complex mathematical concepts, but for those of us who never got past elementary school arithmetic, the simple explanation of a basic premise is that although things look random from up-close, when you zoom out far enough you can see there’s actually order in everything – and with the right skills and tools, you can even predict how it will play out.
Although things look random from up-close, when you zoom out far enough you can see there’s actually order in everything – and with the right skills and tools, you can even predict how it will play out.
One of those concepts is now known as the Furstenberg Boundary. If one looked out at infinity, what would he see? “This is a variant of ‘the line of the horizon,’ far, far out, seemingly ‘at infinity,’” says the professor. “For the renaissance artist, for example, imagining the line of the horizon as an actual line on his canvas is an indispensable aid in his painting. The Furstenberg Boundary has also become a tool for answering questions about infinite groups. This ideal boundary originally arose in dealing with randomly combining elements of the group, and studying the expected behavior as more and more elements are combined. The different points on the boundary represent different modes of approaching infinity, just as the points on the horizon line represent looking far out in different directions.”
Furstenberg is also known for an alternative, more simplified proof of a difficult theorem, originally developed by the Hungarian mathematician Szemerédi. Szemerédi’s theorem asserts that any large set of integers will contain among its terms, a long arithmetic progression and that certain patterns will emerge inside a sufficiently rich structure.
“If Szemerédi’s hands-on approach could be compared to constructing a tall building with bare hands, I had at my disposal machinery comparable to using a crane,” he says. “The machinery here was ‘ërgodic theory,’ a branch of dynamics which also makes use of probability theory. The idea was that we try to see the arbitrary, but not insignificant, subset of numbers as being generated by some mechanism. While we don’t know the laws governing this ‘machine,’ there are laws governing all dynamical systems. In particular there is a phenomenon of ‘repeated recurrence,’ that things return to themselves at some point – the earth, for example, tilts at a certain angle to the sun repeatedly, every 365 days or so.” Furstenberg developed a principle showing that the recurrence phenomenon in dynamics implies Szemerédi’s theorem.
Furstenberg cites a story told by the mathematician John Nash (the subject of the book A Beautiful Mind) who had a reputation for being able to find patterns in random sets. One night he and his wife-to-be took a walk outside on a starry night. He asked her to randomly name an object and she said “umbrella.”
“So he looked up at the sky and showed her a formation of stars in the shape of an umbrella,” Furstenberg recounts, “which shows that you can find any pattern you want, or very close to it, even in the stars.”
Order from Chaos
All this talk about the stars, infinite spaces, and order in the universe has me thinking about Creation and God. Is there a way, through mathematics, to describe the miracle of Creation? And perhaps, further, to determine some kind of order in a world that often seems chaotic and random?
Furstenberg strikes me as a rationalist, and his answer to my question reflects that. There is no one grand formula that describes God and His Creation, but he does raise two ideas – the so-called Bible codes and gematria.
“You shouldn’t be surprised to find patterns in a sufficiently large text,” he says, referring to the codes. One might expect, therefore, to find the word “Torah” spelled out somewhere in some pattern.
“Somewhere in the text, you will find a tav and a vav and a reish and a hei along an arithmetic progression. For instance, if you take the first tav in Bereishit, and then you count 49 letters, you have a vav. Count another 49 letters, and you have a reish. Count another 49 letters and you have a hei. That already indicates that there is some sort of intelligence going on here. A lot of things that didn’t have to happen do happen. You could call that a little miracle.”
A similar principle applies for gematria. We shouldn’t be surprised, he says, to find equivalencies if there are a sufficient number of words at our disposal.
“It turns out that ‘Furstenberg’ in Hebrew and ‘matematekayei’ have the same gematria, 610,” he offers. “It also turns out ‘Zoom’ in Hebrew and the word ‘b’kele’ [in prison] also have the same gematria, 53. If the words are not too long, then you have to expect that some things will coincide. And some of those things will be meaningful.
“This isn’t a scientific proof that the Torah is Divine,” he adds, “but it can convince you that something’s going on here. You could say that God created a world in which this randomness exists. But it has a pattern. Seeming randomness has a pattern.”