Gábor Szegő
Based on Wikipedia: Gábor Szegő
The Tutor Who Wept
In 1908, a fifteen-year-old boy was sent to study advanced calculus with Gábor Szegő, a young Hungarian mathematician who would later become one of the foremost analysts of the twentieth century. That boy was John von Neumann.
Szegő came home from their first session with tears in his eyes.
His wife asked what was wrong. Nothing was wrong, exactly. Szegő had just witnessed something that shook him to his core: mathematical talent so extraordinary, so blazingly fast, that it defied everything he thought he knew about how minds work. For the next several years, Szegő visited the von Neumann household twice a week to tutor the child prodigy. Some of von Neumann's instant solutions to calculus problems—scribbled on his father's stationery—are now preserved in the von Neumann archive in Budapest, artifacts of a mind that seemed to operate on a different plane of existence.
But this is not primarily a story about von Neumann. It's about the man who recognized that genius, nurtured it, and went on to fundamentally reshape our understanding of polynomials and matrices—the mathematical structures that now underpin everything from quantum mechanics to Google's search algorithm.
A Boy from the Hungarian Plains
Gábor Szegő was born on January 20, 1895, in Kunhegyes, a small town on the Great Hungarian Plain in what was then Austria-Hungary. He came from a Jewish family; his father Adolf and mother Hermina raised him in a world that would soon be convulsed by war, revolution, and eventually the Holocaust.
In 1912, Szegő enrolled at the University of Budapest to study mathematical physics. But he didn't limit himself to one institution. During summers, he traveled to the University of Berlin and the University of Göttingen—then the absolute center of the mathematical universe—where he attended lectures by giants like Ferdinand Georg Frobenius and David Hilbert.
Think about what Göttingen meant in that era. It was where the foundations of modern mathematics and physics were being laid, brick by brick. Hilbert was systematically trying to put all of mathematics on rigorous logical foundations. The atmosphere must have been electric: young mathematicians from across Europe converging on this small German university town, arguing in cafés, scribbling proofs on napkins, sensing that they were part of something historic.
In Budapest, Szegő studied under Lipót Fejér, Manó Beke, József Kürschák, and Mihály Bauer. More importantly, he met two young mathematicians who would become lifelong collaborators: George Pólya and Michael Fekete. These friendships would prove intellectually fertile for decades.
War Interrupts Everything
In 1915, World War One interrupted Szegő's studies. He served in the Austro-Hungarian military, cycling through the infantry, artillery, and air corps. The war killed millions and shattered empires. It also scattered the mathematical community that had been flowering in Central Europe.
But even war couldn't completely stop Szegő's mathematical work. In 1918, while stationed in Vienna, he was awarded a doctorate by the University of Vienna for his work on Toeplitz determinants.
What are Toeplitz determinants? To understand them, you first need to understand what a matrix is. A matrix is simply a rectangular array of numbers arranged in rows and columns—think of it like a spreadsheet. Matrices are extraordinarily useful because they can represent systems of equations, transformations of space, and countless other mathematical relationships.
A Toeplitz matrix is a special kind of matrix where each descending diagonal from left to right contains identical values. Imagine a matrix where if you pick any entry and move one step down and one step right, you always find the same number. These matrices are named after Otto Toeplitz, a German mathematician who was Szegő's contemporary and whose work Szegő built upon extensively.
A determinant is a single number calculated from a matrix that captures essential information about it—for instance, whether the transformation represented by the matrix preserves, expands, or flips the orientation of space. Toeplitz determinants, then, are determinants of Toeplitz matrices, and they turn out to have deep connections to signal processing, probability theory, and quantum physics.
The Wandering Scholar
After the war, Szegő's career followed the typical path of a European academic of that era. In 1919, he married Anna Elisabeth Neményi, a chemist, and they would have two children together. In 1921, he received his Privat-Dozent (essentially a license to teach at the university level) from the University of Berlin, where he remained until 1926.
That year, he was appointed as successor to Konrad Knopp at the University of Königsberg, in what is now Kaliningrad, Russia. Königsberg was historically significant in mathematics—it was the city of Immanuel Kant and of the famous Seven Bridges problem that launched the field of topology.
But the 1930s brought catastrophe. The Nazi regime came to power in 1933, and for a Jewish mathematician in Germany, working conditions became, in the dry phrase used by biographers, "intolerable." This understates the reality. Jewish academics were stripped of their positions, humiliated, and eventually murdered. Szegő was fortunate enough to escape.
In 1936, he secured a temporary position at Washington University in St. Louis, Missouri. Two years later, in 1938, he was appointed chairman of the mathematics department at Stanford University in California. He would remain at Stanford for nearly three decades, helping to build the department into a world-class institution.
The Mathematics of Sequences and Approximations
Szegő's mathematical work centered on analysis—the branch of mathematics that deals with limits, continuity, derivatives, and integrals. If you've taken calculus, you've dipped your toe into analysis. But analysis at the level Szegő practiced it is vastly deeper and more abstract.
His most important contributions involved two related areas: orthogonal polynomials and Toeplitz matrices.
A polynomial is an expression like x² + 3x - 7: a sum of terms where a variable is raised to various powers and multiplied by coefficients. Polynomials are among the simplest and most well-behaved mathematical objects, which is why mathematicians love to use them to approximate more complicated functions.
Orthogonal polynomials are families of polynomials that are "perpendicular" to each other in a precise mathematical sense. The word "orthogonal" comes from the Greek for "right angle." Just as perpendicular lines in geometry have a special relationship—they intersect at ninety degrees—orthogonal polynomials have a special property: when you integrate their product over some interval, you get zero (unless they're the same polynomial).
Why does this matter? Because orthogonal polynomials provide an incredibly powerful tool for approximating complicated functions. Instead of trying to work directly with some messy, difficult function, you can express it as a sum of nice, well-understood orthogonal polynomials. This technique underlies vast swaths of applied mathematics, from solving differential equations to processing signals to computing probabilities.
Szegő's 1939 monograph Orthogonal Polynomials became the definitive treatment of the subject. It contained much of his own research and influenced fields as diverse as theoretical physics, stochastic processes (the mathematics of randomness), and numerical analysis (the art of computing approximate solutions to mathematical problems).
Books That Became Bibles
Szegő was unusual among research mathematicians in that he wrote multiple books that became classics in their fields. Research mathematicians typically communicate through journal articles—dense, technical papers proving new theorems. Writing a book requires a different skill: the ability to synthesize a field, to explain it clearly, to create something that will train the next generation.
Szegő did both. He published over 130 research papers in several languages. But his four books achieved something rarer: each became the standard reference in its area.
With George Pólya, his friend from Budapest days, he wrote Problems and Theorems in Analysis, a two-volume work first published in 1925. This book took a distinctive approach: rather than presenting polished proofs, it presented problems arranged to lead the reader toward discovering the mathematics for themselves. It became famous for its pedagogical method.
With Pólya, he also wrote Isoperimetric Problems in Mathematical Physics (1951). Isoperimetric problems ask questions like: among all shapes with the same perimeter, which encloses the greatest area? (Answer: the circle.) These problems have ancient roots—they appear in Virgil's Aeneid—but also deep modern applications.
With Ulf Grenander, he wrote Toeplitz Forms and Their Applications (1958), which became the standard reference on Toeplitz matrices and their uses.
Recognition and Legacy
The honors accumulated over Szegő's long career. In 1928, he received the Julius König Prize from the Hungarian Mathematical Society and became a member of the Königsberger Gelehrten Gesellschaft (the Königsberg Learned Society). In 1960, he became a corresponding member of the Austrian Academy of Sciences in Vienna. In 1965, he was named an honorary member of the Hungarian Academy of Sciences.
He retired from Stanford in 1966 and died in Palo Alto, California, on August 7, 1985, at the age of ninety. Today, the Gábor Szegő Prize honors mathematicians working in his areas, and a primary school and mathematics competition in Hungary bear his name.
The Interconnected World of Early Twentieth-Century Mathematics
Szegő's story illuminates something important about how mathematics developed in the early twentieth century. It was a small world. The same names keep appearing: Hilbert, who Szegő heard lecture at Göttingen, was the most influential mathematician of the era. Toeplitz, whose matrices Szegő studied, was a contemporary and colleague. Pólya and Fekete, whom Szegő met as a student in Budapest, became lifelong collaborators.
And then there was von Neumann, the teenage prodigy Szegő tutored in 1908. Von Neumann would go on to help invent game theory, design the architecture underlying modern computers, and make fundamental contributions to quantum mechanics. The mathematics Szegő taught him—calculus and analysis—formed the foundation for all of it.
When Szegő wept after that first tutoring session, he wasn't just reacting to one extraordinary student. He was glimpsing the future of mathematics itself—a future that he would help create through his own work on the abstract structures that make so much of modern science possible.
The Deep Utility of Abstract Mathematics
It's worth pausing to appreciate how the mathematics Szegő developed—seemingly abstract work on polynomials and matrices—turned out to be practically important.
Toeplitz matrices appear whenever you have a process that depends on time in a consistent way. They're fundamental to signal processing: when your smartphone cleans up a noisy audio recording or a medical imaging machine constructs a picture of your internal organs, Toeplitz matrices are doing the work behind the scenes.
Orthogonal polynomials are used in numerical integration (calculating areas under curves), in solving differential equations, in quantum mechanics, and in statistics. The methods Szegő developed and systematized are coded into the mathematical software that engineers and scientists use every day, usually without knowing whose insights make their calculations possible.
This is often how mathematics works. Someone struggles with abstract problems that seem to have no practical application. Decades later, those problems turn out to be exactly what's needed to solve real-world challenges that hadn't even been imagined when the original work was done. Szegő couldn't have known that Toeplitz matrices would be essential to digital signal processing—the field didn't exist yet. But he worked out their properties anyway, because they were mathematically beautiful and interesting.
Escape and Continuity
Finally, Szegő's story is a reminder of how much mathematical talent was nearly destroyed in the middle of the twentieth century. Hungary produced an astonishing number of brilliant mathematicians in the early 1900s: von Neumann, Paul Erdős, John Kemeny, George Pólya, and many others. Many of them were Jewish. Many had to flee for their lives.
The Nazi regime didn't just murder millions of people. It scattered a mathematical community that had flourished for generations in Central Europe. Budapest, Vienna, Berlin, Göttingen—these had been the centers of mathematical innovation. After the war, that center shifted decisively to America, in large part because America took in refugees like Szegő.
At Stanford, Szegő helped build a department that would become one of the best in the world. His doctoral students, including Paul Rosenbloom and Joseph Ullman, continued his work and trained the next generation. The mathematical traditions that had developed in Central Europe didn't die—they transplanted themselves to new soil and continued to grow.
In this sense, Szegő's tears at that first meeting with von Neumann were prescient. He was witnessing not just one remarkable mind, but the beginning of a transformation in where and how mathematics would be done. He would live long enough to be part of that transformation himself.