Joseph J. Kohn, who played a key role in extending the mathematics of calculus, died on Sept. 13 in Plainsboro, N.J. He was 91.
His death, in a hospital, was confirmed by his son, Eduardo, and by Princeton University, where Dr. Kohn had taught for more than four decades.
Calculus is an old subject, invented independently by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. The idea of what are known as complex numbers — a set of numbers larger than what most people encounter in math class — is even older, although for centuries even mathematicians did not know what to make of them.
Dr. Kohn studied how mathematical functions behave in the realm of complex numbers.
“Kohn was one of the giants of this particular subject,” Charles Epstein, a senior scientist at the Flatiron Institute in Manhattan, said in an interview.
This field is esoteric, even to many mathematicians. But the techniques that Dr. Kohn developed have found use in tackling a wide range of problems in mathematics as well as fundamental equations in physics, including Einstein’s theory of general relativity.
“It is kind of a universal model, and many, many, many equations, many different kinds of physical problems,” can be cast into this mathematical description, Dr. Epstein said. “Kohn’s work is very important for understanding what you do next.”
Complex numbers are indeed complex to understand.
People are most familiar with so-called real numbers: integers like one, two and three; fractions like one-half and one-third; and irrational numbers — like pi, the ratio of the circumference of a circle to its diameter, or the square root of two, the number that when multiplied by itself equals two.
But mathematicians wondered: What is the square root of minus one?
It is not minus one; the product of a negative number multiplied by another negative number is always a positive number. Since the square root of minus one could not be a real number, mathematicians invented what they called an imaginary number, i, which is defined as that square root.
When a real number and an imaginary number are added together, that is a complex number.
Mathematical functions involving a complex variable are more constrained and “have all these remarkable properties that the functions of real numbers didn’t have,” Dr. Epstein said. This led, in the 19th and 20th centuries, to a field called complex analysis — the calculus involving mathematical functions of a single complex variable. It is now a key tool for physicists and engineers in solving a multitude of problems.
In the second half of the 20th century, Dr. Kohn’s work provided fundamental insights into equations involving many complex variables, which are much more difficult to unravel.
“It completely changed the way people thought about the subject of complex analysis in many variables,” said John P. D’Angelo, an emeritus professor of mathematics at the University of Illinois Urbana-Champaign and one of 16 people to receive their doctoral degrees with Dr. Kohn as their adviser.
Born in Prague on May 18, 1932, Joseph John Kohn was the only child of Otto and Ema Kohn. His father was an architect.
In 1939, after Nazi Germany invaded Czechoslovakia, the family fled to Ecuador. The family emigrated again, in 1945, to New York City, where his parents set up a furniture design business.
Joseph received a bachelor’s degree in mathematics from the Massachusetts Institute of Technology in 1953 and a doctorate from Princeton University in 1956.
After working as an instructor at Princeton and then as a postdoctoral researcher at the Institute for Advanced Study in Princeton, N.J., Dr. Kohn joined the faculty at Brandeis University in Massachusetts. He was there for eight years. He returned to Princeton as a professor in 1968 and retired in 2008.
Dr. Kohn was a member of the American Academy of Arts and Sciences and the National Academy of Science.
In addition to his son, he is survived by his wife, Anna Rosa (Di Capua) Kohn, whom he married in 1966; two daughters, Emma and Alicia Kohn; and two grandchildren.
Dr. Kohn’s first major mathematical advances came in the mid-1960s, in the field of partial differential equations — a type of mathematics that describes physical phenomena, including the flow of heat and liquids, the propagation of light and the curvature of space-time.
Scientists often find it easy to write down partial differential equations. But finding exact solutions to those equations is generally impossible, except in a few special cases.
Collaborating with Louis Nirenberg, a mathematician at New York University, Dr. Kohn came up with a process, called microlocal analysis, that could produce approximate solutions. “You wrote down the answer in a sequence of steps, and you got a better and better answer each time,” Dr. Epstein said.
“This is what Kohn and Nirenberg really explained, how you could do in a pretty general setting,” he added. “It became a huge deal afterwards.”
The later work on the calculus of complex variables can also be used to study solutions of certain partial differential equations, which can be recast into mathematical functions of complex variables.
Dr. D’Angelo remembered another math problem that Dr. Kohn tackled, even though Dr. Nirenberg and Lars V. Hormander, another expert on partial differential equations, had said it was hopelessly difficult.
“Now, Joe’s work is absolutely fundamental in that area, but more important is his stubbornness,” Dr. D’Angelo said. “The two top people in the world had told him: ‘This is too complicated. No one will ever figure this out.’ And he started working on it and proving things.”
Dr. Epstein met Dr. Kohn when he was a postdoctoral researcher at Princeton and, he said, not fitting in. “I was miserable there, and nobody noticed at all except Joe Kohn,” Dr. Epstein said. “I didn’t work in his field. I didn’t know anything about super complex variables. He just decided, ‘There’s no reason for this guy to be so unhappy.’”
Dr. Epstein and Dr. Kohn became friends, and Dr. Epstein shifted his research to Dr. Kohn’s field. “Basically for social reasons, I went into the field of several complex variables,” he said. “I just like the people in the subject.”