### Louis Nirenberg

Born 28 February 1925, Hamilton, Ontario, Canada

Abel Prize (2015, together with John Nash) “for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis”.

Louis Nirenberg’s father worked as a Hebrew teacher. Feeling that his son should learn Hebrew as well, he found a teacher for him. However, this man also liked mathematical puzzles and so he infected the young Louis Nirenberg in the Hebrew lessons with a passion for mathematics.

Louis Nirenberg did then undergraduate studies at McGill University (B.A. 1945). He initially planned to specialize in theoretical physics, but he came by chance into contact with Richard Courant who had established the mathematics department at the New York University some years earlier. Courant offered Nirenberg an assistantship in New York and in 1947, Nirenberg got his masters there.

For his Ph.D. thesis, his advisor James J. Stoker suggested him to solve a geometrical problem which Hermann Weyl had already worked on about 30 years ago – with no complete success. Nirenberg gave a complete solution. He finished his thesis in 1949 to stay at the NYU as a research assistant and eventually become a member of the faculty. In 1951/52 Nirenberg received a fellowship to visit Zurich (Switzerland) and Göttingen (Germany). In 1957, he became a full professor at the NYU and retired there in 1999. Between 1970 and 1972 he was director of the math department at the NYU (which is called the Courant Institute since 1964).

Louis Nirenberg has received many honors and awards, among others the Crafoord Prize of the Royal Swedish Academy (1982), the National Medal of Science (1995), the Leroy P. Steele Prizes of the American Mathematical Society both for Lifetime Achievement (1994) and for Seminal Contribution to Research (2014), and the Chern Medal (2010).

He is a member of the National Academy of Sciences, the American Philosophical Society, and the American Academy of Arts and Sciences, and a foreign member of Accademia Nazionale dei Lincei, the Académie des Sciences de Paris, Accademia Mediterranea delle Scienze, Istituto Lombardo Accademia Scienze e Lettere, and the Academy of Sciences of Ukraine.

In his first years at the NYU, Louis Nirenberg was greatly influenced by his colleague Kurt Friedrichs, mainly while diving into the world of partial differential equations. Those equations play an important role in the description of many real world processes, for instance in physics, fluid mechanics or economics. Louis Nirenberg became one of the leading scientists of the 20th century by developing a deep understanding of nonlinear and linear partial differential equations. He is especially interested in a subset of them, the so called partial differential equations of elliptic type. In the last decades, Nirenberg helped develop the theory of these equations by investigating the regularity and solvability of elliptic equations of order 2n. (Regularity means here in essential that solutions of the equations are “smooth” functions.)

To construct differential equations, you need to differentiate functions. The “mathematical machine” for that is the differential operator. Nirenberg developed together with Joseph Kohn a generalization of that operator, the concept of a pseudo-differential operator. This became also a main tool for the investigation of the regularity of solutions of partial differential equations.

Additionally, Louis Nirenberg invented in 1961 together with Fritz John a new tool for harmonic analysis, the “functions of bounded mean oscillation”. Since then, the space of these functions became a standard tool of analysis.

Nirenberg did not only learn from Kurt Friedrichs a lot about partial differential equations, but also that inequalities are often mathematically more meaningful and interesting than equations and identities. He once confessed: “I love inequalities.” Indeed, he solved many problems by proving that certain inequalities have to hold, and several inequalities bear Nirenberg’s name today.