Louis Nirenberg died in New York City on January 26, 2020 at the age of 94.
He was a leading mathematician, whose fundamental contributions in the field of partial differential equations were hugely influential. This area of mathematics provides the language we use to describe—and the techniques we use to analyze—diverse problems from many fields, including geometry, physics, and engineering. Louis' earliest work, in the 1950's, solved two longstanding problems from geometry by proving new estimates for fully-nonlinear elliptic equations. Over the course of his long and productive career his achievements included the solution of many other important problems, and—equally significant—the introduction of many new ideas and techniques.
Louis was born in 1925 in Hamilton, Ontario, but grew up in Montreal. He studied Mathematics and Physics at McGill University, graduating in 1945, then came to New York University as a Mathematics masters student. The postwar years were a remarkable time for mathematics at NYU—his fellow students included Eugene Isaacson, Peter Lax, Joseph Keller, Martin Kruskal, Cathleen Morawetz, Harold Grad, and Avron Douglis. Louis remained at NYU for his entire career: after completing his PhD in 1949 with guidance from James Stoker and Kurt Friedrichs, he held a two-year postdoctoral position then joined the faculty in 1951. His title was Professor of Mathematics from 1957 until 1999, when he retired and became Professor Emeritus. He was Director of the Courant Institute from 1970 to 1972.
Louis' impact was partly due to his exquisite taste in problems. One very successful mode was to recognize, through specific challenges, the need for new tools or estimates. His ability to identify such challenges—and to find the required tools or estimates—was a major driver of his impact. His early work on problems from geometry had this character; other examples include his papers in the 60's with Joseph Kohn on problems from complex differential geometry, and those in the 80's with Haim Brezis on nonlinear elliptic equations with critical exponents.
A different, equally successful mode was to identify tools that were clearly important, then systematically explore their power. His work on the regularity of solutions of linear elliptic equations and systems had this character; it was done in the 50's and 60's with Shmuel Agmon and Avron Douglis. Another example is his work on the symmetry of solutions of nonlinear partial differential equations using the “method of moving planes” and the “sliding method,” developed in the 80's and 90's with Basilis Gidas, Wei-Ming Ni, and Henry Berestycki.
Louis also loved challenges—particularly ones involving estimates or inequalities—and this was the motivation for many projects. One example is his work on solutions of the incompressible Navier-Stokes equations, which describe for example the flow of water. We still don't know whether its solutions are smooth or not, so it is natural to ask about the size of the set where they are not smooth. The estimates Louis proved in the 80's with Luis Caffarelli and Robert Kohn remain the state of the art.
A gifted teacher and mentor, Louis was advisor to 46 PhD students (starting with Walter Littman in 1956 and ending with Kanishka Perera in 1997), and he also had a formative influence on many postdocs and collaborators.
Lively and gregarious, Louis loved music, art, and film almost as much as he loved mathematics. He maintained close friendships with many colleagues around the world. He loved to travel, and to host visitors. He particularly enjoyed working with others; as a result, almost all his papers were coauthored.
Louis received many prestigious awards, including the Abel Prize (2015), the American Mathematical Society's Leroy P. Steele Prizes for Seminal Contribution to Research (2014) and Lifetime Achievement (1994), the International Mathematical Union's Chern Medal (2010), the National Medal of Science (1995), the Canadian Mathematical Society's Jeffery-Williams Prize (1987), the Crafoord Prize (1982), and the American Mathematical Society's Bôcher Prize (1959).
He became a member of the American Academy of Arts and Sciences in 1965, a member of the National Academy of Sciences in 1969, and a Fellow of the American Mathematical Society in 2013. He was also a member of numerous honorary societies in other countries, including Accademia dei Lincei (1978), Accademia Mediterranea della Scienza (1982), Académie des Sciences (1989), Istituto Lombardo Accademia Scienze e Lettere (1991), Ukrainian Academy of Sciences (1994), and Norwegian Academy of Sciences and Letters (2015). He received honorary degrees from McGill University (1986), University of Pisa (1990), Université de Paris IX Paris-Dauphine (1990), McMaster University (2000), University of British Columbia (2010); and he was named Honorary Professor by Nankai University (1987), Zhejiang University (1988), and Peking University (2016).
Louis is survived by his son Marc, his daughter Lisa and her partner, Joseph Ganci, his grandchildren Jimmy and Alma, his sister Deborah, and his partner Nanette.
Additional information about Louis, including video, is available at the Simons Foundation's "Science Lives" website, via this link: https://www.simonsfoundation.org/2014/04/21/louis-nirenberg/
An autobiography is included in a recent book The Abel Prize 2013-2017 (H. Holden and R. Piene eds, Springer-Verlag, 2019, pp 379-389).