Luis Ángel Caffarelli


Quick Info

Born
8 December 1948
Buenos Aires, Argentina

Summary
Luis Caffarelli is an Argentine mathematician who has won all the major prizes for his work in the field of partial differential equations and their applications. These major mathematical prizes include the Shaw Prize, the Wolf Prize and, most prestigious of all, the Abel Prize.

Biography

Luis Caffarelli's parents were Luis Caffarelli (born 18 March 1915 in Buenos Aires, Argentina) and Hilda Delia Cespi (born 9 September 1922 in Buenos Aires, Argentina). Luis Caffarelli Sr was a mechanical engineer who worked in the shipping industry in the Río de la Plata bay. Luis and Hilda had three children, Maria Luisa Caffarelli, Luis Ángel Caffarelli (the subject of this biography) and Alicia Caffarelli. Luis spoke about his early years in the interview [19]:-
We lived in a nice middle class area of Buenos Aires. At the time when I was a kid I would play soccer a lot with my friends. Another game we enjoyed a lot was to throw a ball, or something else, and see who got it closest to a wall or to a line.
He attended the Colegio Nacional de Buenos Aires which began its existence as a Jesuit school in the 17th century. As a National College it dates from 1863, and was incorporated into the University of Buenos Aires in 1911. By the time Caffarelli studied there it had gained an excellent reputation for both Humanities and Sciences. Excellent teachers of mathematics and science led to these topics becoming the subjects he most enjoyed. He graduated from the Colegio Nacional in 1966 and entered the University of Buenos Aires in March 1967. He wrote [12]:-
The military led the country at that time and many of the top scientists were abroad. Still, the remaining faculty and students at the university were very dedicated.
He spoke in [19] about his lecturers at the University:-
As a student I was heavily influenced and inspired by Luis Santaló (1911-2001), Manuel Balanzat (1912-1994) and Carlos Segovia (1937-2007). Santaló and Balanzat were both Spanish mathematicians who moved to Argentina as a consequence of the Spanish Civil War. Santaló made important contributions to integral geometry and geometric probability, while Balanzat worked in functional analysis. They built, jointly with Rey Pastor (1888-1962) and Pi Calleja (1907-1986), a superb undergraduate and graduate mathematics programme at the University of Buenos Aires, generating a very strong group in analysis, geometry and algebraic geometry. The harmonic analyst Segovia was a prominent graduate from Universidad de Buenos Aires who did his PhD at the University of Chicago in 1967 with Alberto Calderón (1920-1988). While closer to me in age than Santaló and Balanzat, Segovia was always a strong support.
For about a year and a half he studied both mathematics and physics but finally he chose to concentrate totally on mathematics. He was awarded his Master of Science degree in 1969 and continued undertaking research for a Ph.D. degree. He wrote [15]:-
I was trained in the Calderón school of real analysis and wrote my Ph.D. dissertation and some other articles on summability and conjugation of series of special polynomials.
We should say a little about the 'Calderón school'. There are two half-brothers named Calderón, namely Calixto Pedro Calderón and his much older half-brother Alberto Pedro Calderón. Both had the same father, Pedro Calderón, but after Alberto's mother died, Pedro remarried Matilde Garcia who was Calixto's mother. Calixto, born in December 1939, was 20 years younger than Alberto Calderón. He was awarded his doctorate in 1969 from the University of Buenos Aires, only two years before Luis Caffarelli who was his first Ph.D. student. In 1974 Calixto Calderón went to the University of Illinois at Chicago where he worked until he retired in 2000.

In 1971 Caffarelli submitted his thesis Sobre conjugación y sumabilidad de series de Jacobi to the University of Buenos Aires and he was awarded a Ph.D. in 1972. He undertook joint research with his thesis advisor Calixto Pedro Calderón and they wrote two joint papers: Weak type estimates for the Hardy-Littlewood maximal functions (1974-74); and On Abel summability of multiple Jacobi series (1974).

While in Buenos Aires he married Irene Martínez Gamba (born 1957) who is also an exceptional mathematician; they have three children Alejandro Caffarelli, Nicolas Caffarelli and Mauro Caffarelli.

In 1973, having been awarded a postdoctoral fellowship, he went to the University of Minnesota to work with Eugene Fabes and Calixto Calderón; he remained there until 1983. He said [56]:-
From the hot Buenos Aires summer to the Minnesota winter was a dramatic change, but with the generous help of Fabes and Walter Littman I got to love Minneapolis.
In 2009 Caffarelli spoke about his arrival at the University of Minnesota in 1973 [1]:-
I came to the United States to the University of Minnesota in January of 1973. There was no email, no fax, and even the telephone was very expensive. But I found at Minnesota and in the midwest an extraordinary group of people. My colleagues were extremely generous, dedicated, and friendly, and they taught me much of what I know. They shared their ideas and gave me guidance as I began my research programme.
After the Postdoctoral Fellowship which he held at the University of Minnesota in 1963-74, he was appointed as an Assistant Professor in 1975, an Associate Professor in 1977, and a full professor in 1979. He wrote in [15] about the research he undertook while at the University of Minnesota attempting to explain his high-powered results to a general audience:-
Shortly after my arrival, I attended a fascinating series of lectures by Hans Lewy and became interested in nonlinear partial differential equations, variations inequalities and free-boundary problems. An elementary example of this type of a problem would be a balloon inside a box (or a drop inside a cavity). If the balloon were suspended freely in the air, a first approximation to its shape would be given by a prescribed mean curvature equation (a mildly non-linear equation) that we could deduce from the fact that the balloon tries to minimise the energy of the configuration (an unconstrained variational problem). If constrained to lie inside the box, the surface of the balloon would behave differently when it is free than when it presses against the wall (a strongly nonlinear differential equation) creating a separation curve (the free boundary) between both regions. In this area, I investigated extensively the mathematical problems associated with solid-liquid interphases, jet and cavitational flows, and gas and liquid flow in a porous media.
Caffarelli's paper Surfaces of minimum capacity for a knot was published in 1975 but it was the following year that proved a remarkable one for Caffarelli in terms of publications for in that year six of his papers were published: Certain multiple valued harmonic functionOn the Hölder continuity of multiple valued harmonic functionsThe regularity of elliptic and parabolic free boundaries(with Néstor M Rivière) On the rectifiability of domains with finite perimeter(with Néstor M Rivière) Smoothness and analyticity of free boundaries in variational inequalities; and The smoothness of the free surface in a filtration problem. He continued his remarkable publication record with three papers in 1977, two of which were written jointly with Néstor M Rivière, and five papers in 1978, four of which were written jointly with Avner Friedman.

Although Caffarelli continued to hold his professorship at the University of Minnesota until 1983, he spent the two years 1980-82 as a Professor at the Courant Institute. This brought about a change in his research interests as he explained in [15]:-
In 1980 I was invited to join the faculty at the Courant Institute, where I developed new interests: fluid dynamics and fully nonlinear equations under the advice and in collaboration with Louis Nirenberg. A standing area of research we pursued was the three-dimensional Navier-Stokes flows (a model for the evolution of viscous, incompressible fluid flows) where we showed that the speed of the flow could become infinite at most on a set of zero one-dimensional measure (that is less than a curve) in space and in time. (A nearly optimal result, according to the recent examples of Shefer). We also extensively investigated the properties of hypersurfaces in n-dimensional Euclidean space for which elliptic relations among their principal curvatures are prescribed (i.e. mean curvature equation, Monge-Ampère equation in real or complex space, etc.).
In 1982 Caffarelli was awarded the Guido Stampacchia Prize by the Scuola Normale Superiore di Pisa (see [62] for details). This prize was jointly awarded to six mathematicians, Caffarelli and his co-authors, for the series of six papers: Brownian Motion and Harnack inequality for Schrödinger operators; Existence and regularity for a minimum problem with free boundary; Axially symmetric jet flows; Asymmetric jet flows; Jet flows with gravity; and Asymptotic behaviour of minimum problems with bilateral obstacles. It was the first of many awards and prizes given to Caffarelli.

In 1983 he was appointed as a professor at the University of Chicago, a position he held for three years. During these years he continued to receive major awards for his outstanding contributions. In 1984 he was awarded the Bôcher Prize by the American Mathematical Society [44]:-
... for his deep and fundamental work in nonlinear partial differential equations, in particular his work on free boundary problems, vortex theory and regularity theory.
In 1986 Caffarelli left Chicago when he was appointed Professor at the Institute for Advanced Study in Princeton. On 31 October 1988 Pope John Paul II presented Caffarelli with the Pontifical Academy of Sciences Pius XI Gold Medal. We have quoted several times above from the speech that Caffarelli delivered on that occasion. Let us quote here what he said about his current research interests (in 1988) [15]:-
My current interests include uniform estimates on singular perturbations when approaching a singular limit, for instance for the level surfaces approaching a surface of discontinuity and its implications on the stability of both the surface of discontinuity and numerical methods employed to simulate the limiting problem.
For ten years Caffarelli worked at the Institute for Advanced Study, then in 1994, because he missed working with graduate students, he returned to the Courant Institute where he spent three years. In 1997 he moved to the University of Texas at Austin where he was appointed to the Sid W Richardson Foundation Regents Chair. Awards continued to acknowledge the extremely high quality of his research contributions. In 2003 he was awarded the Premio Konex, Platino y Brillantes, Argentina [43]:-
... as the most important scientist of his country in the last decade.
The Swedish Academy of Sciences awarded him their Rolf Schock Prize in 2005 [52]:-
... for his important contributions to the theory of nonlinear partial differential equations.
The American Mathematical Society awarded him their Leroy P Steele Prize for Lifetime Achievement in Mathematics in 2009. The citation begins [1]:-
Luis Caffarelli is one of the world's greatest mathematicians studying nonlinear partial differential equations (PDE). This is a difficult field: there are rarely exact formulas for solutions of nonlinear PDEs, and rarely will exact algebraic calculations yield useful expressions. Instead researchers must typically invoke functional analysis to build "generalised" solutions for many important equations. What remains is the profound and profoundly technical problem of proving regularity for these weak solutions and, by universal acclaim, the greatest authority on regularity theory is Luis Caffarelli. His breakthroughs are so many, and yet so technical, that they defy any simple recounting here. But it was certainly Caffarelli's work on "free boundary" problems that first showed his deep insights. Free boundary problems entail finding not only the solution of some PDEs, but also the very region within which the equation holds. Luis Caffarelli's vast work totally dominates this field, starting with his early papers on the obstacle problem. ...
In 2012, he was made a Fellow of the American Mathematical Society. In the same year Caffarelli received the prestigious Wolf Prize in Mathematics [23]:-
... for his work on partial differential equations.
His work leading to the award of the Wolf Prize is described in [23]:-
Cafferelli has repeatedly made very deep breakthroughs. His early work on free boundary problems was the first place where his extraordinary talent and intuition began to show. Free boundary problems are about finding the solution to an equation and the region where the equation holds. In a series of pioneering papers, Caffarelli put forward a novel methodology which eventually leads, after several truly amazing technical estimates that step by step improve the regularity of the solutions and the boundary, to full regularity under very mild assumptions. Although the theory is complicated, the arguments are elementary and full of beautiful geometric intuition and mastery of analytic technique. A second fundamental contribution by Caffarelli is the study of fully nonlinear elliptic partial differential equations (including the famous Monge-Ampère equation), which he revolutionised. The upshot is that, although the equations are nonlinear, they behave for purposes of regularity as if they were linear. ... Another fundamental contribution by Caffarelli is his joint work with Kohn and Nirenberg on partial regularity of solutions of the incompressible Navier-Stokes equation in 3 space dimensions. Although the full regularity of solutions is still unknown and likely very hard, Caffarelli-Kohn-Nirenberg showed that the singular set must have parabolic Hausdorff dimension strictly less than one. In particular, singular fibres cannot occur. ... Caffarelli has also produced deep work on homogenisation and on equations with nonlocal dissipation. The list could be continued. Caffarelli is the world's leading expert on regularity of solutions of partial differential equations.
In 2018 Cafferelli was made a fellow of the Society for Industrial and Applied Mathematics (SIAM). The citation states that [47]:-
Luis Angel Caffarelli, The University of Texas at Austin, is being recognised for seminal contributions in regularity theory of nonlinear partial differential equations, free boundary problems, fully nonlinear equations, and nonlocal diffusion.
In the same year of 2018 he was awarded the Shaw Prize in Mathematics by the Hong Kong-based Shaw Foundation [8]:-
... for his ground-breaking work on partial differential equations, including creating a theory of regularity for nonlinear equations such as the Monge-Ampère equation, and free-boundary problems such as the obstacle problem, work that has influenced a whole generation of researchers in the field.
This prize is one of the world's largest mathematics prizes with a monetary value of US$1.2 million.

Caffarelli has received many honours in addition to those mentioned above. He has been elected to the American Academy of Arts and Sciences (1986); the National Academy of Sciences (1991); the Pontifical Academy of Sciences; the Argentina Mathematical Union; the Accademia Nazionale dei Lincei; and several others. He was invited to deliver the Fermi Lectures at the Scuola Normale di Pisa (1998) and was the American Mathematical Society Colloquium Lecturer in 1993. A summary of his research by the Pontifical Academy of Sciences is as follows [32]:-
Luis Caffarelli works in non linear analysis, mainly on non linear partial differential equations arising from geometry and mechanics. He has conducted extensive research into free boundary and singular perturbation problems. He has worked on free boundary problems that arise naturally when a constitutive relation or a conserved quantity (a temperature, a pressure, a density) changes discontinuously its behaviour across some value of the variables under consideration. Typical examples are solid-liquid interphases, burnt-unburnt mixtures in flame propagation, and flow in porous media. Understanding of the geometry and stability of the solution and its interphase is important in selecting and evaluating simulation methods, as well as understanding the models themselves. Another area of research is fully non linear equations and optimal transportation. Fully non linear equations arise in optimisation and optimal control. They have also been recently studied in relation to optimal transportation and optimal antenna design. Other areas of interest are incompressible flows, harmonic maps, and minimal surface theory and more recently, on non linear random homogenisation.
The Abel Prize is recognised as the highest possible award to a mathematician. It was presented to Luis Caffarelli in 2023 [16]:-
... for his seminal contributions to regularity theory for nonlinear partial differential equations including free-boundary problems and the Monge-Ampère equation.
For the Press Release, the Citation and other information about the award of this prize to Caffarelli, see THIS LINK.

In the interview [57], Louis Nirenberg was asked, "What is Caffarelli like as a mathematician?" He replied:-
Fantastic intuition, just remarkable. We haven't worked together for several years now, but when we worked together, I had a hard time keeping up with him. He somehow immediately sees things that other people don't see, but he has trouble explaining them. He says things and writes very little, so when we were working at the board, I would always say, "Luis, please write more, write down more." Once I said to him, "Luis, to use a Biblical expression, 'Where is it written?'" Somebody said he once heard a talk in which Luis proved something in partial differential equations - using nothing! Just somehow out of thin air, he can come up with ideas. He's really fantastic - and a very nice person.
Both the quantity and quality of Caffarelli's work is astounding. We give data up to the end of September 2023. MathSciNet lists 322 publications by Caffarelli, three of which appeared in 2022 and one in 2023. These papers have 20,200 citations in 11,499 publications by 7,236 distinct authors. He has advised more than 30 PhD students. While giving this data, we note that his wife, Irene Martínez Gamba, had 99 publications listed in MathSciNet with the earliest publication being in 1986. These 99 papers have 1,715 citations in 954 publications by 975 by distinct authors. Her Ph.D. was from the University of Chicago in 1989 with the thesis Asymptotic behavior at the boundary of a semiconductor device in two space dimensions. She holds the W A Tex Moncrief, Jr. Chair in Computational Engineering and Sciences III at the University of Texas.

Given Caffarelli's incredible output of mathematics it is hard to see how he has much time for hobbies. When asked about these in the interview [19] he said:-
I like to cook, but my wife is a better chef than I am. I also like to play some soccer if I have time. I play the piano. I prefer mostly classical music.
Let us end with a quote by Caffarelli from [1]:-
Through the years, I have had the opportunity to belong to wonderful institutions and to befriend and collaborate with extraordinary scientists all over the world. This led to further opportunities to mentor very talented young people who have invigorated my research with new ideas.


References (show)

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  2. 2014 Steele Prizes, Notices of the American Mathematical Society 61 (4) (2014), 393-397.
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  60. S Talukdar, Luis Caffarelli may have a mathematical answer to how ice cubes melt in the juice glass, Newsclick (29 March 2023).
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  62. Stampacchia Prize shared by six mathematicians, Notices of the American Mathematical Society 29 (6) (1982), 520-521.

Additional Resources (show)

Other pages about Luis Caffarelli:

  1. 2023 Abel Prize awarded to Luis Ángel Caffarelli

Honours (show)


Cross-references (show)


Written by J J O'Connor and E F Robertson
Last Update December 2023