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 who is also an exceptional mathematician; they have three children Alejandro Caffarelli, Nicolas Caffarelli and Mauro Caffarelli.
He wrote in  how his career developed:-
In 1973 I went with a postdoctoral fellowship to the University of Minnesota, where I was to remain until 1983.In 2009 Caffarelli spoke about his arrival at the University of Minnesota :-
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  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 years 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 function; On the Hölder continuity of multiple valued harmonic functions; The 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 :-
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. 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:-
... 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) :-
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 he returned to the Courant Institute when 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 Primio Konex, Platino y Brillantes, Argentina:-
... 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: -
... 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 and, in 2012, he was made a Fellow of the American Mathematical Society. In 2012 Caffarelli received the prestigious Wolf Prize in Mathematics: -
... for his work on partial differential equations.His work leading to the award of the Wolf Prize is described in :-
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 :-
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 : -
... for his groundbreaking 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 :-
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 behavior 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 optimization 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 homogenization.Let us end with a quote by Caffarelli from :-
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.
Article by: J J O'Connor and E F Robertson