Asia Pacific Mathematics Newsletter

Shaw Prize 2018

The Shaw Prize in Mathematical Sciences 2018
was awarded to Luis A Caffarelli, Professor ofMathematics at the University of Texas atAustin, USA for his groundbreaking work on partial
differential equations, including creating a theory ofregularity for nonlinear equations such as the Monge–
Ampère equation, and free-boundary problems suchas the obstacle problem, work that has influenced awhole generation of researchers in the field.
The Shaw Prize, now in its 15th year, is an international award to honour those who have made significant
advances or greatly contributed to the research in thefields of astronomy, life sciences and medicine, ormathematical sciences. It is widely regarded as the“Nobel Prize of the East”, and the monetary reward foreach award is US$1.2 million.

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Luis A Caffarelli was born in 1948 in Buenos Aires,Argentina and is currently Professor of Mathematics
at the University of Texas at Austin, USA. He obtained
his Master of Science in 1969 and his PhD in Mathematics in 1972 from the University of Buenos Aires,Argentina. He joined the University of Minnesota, USA,
where he was successively Postdoctoral Fellow(1973–1974), Assistant Professor (1975–1977), Associate Professor (1977–1979) and Professor (1979–1983).
He was a Professor at the Courant Institute of Mathematical Sciences, New York University, USA (1980–1982), the University of Chicago, USA (1983–1986),
the Institute for Advanced Study in Princeton, USA(1986–1996) and the Courant Institute, New York
University, USA (1994–1997). He is a member of theUS National Academy of Sciences and a Fellow of theAsia Pacific Mathematics Newsletter

American Academy of Arts and Sciences.
Partial differential equations are fundamental tolarge parts of mathematics, physics, and indeed all thesciences. They are used to model heat flow, fluid motion,
electromagnetic waves, quantum mechanics, the shapeof soap bubbles, and innumerable other physicalphenomena.
A few very simple equations can be solved explicitly— that is, one can find an exact formula for their solutions — but this is very much the exception rather than
the rule. Instead, one has to be content with being able
to show that solutions exist, and with being able to saysomething about how they behave.
A very important example of this is the Navier–Stokes equation, which describes the motion of aviscous fluid. It is not known whether, given appropriate
initial conditions, there must be a solution to theNavier–Stokes equation that remains well-behaved
forever, or whether singularities will necessarily
develop. To put it more graphically, if you stir a bucketof water, is there a danger that a week later it will blow
up? Probably not, but nobody knows how to prove this,and it is one of the major unsolved problems of mathematics.
Although it is not known how to solve the Navier–Stokes equations, one can find so-called “weak solutions”, which are abstract objects that solve the equations, but not in quite the sense one wants. If one could
show that these solutions were “regular”, then theNavier–Stokes problem would be solved. A famous
result of Caffarelli, Kohn and Nirenberg is the closest
anybody has come to that: it shows that weak solutions
exist that are regular except on a set of singularities thathas to be very small, in a precise mathematical sense.
Another area in which Caffarelli has created a newand highly influential theory is obstacle problems. Hereone would like to know the shape that will be taken by
an elastic membrane with a given boundary if it has tolie above a certain obstacle. The shape taken will be theone that minimises its energy, but the important questions concern how well-behaved, or “regular”, a solutionof this kind will be. As with all important problems inpartial differential equations, this one arises in many
contexts, including fluid filtration in porous media, andfinancial mathematics.
In general, because one does not usually have
explicit formulae for solutions to partial differentialequations, the analysis of their properties is very hard,and depends on extremely delicate estimates. Caffarelli
is a master at this, frequently coming up with argumentsthat have left other researchers wondering how he could

possibly have thought of them. He continues to workat the forefront of the field and has had a huge influence,both through his own work and that of his doctoral
students, many of whom have themselves become
extremely distinguished mathematicians. In a way thatfew mathematicians achieve even once, he has repeatedly created important areas almost from scratch thatare extremely active to this day.
Sources:
http://www.shawprize.org/en/

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