2023 Abel Prize awarded to Luis Ángel Caffarelli

We present below the Press Release for the award of the 2023 Abel Prize to Luis Caffarelli, the Citation for the award, A Glimpse of the Laureate's Work by Alex Bellos and the article Luis Caffarelli. Mathematics of mathematical models by Arne B Sletsjøe.

1. Press Release.

Luis Ángel Caffarelli awarded the 2023 Abel Prize:

Differential equations are tools scientists use to predict the behaviour of the physical world. These equations relate one or more unknown functions and their derivatives. The functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Partial differential equations arise naturally as laws of nature, to describe phenomena as different as the flow of water or the growth of populations. These equations have been a constant source of intense study since the days of Isaac Newton and Gottfried Leibniz. Yet, despite substantial efforts by numerous mathematicians over centuries, fundamental questions concerning the existence, uniqueness, regularity, and stability of solutions of some of the key equations remain unresolved.

Technically virtuous results

Few other living mathematicians have contributed more to our understanding of partial differential equations than the Argentinian-American Luis Caffarelli. He has introduced ingenious new techniques, shown brilliant geometric insight, and produced many seminal results. Over a period of more than 40 years, he has made ground-breaking contributions to regularity theory. Regularity - or smoothness - of solutions is essential in numerical computations, and absence of regularity is a measure of how wildly nature can behave.

Helge Holden, the chair of the Abel Committee says

Caffarelli's theorems have radically changed our understanding of classes of nonlinear partial differential equations with wide applications. The results are technically virtuous, covering many different areas of mathematics and its applications.

A large part of Luis A Caffarelli's work concerns free-boundary problems. Consider, for instance, the problem of ice melting into water. Here the free boundary is the interface between water and ice; it is part of the unknown that is to be determined. Another example is provided by water seeping through a porous medium - again the interface of water and the medium is to be understood. Caffarelli has given penetrating solutions to these problems with applications to solid-liquid interfaces, jet and cavitational flows, and gas and liquid flows in porous media, as well as financial mathematics.

Enormous impact on the field

Caffarelli is an exceptionally prolific mathematician, with more than 130 collaborators and more than 30 PhD students over a period of 50 years.
"Combining brilliant geometric insight with ingenious analytical tools and methods he has had and continues to have an enormous impact on the field," says Helge Holden.

Luis A Caffarelli has won numerous awards, among them the Leroy P Steele Prize for Lifetime Achievement in Mathematics, the Wolf Prize and the Shaw Prize.

About the Abel Prize:

The Abel Prize will be presented to Luis A Caffarelli at the award ceremony in Oslo on 23 May

The Abel Prize is funded by the Norwegian government and amounts to NOK 7.5 million

The prize is awarded by the Norwegian Academy of Science and Letters and presented by His Majesty King Harald

The choice of the Abel laureate is based on the recommendation by the Abel Committee, which is composed of five internationally recognised mathematicians

2. Citation for the 2023 Abel Prize awarded to Luis Ángel Caffarelli.

The Norwegian Academy of Science and Letters awards the Abel Prize 2023 to Luis A Caffarelli, The University of Texas at Austin, USA

for his seminal contributions to regularity theory for nonlinear partial differential equations including free-boundary problems and the Monge-Ampère equation.

Partial differential equations arise naturally as laws of nature, whether to describe the flow of water or the growth of populations. These equations have been a constant source of intense study since the days of Newton and Leibniz. Yet, despite substantial efforts by mathematicians over centuries, fundamental questions concerning stability or even uniqueness, and the occurrence and type of singularities of some key equations, remain unresolved.

Over a period of more than 40 years, Luis Caffarelli has made ground-breaking contributions to ruling out or characterising singularities. This goes under the name of regularity theory and captures key qualitative features of the solutions beyond the original functional analytic set-up. It is conceptually important for modelling - is for instance the assumption of macroscopically varying fields self-consistent? - and informs discretisation strategies and is thus crucial for efficient and reliable numerical simulation. Caffarelli's theorems have radically changed our understanding of classes of nonlinear partial differential equations with wide applications. The results go to the core of the matter, the techniques show at the same time virtuosity and simplicity, and cover many different areas of mathematics and its applications.

A large part of Caffarelli's work concerns so-called free-boundary problems. Consider for instance the problem of ice melting into water. Here the free boundary is the interface between water and ice; it is part of the unknown that is to be determined. Another example is provided by water seeping through a porous medium - again the interface between the saturated and unsaturated part of the medium is to be understood.

A particular class of free-boundary problems are denoted as obstacle problems. An example is given by a balloon pressing against a wall or an elastic body resting on a surface. Caffarelli has given penetrating solutions to these problems with applications to solid-liquid interfaces, jet and cavitational flows, and gas and liquid flows in a porous media, as well as financial mathematics. Caffarelli's regularity results rely on zooming in on the free boundary, and classifying the resulting blow-ups, where non-generic blow-ups correspond to singularities of the free boundary.

The incompressible Navier-Stokes equations model fluid flow, such as water. The regularity of solutions of these equations in three dimensions is one of the open Clay Millennium Problems. In 1983, based on Scheffer's previous work, Caffarelli, with Kohn and Nirenberg, showed that sets of singularities of suitable weak solutions cannot contain a curve, that is, they have to be very "small".

Caffarelli's regularity theorems from the 1990s represented a major breakthrough in our understanding of the Monge-Ampère equation, a highly nonlinear, quintessential partial differential equation, that for instance is used to construct surfaces of prescribed Gaussian curvature. Important existence results were established by Alexandrov, and earlier central properties had been shown by Caffarelli in collaboration with Nirenberg and Spruck, with further key contributions by Evans and Krylov. Caffarelli however closed the gap in our understanding of singularities by proving that the explicitly known examples of singular solutions are the only ones.

Caffarelli has - together with collaborators - applied these results to the Monge-Kantorovich optimal mass transportation problem, based on previous work by Brenier. Caffarelli and Vasseur gave deep regularity results for the quasi-geostrophic equation in part by applying the exceptionally influential paper by Caffarelli and Silvestre on the fractional Laplacian.

Furthermore, Caffarelli has made seminal contributions to the theory of homogenisation, where one seeks to characterise the effective or macroscopic behaviour of media that have a microstructure, for instance because they are formed by a composite material. A typical problem regards a porous medium - like a hydrocarbon reservoir - where one has a solid rock with pores, posing a complex and - to a large degree - unknown structure through which fluids flow.

Caffarelli is an exceptionally prolific mathematician with over 130 collaborators and more than 30 PhD students over a period of 50 years. Combining brilliant geometric insight with ingenious analytical tools and methods, he has had and continues to have an enormous impact on the field.

3. A Glimpse of the Laureate's Work By Alex Bellos.

The Abel Committee has awarded Luis A Caffarelli the 2023 Abel Prize for his seminal contributions to the study of nonlinear partial differential equations. The text below gives a brief explanation of some of the work he has done in this area.

The discovery in the seventeenth century that the universe can be described by mathematical equations marked the beginning of modern science. Isaac Newton's second law of motion was an early example of such an equation: it states that the force on an object is equal to the mass of that object times its acceleration, usually expressed by the formula F = ma. Not only were Newton's laws a conceptual leap forward, but they also required a new type of mathematics.

This new mathematics, developed by Newton and Gottfried Leibniz, became popularly known as the infinitesimal calculus, and introduced the idea of
an instantaneous "rate of change". Since these instantaneous rates of change are calculated by considering infinitesimal differences, equations in calculus became known as differential equations. They come in two classes: ordinary differential equations, which feature a single variable, and partial differential equations (PDEs), which feature more than one variable.

PDEs are ubiquitous across science. They underlie our understanding of the physical world, beautifully modelling phenomena from heat to sound to electromagnetism to quantum mechanics. PDEs also arise in the social sciences, explaining the behaviour, for example, of epidemics, interest rates and stock options. Indeed, wherever there is a system involving multiple variables undergoing continuous change, you will find PDEs.

The power of a differential equation is that it predicts the future. For example, if I throw a ball I know - thanks to Newton's second law - that it will travel through the air in the shape of a parabola. I also know that if I throw the ball with a bit more power, or at a slightly different angle, that the ball will still travel in the shape of a parabola, even though it may be a slightly bigger or smaller one. In other words, Newton's second law is well-behaved: if I adjust the input values slightly, there are no nasty surprises in the output. The research of Luis Caffarelli asks similar questions of PDEs: when you adjust the input, do they also always act in the expected way, or are there values that trigger unstable, irregular or erratic behaviour?

Caffarelli's first area of investigation was the obstacle problem, a classic example in the field of nonlinear PDEs, which asks what is the equilibrium position when an elastic membrane pushes against a rigid obstacle, such as, for example, when a balloon presses against a wall. This work led him to the wider area of "free boundary problems", so-called because the boundary under discussion - such as where the membrane meets the obstacle, or where the balloon meets the wall - is unknown at the outset and is what needs to be determined. Other examples of free boundary problems include ice melting into water, or water seeping through a porous medium. In the case of ice melting into water, the free boundary is the interface between the ice and the water, and can be used to model other examples of phase transitions in physics, biology and finance.

Caffarelli revolutionised the study of free boundary problems in the 1970s, after which he turned his attention to probably the most famous PDEs in all mathematics, the Navier-Stokes equations. Formalised in the mid-nineteenth century, these two equations describe the motion of viscous fluids, such as how water flows down a stream or oil down a pipe. The first equation, marked (i) below, states the fluid is incompressible.

(i)

\nabla . \textbf{V}\ = 0

The first Navier-Stokes equation

(ii)

\rho \Large\frac{DV}{Dt}\normalsize = -\nabla p + \mu\nabla^2 + \rho g

The second Navier-Stokes equation

The second equation, marked (ii), is an application of Newton's second law: it says that the mass times the acceleration (on the left of the equals sign) is equal to the force (on the right), which is broken down into the internal forces (pressure and viscosity) and external forces (usually gravity, hence the abbreviated

g

).

Physicists and engineers use the Navier-Stokes equations to predict the behaviour of fluid flows every day, and they work extremely well. Yet despite their practical importance, the equations are not fully understood. For example, it is an open question whether or not the equations are always 'smooth' or whether they will sometimes 'blow up', meaning that if you smoothly tweak the pressure, viscosity and so on, it is not known whether the velocities within the fluid will always change smoothly, or if the equations may throw up a point at which the velocity spikes to infinity. In the real world, velocity can never be infinity, so the discovery of singularities with infinite velocity would mean that the equations are somehow inadequate models of physical behaviour. The question of the smoothness of the Navier-Stokes equations has gained notoriety in recent years, since it is a Millennium Problem, one of the six problems the Clay Mathematics Institute has decided it will give a $1 million prize to the first person to provide a solution.

In 1982 Caffarelli, together with Robert Kohn and Louis Nirenberg, proved that if the Navier-Stokes equations do produce singularities, they will disappear instantly because the singularities produced cannot fill a curve in space time (meaning the three dimensions of space and the one dimension of time treated as four dimensions.) The 1982 paper remains the closest anyone has got to proving or disproving the smoothness of the Navier-Stokes equations, even after another four decades of intense research in this area.

PDEs arise when scientists try to describe natural laws, but they are studied by mathematicians for their own internal consistency and beauty. Luis Caffarelli has made his life's work the desire to establish that these tools have a rigorous mathematical foundation. He has been hugely influential in taming their wildness, making sure PDEs are meaningful representations of reality.

4. A B Sletsjøe, Luis Caffarelli. Mathematics of mathematical models.

Mathematics of Mathematical Models

If you predict that today's weather will be the same as yesterday's weather, you will succeed with a probability of approximately 0.5. If you in addition incorporate some old weather sayings, like "Red sky at night, sailors delight. Red sky in morning, sailors take warning," you will noticeably increase your fortune-telling abilities. But still, if you plan to cross the sea, you would probably prefer a more knowledge-based forecast. So, you look up at the Internet or listen to your radio to read or hear what the meteorologist can tell you about challenges you will face on your imminent journey.

Putting on your sunglasses and rejoice that the wind is blowing just right in the mainsail, you send a warm thought to the meteorologist who fed her computer with a lot of data and some physical laws to give you a close to perfect description of today's nice weather.

Weather forecast, as well as many other models for how nature behaves, is based on what is called a "partial differential equation", or a PDE for short. The general setup for a PDE as a mathematical model for something we observe in nature, is the correspondence between an acting force and a resulting reaction. The reaction can either be described as a time-based process, or just as a geometric configuration with no time involved. The acting force can originate from different sources, a gravitational field, pressure or a temperature gradient. In additional to the source of the acting force, the geometry of the object in study plays a major role.

The behaviour of a gas flowing through a pipe is highly dependent of the shape of the pipe. If we narrow the pipe, the gas will flow faster, and obstacles will cause turbulence. Or, dipping a metal frame into a soapy water, suitable for blowing soap bubbles, the soap film will form a surface of minimal area with the shape of the frame as the only constraint.

The gas flow through a pipe is modelled by what is called the Navier-Stokes equations. The equations come from applying Isaac Newton's second law to fluid motion, together with assumptions about how the molecules in the fluid interacts.

A solution of the Navier-Stokes equations is a velocity field, describing the velocity of the fluid at any time and any point. Boundedness of this solution means that the speed of any part of the fluid will never exceed a certain given value. In the physical world this not an issue since infinite speed is an impossibility. But we can still ask the truly physical question; does there exist a limit for the speed of a tornado, or will we constantly observe new speed records? Mathematically this question is rephrased as a question of boundedness of solutions. Do the Navier-Stokes equations develop unbounded solutions in finite time? This question is called the Navier-Stokes existence and smoothness problem. The Clay Mathematics Institute has called this problem one of the seven most important open problems in mathematics and has offered a US$1 million prize for a solution or a counterexample.

The Navier-Stokes equations involve time as a parameter, i.e. the equation models what happens in time based on acting forces and geometry. An example of a physical phenomenon which is modelled by a PDE, but which not involves time as a parameter, is the minimal surface phenomenon. When you dip a steel wire frame in soapy water, if you are lucky, you will catch a soap film attached to the frame. Since the soap molecules prefer to stick together as much as possible, the soap film will always exhibit a minimal surface. This means that provided the film is attached to the frame, it is not possible to find a surface of smaller area.

Even if we consider the formation of the minimal surface as a purely geometric result, there is of course a time-dependent process that takes place after dipping the steel wire frame in the soapy water. The soap molecules work fast, reaching their final position in just a fraction of a second. In the process they move a bit around, playing their game with all the other molecules, and finally settle down. They have reached a state of equilibrium. Thus, the solution of the time-independent model can also be considered as the equilibrium state of a time-dependent process.

A common problem for all mathematical models is to find solutions. Suppose you put up a mathematical model for something that takes place in nature and you are clever enough to find solutions to the involved equations. Then you actually will be in position to predict something about the future based on scientific reasoning. Your predictions will be far more valuable than if you were just guessing.

In general it is very difficult to find the solutions. In many cases it is hard even to prove that there exist solutions. And if you know that there exists solutions, they may have bad properties. Luis Caffarelli has dedicated his professional life to study the nature of the solutions of various partial differential equations. Thanks to his efforts and to the contribution of many other mathematicians, our insight in the nature of the solutions has increased significantly over the last 50 years. The Abel Prize committee writes in their citation for this year's Abel Prize: "Combining brilliant geometric insight with ingenious analytical tools and methods, he [Caffarelli] has had and continues to have an enormous impact on the field."

Last Updated December 2023