Peter Lax, an Inspired and Inspiring Life in Mathematics

By Gilbert Strang

BOOK REVIEW: Peter Lax, Mathematician: An Illustrated Memoir. By Reuben Hersh, American Mathematical Society, Providence, Rhode Island, 2015, 253 pages, $35.00.

There are hundreds of biographies of Churchill and Roosevelt. They are great men on a world scale. They imposed their vision on the rest of us, or perhaps sometimes it was just their will.  Either way, millions were affected.

Closer to us, there are biographies of von Neumann, Turing, Courant, and Erdős—and now we have a biography of Peter Lax. These are great men on our scale. Their vision has affected every reader of SIAM News. To some degree, we can understand what they did and how they did it. In Peter’s case we could even ask him—but magic remains.

Peter Lax’s life story is remarkable in itself; he was special from the start. Here are four parts, three of them very early:

• Fleeing Budapest with his family in 1941
• Meeting Courant and von Neumann as a New York schoolboy
• Working on the Manhattan Project in Los Alamos (as a corporal in the Army)
• Receiving the Wolf and Abel Prizes

Between the third and the fourth came Lax’s life work. That barely fits into Reuben Hersh’s book, and it certainly won’t fit into this review. Perhaps from just three topics, one linear and two nonlinear, readers can extrapolate. My goal is to show the person behind this work—we all learn by example. Peter is an inspiration to his students and friends (as von Neumann was to him).

The first topic is stability; everyone knows its importance—computations make that clear. Lax’s equivalence theorem (convergence \(\leftrightarrow\) stability) demonstrated that stability is not just desirable but indispensable. When an approximation method is unstable, a complete space will contain the limit of functions that grow faster and faster. The opposite is uniform boundedness (stability). That key point is part of pure mathematics, in the service of numerical analysis.

The equivalence theorem gave us work to do, precise results to prove, something to accomplish with our analysis and our lives. Those words sound personal and I suppose they are—by random chance Peter Henrici assigned me the Lax–Richtmyer paper to present in his graduate student seminar. And the stability requirement led Heinz-Otto Kreiss in 1962 to one of the great achievements of all time in matrix analysis. At the International Congress in Stockholm that year, Peter changed his invited address to present the Kreiss matrix theorem.

The second topic is CFD, computational fluid dynamics. Los Alamos was an enormous impetus; scientific computing came to life. An essential question was to understand shock waves and how to compute their movement. This took Peter (and von Neumann before him) into a different world, beyond the smaller and safer domain where talent in pure analysis wins.

Applied mathematics requires a combination of skills, not perfection of one. Lax identified and solved the crucial problem at the moment of discontinuity, to apply the jump equations (Rankine–Hugoniot) to start the solution again beyond the shock. With a little viscosity you can go suddenly but smoothly through that singularity, or you can achieve the same result with finite differences. Here the Lax–Wendroff method combined stability with second-order accuracy—the upgrade from simple but inaccurate first-order approximation that scientific computation always demands.

The third topic is Lax pairs. They are the ultimate example of an insight in algebra that rewrote the theory of nonlinear integrable PDEs. The key idea is that eigenvalues are preserved as the solution evolves and the differential operator changes. Special cases like Korteweg–deVries had been seized on, one at a time, but the pattern was unknown. Lax found it in the equation \(dL/dt = BL – LB\), with solution \(e^{Bt} L(0) e^{–Bt}\). If we can write a PDE in this form, then \(L(t)\) is similar to \(L(0)\). An amazing number of special nonlinear equations do have that form—they come from Lax pairs \((B,L)\) and conservation laws follow.

One more note about Lax’s papers, an important one. He didn’t close out problems, he opened them. After reading a paper, you had something to think about. I am sure this was appreciated by his students, one of whom was Reuben Hersh (who is also a celebrated writer). That connection allows this book to include many insights on Peter’s life—ordinary as well as extraordinary, exciting and quiet, part happy and also part sad. His first wife Anneli and his first son Johnny are unforgettable.

As long as I am using more space than the editor expected, let me look back to other early leaders of numerical analysis, contemporaries of Peter. For a well-posed (and stable) question, let’s exclude those who are still alive. Jim Wilkinson in England and Germund Dahlquist in Sweden come immediately to mind. In France the choice must be Jacques-Louis Lions; he deserves a full biography. Those three were great men, world class. In the U.S. I will name Gene Golub, I hope not too controversially. Gene was active with SIAM but disappointed in a SIAM decision about journals. Then, by a happy twist of fate, the Summer School that flourishes today became his legacy (along with the SVD).

Peter Henrici was Swiss, and before him was Stiefel. China gave us Feng Kang, a wonderful character and a creator of the finite element method. The Soviet Union was too big for a single name—Olga Ladyzhenskaya was a SIAM von Neumann lecturer on wave equations, and Olga Oleinik analyzed conservation laws. We have Lanczos from Ireland, Ron Mitchell from Scotland, Fujita from Japan. But powerhouses of research like Germany, Italy, Spain, and Australia developed a little later. I don’t have a conclusion to draw, except to say how fortunate we are to work together.

And I hope that another Peter Lax comes along soon.

Gilbert Strang is a professor of mathematics at the Massachusetts Institute of Technology.