Summary
We look at 5 results of Peter Lax (1926-2025) in a youtube short (1 minute clip)
1) 1956: Lax Pairs for integrable system
2) 1990: Pedal Map in geometry: a chaotic system
3) 1971: Approximation theorem of volume preserving continuous maps
4) 1954: Lax Milgram theorem generalizing the Riesz representation theorm
5) 1956: Equivalence theorem for numerical schemes
Remarks
The mathematician Peter Lax had influenced me since graduate school even so I never met him personally. (I saw him in person once during the 2009 Ahlfors lectures by him however). While much of applied mathematics by nature is rather ugly due to the fact that one has to deal with real models rather than toy models, Lax demonstrated that there is beauty in applied mathematics too. The 5 results chosen by me are of course personal: as an undergraduate, I took 3 dynamical systems courses (a standard dynamical systems, celestial mechanics and calculus of variations) and 2 functional analysis courses (functional analysis and partial differential equations) from Moser and read rather well the script of a graduate student colleague Andreas Stirnemannn about integrable systems (which took place before I could choose more freely courses). In graduate school, I explored integrable systems to tackle ergodic theoretic questions (the entropy functional log(det(L)) of the Hessian L of a variational problem describing a symplectic map is a spectral property and invariant under isospectral deformations so that one might hope to use scattering to deform to a situation where the entropy can be estimated. This did not work but it produced three papers and part of my PhD Thesis. The isospectral deformation of L is of course a Lax pair L’= [B,L] where B is a skew symmetric matrix related to the matrix L). Lax paper in the monthly about the Pedal map was mind-blowing for me as it showed how chaotic systems naturally occur in situations which Euclid already tackled! Euclid did not study yet what happens if one maps a general triangle to its Pedal triangle. Lax-Milgram is a result that probably appears in any functional analysis course. The Lax equivalence theorem got my attention not directly but because of my wife Ruth who was then doing numerical PDE stuff. It is a fundamental (rather ugly) problem in general to design numerical PDE schemes that work. Either they are elegant and do not work or they work and are ugly. A basic dilemma in computer science context is that beauty and effectiveness not always align. [I told that story a couple of times before but traumas need to be worked through: my computer code to an execercise was once chosen in a recitation session of our mandatory CS course (CS 50 analog but probably more time consuming as we had even just for appreciation write code for solving an ODE using punch cards! One typo and you had to make the treck over to the machine and produce a new punch card!): my code was chosen by the course assistant as an example of “how not to program”. The problem was that my code had been efficient but ugly. If you want to write a code to factor an integer, you can do that in one line using Baby test but tens of thousands of lines of code are needed to build a cutting edge factorization code. (I know that because we had in military to do just that, in Pascal using house written large integer code which a CS part of the cryptology group tuned and maintained and where you had to write in a language that is in the same time also developed.] The last of the chosen topics of Lax is the beautiful approximation theorem for volume preserving continuous maps on a torus. This is very much a quantum calculus topic because the theme of approximating maps by permutations is very much in the quantum calculus spirit. Coincidencially, also Moser (my undergradate advisor) and Lanford (my graduate advisor) were interested very much in finite approximations of dynamical systems. Moser certainly also due to his exposure to Siegel, who was very much also into number theory and celestial mechanics (there were quite a few astronomers like Rannou or Vivaldi interested in finite approximations of symplectic maps) and Lanford due to computer assisted proofs which require to understand finite approximations of infinite structures. Finally, I should mention that Lax is also interesting for me as a teacher as he wrote textbooks in areas I taught much: there are two textbooks with Maria Terrell (who by the way more than 20 years ago talked in the department about “good problems” in our group) and then a linear algebra textbook which is is also quite original. Original approaches to calculus or linear algebra seldom make it to larger service courses taught by larger teams simply also because of momentum.
