Lax has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields.

In a 1958 paper Lax stated a conjecture about matrix representations for third order hyperbolic polynomials which remained unproven for over four decades.

Interest in the "Lax conjecture" grew as mathematicians working in several different areas recognized the importance of its implications in their field, until it was finally proven to be true in 2003.

[2] He began displaying an interest in mathematics at age twelve, and soon his parents hired Rózsa Péter as a tutor for him.

During this time, he met with John von Neumann, Richard Courant, and Paul Erdős, who introduced him to Albert Einstein.

At Los Alamos, he began working as a calculator operator, but eventually moved on to higher-level mathematics.