THE ONSAGER FORMULA, THE FISHER-HARTWIG CONJECTURE, AND THEIR INFLUENCE ON RESEARCH INTO TOEPLITZ-OPERATORS

This is not primarily a paper about applications of mathematics to statistical physics, but rather a report on how a particular problem of statistical physics has resulted in an extensive mathematical theory. The problem alluded to is the computation of the spontaneous magnetization M(o)(T) of the two-dimensional Ising model with nearest-neighbor interactions, whose solution for temperatures T below the Curie point T-C was given by the famous formula of Lars Onsager in 1948. The theory grown out of this formula is the edifice of Toeplitz determinants, matrices, and operators.