Hörmander

Lars Hörmander (1931–2012) was a Swedish mathematician renowned for fundamental contributions to the analysis of partial differential operators. His work helped establish the modern theory of linear PDEs, distribution theory, and microlocal analysis.

Among his core achievements is the development of the calculus of pseudodifferential operators and the symbol classes now known as Hörmander classes, which provide a rigorous framework for linear operators with non-constant coefficients. He introduced the concept of the wavefront set to describe the precise location and directions of singularities of distributions, and he formulated hypoellipticity criteria for differential operators, including the bracket-generating (Hörmander) condition for sums of squares of vector fields. This condition guarantees regularity of solutions: if the operator is a sum of squares of vector fields that satisfy the condition, then distributions that are acted on by the operator become smooth where the operator applies.

Hörmander authored foundational texts, most notably The Analysis of Linear Partial Differential Operators I–IV, which systematize distribution theory, Fourier analysis, and the pseudodifferential techniques essential to the field. His work laid the groundwork for much of modern PDE theory, microlocal analysis, and related areas in geometry and mathematical physics. He held professorships at several institutions in Sweden and abroad, and his research has influenced generations of mathematicians and subsequent developments in analysis.