Gaspard Monge (1746-1818): Application of Geometry to Analysis

Two centuries after the death of the mathematician Gaspard Monge, his main results are presented, for the period preceding his political and organizational commitment during the French Revolution and the Napoleonic era in Italy and Egypt. After an examination of the fortune of his works in Russia, Monge’s main contributions to the theory of partial differential equations of the first order, to the solution of the minimal surface equation (with his pupil Meusnier) and to the first study of the optimal transport problem (Monge-Ampère equation) are underlined. These subjects are well developed in the current mathematical research in analysis and in differential geometry. This paper is divided into three parts: the first presents Monge's main results relating to the partial dfferential equations of first order, with particular attention to the geometric aspects. The second part concerns Monge's contributions to the minimal surface equation that support the solution of this equation found by his pupil Meusnier. The third part concerns the problem of optimal transport which gave rise to the study of the Monge-Ampère equation and to important questions of differential geometry, linked to what is called the Gaussian curvature of a surface. The geometric aspects of these results will be highlighted, justifying our title: "Application of geometry to analysis", which reverses the name of "Application of analysis to geometry" given by Monge himself to the collection of his results published in 1807 and in 1809 for use by the Polytechnic school.