The study of the structural properties of the set of points at which the viscosity solution of a first order Hamilton–Jacobi equation fails to be differentiable—in short, the singular set—started with the paper On the Singularities of Viscosity Solutions to Hamilton–Jacobi–Bellman Equations, Indiana Univ. Math. J. 36 (1987), 501–524 by Mete Soner and myself. These thirty years have registered enormous progress in the comprehension of the way how singularities propagate: We can now provide a fine topological analysis of the singular set, describe singular dynamics by generalized characteristics, and even study the long time behavior of singular curves. In my talk, I will revisit the milestones of this theory and discuss possible developments and open problems.

Piermarco Cannarsa graduated in Mathematics from the Scuola Normale Superiore and University of Pisa in 1979. Since 1990, he is a Professor of Mathematical Analysis at the University of Rome Tor Vergata, where he has been the advisor for 14 Ph.D. students. Cannarsa’s scientific interests include partial differential equations, control theory, and nonsmooth analysis. His current research is concerned with control of partial differential equations, as well as regularity of viscosity solutions to Hamilton–Jacobi–Bellman equations and mean field games. He authored about 130 papers in international journals and 2 books including Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control(with C. Sinestrari). Cannarsa has given plenary talks at a number of international conferences. Since 2018, Cannarsa is the President of the Italian Mathematical Union and the Italian coordinator of the LIA COPDESC (Control, Calculus of Variations, Nonlinear PDEs, and Scientific Computing) on Applied Mathematics. Cannarsa is the Editor-in-Chief of the Bollettino dell’Unione Matematica Italiana (Springer) and an editor of several international journals including Evolution Equations and Control Theory (AIMS), Mathematical Control and Related Fields (AIMS), Minimax Theory and its Applications (Heldermann), Nonlinear Analysis: Real World Applications (Elsevier), and Nonlinear Differential Equations and Applications (Birkhäuser).