Mathematics

The concept of maximal slope curves originates from the theory of abstract gradient flows and has significant applications in fields such as optimal transport. Generalized characteristics stem from the study of shock wave evolution in hyperbolic conservation laws. The generalized characteristics for the viscosity solutions of Hamilton-Jacobi equations provide a precise description of singular dynamics for viscosity solutions. For any pair of semiconcave functions and convex Hamiltonians, we have established a theory of maximal slope curves, proving that they are exactly a special class of generalized characteristics, namely strict singular characteristics. We also explore transport problems on cut loci and singular sets, providing corresponding continuity equations. This theory resolves several important open problems in this field and offers some new perspectives in the study of certian porblems on cut locus in various fields such as Hamiltonian dynamical systems, optimal transport and so on. This lecture will briefly introduce some applications of this theory. These works are based on recent collaborations with Piermarco Cannarsa, Jiahui Hong, et al.