Geometry & Topology Seminar

Ali Aleyasin, Department of Pure Mathematics, University of Waterloo

"Riemannian cones in disguise: singular elliptic PDEs"

It is well-known that several non-linear elliptic partial differential equations have applications in various fields of geometry and analysis, including but not limited to the Calabi problem, the Weyl and Minkowski problems, and optimal transport. An important class of such non-linear equations are the real and complex Monge-Amp\`ere equations. Although the case of strictly elliptic equations with smooth source term has been rather well-understood, the behaviour of solutions in the vicinity of possible singularities or degeneracies of the source term is far from being understood. This corresponds to the vanishing or blowing up of the prescribed curvature in the Well problem. In this talk, I will present an application of a differential geometric approach to the study of certain singularities and degeneracies of elliptic complex Monge-Amp\`ere equations. This  approach will allow deriving new estimates for solutions. I shall also outline how the idea works in case of several other important geometric partial differential equations.

M3-3103