Geometric Structures arising from Partial Differential Equations

Abstract

We study several geometric problems arising from partial differential equations: (1) the geometry of Laplacian Eigenfunctions on smooth, compact manifolds without boundary (2) the energy landscape of the conservation laws of the defocusing generalized Korteweg de Vries equation (3) level sets of harmonic functions in the unit disc (4) periodic solutions of ODEs in Banach spaces and (5) packing problems related to nodal domain estimates for Laplacian eigenfunctions in the plane in the style of Pleijel.