Variational methods in singular geometry

Many physical phenomena can be described by the principle of least action. This involves studying minima of certain functionals, called Lagrangians, named after the French mathematician and astronomer J-L Lagrange (1736-1813), that describe the energy of the system under consideration. For example, it is possible to derive Newton's laws of classical mechanics from the principle of least action. The principle can be applied also to more complicated systems, even infinite dimensional configuration spaces. One famous such example is the case of geodesics, paths minimizing the distance between two points in a smooth space. Another, more involved example is the case of harmonic maps. Here the Lagrangian energy is the total stretch of a map between two smooth spaces. In this project the PI proposes to study analogous situations for more complicated Lagrangians that have important applications. The project has also an educational component where the PI is planning to supervise graduate students towards their Ph.D. theses, undergraduates through seminar courses, and write expository notes for a wider audience. In slightly more technical terms, harmonic maps are critical points of the L-2 norm of the gradient (Dirichlet integral) of a map between two Riemannian manifolds. The PI proposes to study the calculus of variations of functionals associated with other function space norms like the L-infinity and L-1 norms. Minimizing functionals associated to the L-infinity norm yield solutions of fully non-linear degenerate elliptic PDE’s with very challenging regularity properties. Similarly, solutions of the dual non-linear problem for functionals involving the L-1 norm are equally challenging. The singular sets of these solutions provide geometric realizations of topological objects, like geodesic foliations and laminations, studied in topology. The PI proposes to develop the analytic methods to study these topological objects as well as others studied in Thurston theory. Examples are earthquakes and cataclysms. There is very little known in the literature about the analytic underpinnings of the theory so the PI has to develop most of the techniques from scratch. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.