I am interested in quantitative aspects of Geometric Calculus of Variations. In 1951 J.P. Serre proved that every two points on a closed Riemannian manifold can be connected by an infinite set of distinct geodesics. I would like to prove that the lengths of the first k of them admit an upper bound of the form f(n,k)d, where n is the dimension and d is the diameter of the manifold. I am interested in similar curvature-free upper bounds for the length of the shortest periodic geodesic and the smallest area of a minimal surface. I am also interested in distribution of geodesic segments between a fixed pair of points and geodesic nets on a manifold. In another direction I would like to extend my previous results on fractal features of Morse landscapes of Riemannian functionals to scale-invariant Riemannian functionals involving a lower bound for the Ricci curvature. This would involve proving some new results about Riemannian manifolds with Ricci curvature bounded from below.
The notion of a closed Riemannian manifold is a higher dimensional generalization of a closed surface, like the surface of a donut, or a sphere. We plan to study connections between ``sizes" of various extremal objects on a closed Riemannian manifold and the ``size" of the manifold. Examples of extremal objects include geodesic segments (i.e. straightest curves between two points), periodic geodesics (i.e. straight curves on manifolds that smoothly close on themselves), geodesic nets (objects that arize when one tries to connect three or more points by a shortest tree) and minimal surfaces (i.e. mathematical models of soap bubbles). In another direction we plan to study "optimal" shapes of higher dimensional manifolds. Our approach to this last question involves ideas coming from different areas of Mathematics, including Computability Theory.
