Quantitative Homotopy Theory and Geometric Inequalities

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My field of interest is Riemannian Geometry, which is a generalization of the geometry of surfaces. I specifically am interested in stationary objects in Riemannian geometry, such as geodesic segments, geodesic loops, periodic geodesics, stationary geodesic nets and cycles, minimal surfaces, etc. A great deal of my work is aimed at analyzing the connection between the "size'' of various extremal objects and the "size" of the ambient space, as well as their existence. Geodesics are the "straightest" curves on a manifold. For example, in a Euclidean geometry, they are simply the straight lines. On the standard sphere, geodesics are the big circles, like the equator of the sphere. So, on a manifold, it is possible to walk straight, only to return to the starting point. If this happens, it means that we have encountered a geodesic loop. At any point of a closed Riemannian manifold there exists infinitely many directions, such that we will return to the starting point, if we continue walking in this direction. Will we have to walk a long distance? How does it connect to the size of the manifold? This is an example of questions that are of interest to me. Now imagine that you are walking along a geodesic on a manifold, which means that your direction is not changing, and after some time you notice that you are going along the same path. This is something that would happen if you walk along the equator. That means that you have encountered a periodic geodesic. A periodic geodesic does not only return to the starting point, but does so smoothly. Periodic geodesics play a special role in Geometry, Analysis, Dynamical Systems. Minimal geodesic nets are homological analogs of periodic geodesics. Similar concepts were first encountered in a study of optimal telephone networks, and later appeared in other applications. Minimal surfaces are used in Physics as mathematical models of soap bubbles, as well as in String Theory.