Quantitative Homotopy Theory and Geometric Inequalities

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Most of my results are proofs of various geometric inequalities. While these inequalities are proven in the setting of Riemannian Geometry, a lot of times they can be generalized to more general metric spaces, i. e. spaces in which we can measure distances between points. Riemannian Geometry is the generalization of the geometry of surfaces that lie in a Euclidean three dimensional space. Some of the examples of such surfaces are a round sphere, (as we all know the surface of the Earth has the shape of the sphere), or a torus, the shape that looks like a donut. If we look at the neighbourhood of a point of a surface under the magnifying glass, it will look like a plane. Therefore, a lot of our intuition comes from the Euclidean geometry. At each point of a surface there is a way to measure an inner product between the tangent vectors. This notion of the inner product on a surface generalizes as the Riemannian metric. It gives us a way to measure the distance between points, to define the diameter of a manifold, (i. e. the maximal distance between pairs of points on a manifold), its volume, its curvature, etc. It also give us a way to generalize the notion of a straight line to Riemannian manifolds, obtaining a notion of geodesics, i.e. the straightest curves on a manifold. While the straight line stretches infinitely in both directions, geodesics can close on themselves, resulting in a periodic geodesic, when they do so smoothly, just like the Equator on a sphere, or in a geodesic loop. Many of my previous results, as well as the proposed problems deal with how the lengths of geodesic segments, geodesic loops, or periodic geodesics relate to the other parameters of the manifold, such as its diameter, volume, or some times its curvature. Other minimal objects that are of interest to me are minimal surfaces. A prototype of a minimal surface is a surface of a soap bubble. I am also interested in estimates of the areas of minimal surfaces. Another direction is to prove the existence of various minimal objects, when it does not immediately follow for the topological reasons. It turns out that some of the methods that we use to estimate the "size" of minimal objects some times also works when we try to establish the existence of minimal objects on noncompact manifolds, i.e. the manifolds that stretch to infinity, under certain natural geometric restraints. For example, one of the natural questions that we want study is the following question attributed to V. Bangert: Let M be a complete noncompact Riemannian manifold of finite volume, is there always a periodic geodesic on M?