Minimal surfaces and quantitative topology of the space of cycles

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The proposed research program concerns existence and properties of minimal and constant curvature surfaces in Riemannian manifolds. Soap bubbles and trajectories of charged particles in a magnetic field are some of the real world examples of these phenomena. More generally, problems in minimal surface theory are closely related to many problems in partial differential equations, general relativity, engineering and many other areas. ******In 1960s Almgren suggested a new technique for the construction of minimal surfaces in closed manifolds, which is now known as the Almgren-Pitts Min-Max Theory. In the last several years the Min-Max Theory has experienced a renaissance. The proof of the Willmore conjecture by Marques and Neves was one remarkable achievement with many other important results that followed, answering a number of long-standing conjectures. Current research proposal is focused on some fundamental questions about families of cycles in connection with problems in Min-Max Theory and with applications in geometry and topology.******This research project lies at the interface of Geometric calculus of variations and quantitative topology. It contains a number of problems related to the geometric properties of the space of flat cycles in a Riemannian manifold and existence of families of cycles satisfying certain special conditions with the goal of obtaining information about regularity, geometry and topology of min-max minimal hypersurfaces. The project relies on techniques and ideas developed by Gromov, Guth, Marques and Neves, as well as other ideas from topology, geometric measure theory and analysis.**