Generic Flows, Ricci Curvature, Heegaard Splittings, and Nodal Sets

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A major topic under study in this project is mean curvature flow. Mean curvature flow (MCF) has been used and studied in materials science for almost a century to model phenomena such as cell, grain, and bubble growth. MCF's computational and theoretical study, as well as that of other similar flows, have had enormous impact in diverse areas of pure and applied mathematics, theoretical physics, materials science, and engineering. A particular emphasis of this project is to understand the evolution of a typical (or generic) surface. Are the singularities of flow and the structure of the set where the singularities occur simpler and nicer in the typical case than in bad examples?This research project is divided into four parts. The first part concerns what happens to a hypersurface under the mean curvature flow. This part is mostly concerned with hypersurfaces that are in general or generic position before the flow starts. The mean curvature flow (MCF) is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. The project further develops the theory of generic MCF, extending recent results to other flows. A second part of the project concerns Ricci curvature. Previously estimates with applications have been obtained by the PI and Naber, and recently monotonicity formulas have been proved by the PI and used with Minicozzi to settle a long standing open problem about uniqueness of tangent cones of Einstein metrics. This part of the project discusses possible extensions and related conjectures. The third part of the project concerns the longstanding classification problem in the theory of Heegaard splittings of 3-manifolds: to exhibit for each closed 3-manifold a complete list, without duplication, of all its irreducible Heegaard splittings, up to isotopy. The final and smaller part of the proposal is about bounds for nodal sets (or zero sets) of eigenfunctions.