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Mean Curvature Flow describes the evolution of a family of embedded surfaces in Euclidean space that move in the direction of the mean curvature vector. It is the gradient flow of the area functional and a natural analog of the heat equation for an evolving surface. Initially, this flow tends to smooth out geometries over brief time-intervals. However, due to its inherent non-linearity, the Mean Curvature Flow equation frequently leads to the formation of singularities. The analysis of such singularities is a central goal in the field.

A long-standing conjecture addressing this goal has been the Multiplicity One Conjecture. Roughly speaking, the conjecture asserts that singularities along the flow cannot form by an "accumulation of several parallel sheets”. In recent joint work with Bruce Kleiner, we resolved this conjecture for surfaces in R^3. This had several applications. First, combining our work with previous results, we obtain that the problem of evolving embedded 2-spheres via the Mean Curvature Flow equation is well-posed within a natural class of singular solutions. Second, we remove an additional condition in recent work of Chodosh-Choi-Mantoulidis-Schulze to show that the Mean Curvature Flow starting from any generically chosen embedded surface only incurs cylindrical or spherical singularities. Third, our approach offers a new regularity theory for solutions of general Mean Curvature Flows that flow through singularities.

This talk is based on joint work with Bruce Kleiner.