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Award ID contains: 2304432

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  1. ; ; (, Inventiones mathematicae)
    Abstract We prove Ilmanen’s resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface$$M^{n}\subset \mathbb{R}^{n+1}$$ M n ⊂ R n + 1 ,$$2\leq n\leq 6$$ 2 ≤ n ≤ 6 , with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens. 
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    Free, publicly-accessible full text available December 1, 2025
  2. ; ; ; (, Inventiones mathematicae)
    Abstract We show that the mean curvature flow of generic closed surfaces in$$\mathbb{R}^{3}$$ R 3 avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in$$\mathbb{R}^{4}$$ R 4 is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons. 
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  3. ; (, Communications on Pure and Applied Mathematics)
    We prove that round balls of volume uniquely minimize in Gamow's liquid drop model. 
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    Free, publicly-accessible full text available February 1, 2026
  4. ; ; (, Ars inveniendi analytica)
    Free, publicly-accessible full text available January 1, 2026
  5. ; (, Annals of Mathematics)
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  6. ; ; (, Ars Inveniendi Analytica)
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