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Abstract: In this talk, we will talk about the proof of that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. As a corollary, it implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. This addresses a question posed by Evans-Spruck and De Giorgi. The talk is based on the joint work with Tang-Kai Lee.

If A is a cluster algebra then, by the Laurent phenomenon, every cluster determines an open torus in the cluster variety Spec(A) called a cluster torus. In general, the union of cluster tori only covers Spec(A) up to codimension 2, and the complement of the union of cluster tori in Spec(A) is called the deep locus. Any “bad” (e.g. singular) point in the cluster variety must belong to the deep locus, but the deep locus may be nonempty even when Spec(A) is nice. In joint work with Marco Castronovo, Mikhail Gorsky, and David Speyer, we conjecture that the deep locus may be characterized as those points with nontrivial stabilizer under a natural action of a group on Spec(A). We are able to prove this conjecture for algebras of finite cluster type and for algebras associated to Grassmannians Gr(3,n), that are typically of infinite cluster type. An essential tool in our approach is the realization of the corresponding cluster varieties as braid varieties. In particular, for braid varieties the geometry of the deep locus should be related to properties of the link obtained when closing the braid. I won’t assume previous knowledge of cluster algebras or braid varieties.