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We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and generalized valuations on $M$ are modules over the algebra of compactly supported generalized valuations on $G$ satisfying some technical condition of tameness.
The case of a vector space acting on itself is studied in detail. We prove explicit formulas in this case and show that the new convolution is an extension of the convolution on smooth translation invariant valuations introduced by J. Fu and the second named author.
We prove some epsilon regularity results for $n$-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one $n$-dimensional planes, and in a neighbourhood of any point at which at which at least one tangent cone is equal to a union of four distinct multiplicity one $n$-dimensional half-planes that meet along an $(n-1)$-dimensional axis. The key ingredient is a new Excess Improvement Lemma obtained via a blow-up method (inspired by the work of L. Simon on the singularities of ‘multiplicity one’ classes of minimal submanifolds) and which can be iterated unconditionally. We also show that any tangent cone to an $n$-dimensional minimal two-valued Lipschitz graph that is translation invariant along an $(n-1)$ or $(n-2)$-dimensional subspace is indeed a cone of one of the two aforementioned forms, which yields a global decomposition result for the singular set.
We prove the nonequivariant coherent-constructible correspondence conjectured by Fang–Liu–Treumann–Zaslow in the case of toric surfaces. Our proof is based on describing a semi-orthogonal decomposition of the constructible side under toric point blow-up and comparing it with Orlov’s theorem.