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The LaplaceBeltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the LaplaceBeltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (unique up to a scaling) are mutually determined by each other. Given a Euclidean polyhedral surface, its Riemannian metric is represented as edge lengths, satisfying triangle inequalities on all faces. The LaplaceBeltrami operator is formulated using the cotangent formula, where the edge weight is defined as the sum of the cotangent of angles against the edge. We prove that the edge lengths can be determined by the edge weights unique up to a scaling using the variational approach.

We prove that for any free ergodic probability measure-preserving action $${\mathbb{F}_n \curvearrowright (X, \mu)}$$ of a free group on$n$generators $${\mathbb{F}_n, 2\leq n \leq \infty}$$ , the associated group measure space II_{1}factor $${L^\infty (X)\rtimes \mathbb{F}_n}$$ has$L$^{∞}($X$) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II_{1}factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II_{1}factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.

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Durhuus and Jonsson (1995) introduced the class of “locally constructible” (LC) 3-spheres and showed that there are only exponentially many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity.