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We consider weighted double Hurwitz numbers, with the weight given by arbitrary rational function times an exponent of the completed cycles. Both special singularities are arbitrary, with the lengths of cycles controlled by formal parameters (up to some maximal length on both sides), and on one side there are also distinguished cycles controlled by degrees of formal variables. In these variables the weighted double Hurwitz numbers are presented as coefficients of expansions of some differentials that we prove to satisfy topological recursion. Our results partly resolve a conjecture that we made in [Comm. Math. Phys. 402 (2023), pp. 665–694] and are based on a system of new explicit functional relations for the more general (m,n)-correlation functions, which correspond to the case when there are distinguished cycles controlled by formal variables in both special singular fibers. These (m,n)-correlation functions are the main theme of this paper and the latter explicit functional relations are of independent interest for combinatorics of weighted double Hurwitz numbers. We also put our results in the context of what we call the "symplectic duality", which is a generalization of the x-y duality, a phenomenon known in the theory of topological recursion. [ABSTRACT FROM AUTHOR]

We prove that for each positive integer n, the Rips complexes of the n-dimensional integer lattice in the d_1 metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above n^2(2n-1), with the bounds arising from the Jung constants. We introduce a new concept of locally dominated vertices in a simplicial complex, upon which our proof strategy is based. This allows us to deduce the contractibility of the Rips complexes from a local geometric condition called local crushing. In the case of the integer lattices in dimension n and a fixed scale r, this condition entails the comparison of finitely many distances to conclude that the corresponding Rips complex is contractible. In particular, we are able to verify that for n=1,2,3, the Rips complex of the n-dimensional integer lattice at scale greater or equal to n is contractible. We conjecture that the same proof strategy can be used to extend this result to all dimensions n. [ABSTRACT FROM AUTHOR]

A classical theorem in conformal geometry states that on a compact manifold with nonpositive Yamabe invariant \sigma _0, a smooth metric achieving the invariant must be Einstein. In this work, we extend it to the singular case and show that in all dimensions, if a continuous metric is smooth outside a compact set \Sigma of co-dimension 2+a for some a>0 with unit volume and with scalar curvature bounded below by \sigma _0, then the metric is Einstein away from \Sigma and is isometric to a smooth Einstein manifold of scalar curvature \sigma _0. In this sense, singular metric can be extended to be smooth on the manifold in a suitable sense. This is related to a conjecture of Schoen in the continuous setting. As an application of the method, we prove a positive mass theorem for asymptotically flat manifolds with analogous singularities. This is based on a new maximum principle on Ricci-DeTurck flows which allow initial data to be singular. [ABSTRACT FROM AUTHOR]