Dimers on Riemann surfaces and compactified free field

We consider the dimer model on a bipartite graph embedded into a locally flat Riemann surface with conical singularities and satisfying certain geometric conditions in the spirit of the work of Chelkak, Laslier and Russkikh, see arXiv:2001.11871. Following the approach developed by Dub\'edat in his work ["Dimers and families of Cauchy-Riemann operators I". In: J. Amer. Math. Soc. 28 (2015), pp. 1063-1167] we establish the convergence of dimer height fluctuations to the compactified free field in the small mesh size limit. This work is inspired by the series of works of Berestycki, Laslier and Ray (see arXiv:1908.00832 and arXiv:2207.09875), where a similar problem is addressed, and the convergence to a conformally invariant limit is established in the Temperlian setup, but the identification of the limit as the compactified free field is missing

On uniqueness in Steiner problem

We prove that the set of $n$-point configurations for which solution of the planar Steiner problem is not unique has Hausdorff dimension is at most $2n-1$. Moreover, we show that the Hausdorff dimension of $n$-points configurations on which some locally minimal trees have the same length is also at most $2n-1$. Methods we use essentially requires some analytic structure and some finiteness, so that we prove a similar result for a complete Riemannian analytic manifolds under some apriori assumption on the Steiner problem on them

Inverse maximal and average distance minimizer problems

Consider a compact $M \subset \mathbb{R}^d$ and $r > 0$. A maximal distance minimizer problem is to find a connected compact set $\Sigma$ of the minimal length, such that $$\max_{y \in M} dist (y, \Sigma) \leq r.$$ The inverse problem is to determine whether a given compact connected set $\Sigma$ is a minimizer for some compact $M$ and some positive $r$. Let a Steiner tree $St$ with $n$ terminals be unique for its terminal vertices. The first result of the paper is that $St$ is a minimizer for a set $M$ of $n$ points and a small enough positive $r$. It is known that in the planar case a general Steiner tree (on a finite number of terminals) is unique. It is worth noting that a Steiner tree on $n$ terminal vertices can be not a minimizer for any $n$ point set $M$ starting with $n = 4$; the simplest such example is a Steiner tree for the vertices of a square. It is known that a planar maximal distance minimizer is a finite union of simple curves. The second result is an example of a minimizer with an infinite number of corner points (points with two tangent rays which do not belong to the same line), which means that this minimizer can not be represented as a finite union of smooth curves. Our third result is that every injective $C^{1,1}$-curve $\Sigma$ is a minimizer for a small enough $r>0$ and $M = \overline{B_r(\Sigma)}$. The proof is based on analogues result by Tilli on average distance minimizers. Finally, we generalize Tilli's result from the plane to $d$-dimensional Euclidean space

Homology of the Pronilpotent Completion and Cotorsion Groups

For a non-cyclic free group F, the second homology of its pronilpotent completion H-2((F) over cap) is not a cotorsion group.Peer reviewe