Mikhail Basok’s research while affiliated with University of Helsinki and other places

For a non-cyclic free group F , the second homology of its pronilpotent completion ${H_2}(\widehat F)$ H 2 ( F ^ ) is not a cotorsion group.

For a non-cyclic free group F, the second homology of its pronilpotent completion $H_2(\widehat F)$ is not a cotorsion group.

Building upon recent results of Dubédat [7] on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations \Omega^\delta to a simply connected domain \Omega\subset\mathbb C we prove the convergence of probabilities of cylindrical events for the double-dimer loop ensembles on \Omega^\delta as \delta\to 0 . More precisely, let \lambda_1,\dots,\lambda_n\in\Omega and L be a macroscopic lamination on \Omega\setminus\{\lambda_1,\dots,\lambda_n\} , i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities P_L^\delta that one obtains L after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on \Omega^\delta converge to a conformally invariant limit P_L as \delta \to 0 , for each L . Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety Hom (\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb C) and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do not use any RSW-type arguments for double-dimers. The limits P_L of the probabilities P_L^\delta are defined as coefficients of the isomonodromic tau-function studied in [7] with respect to the Fock–Goncharov lamination basis on the representation variety. The fact that P_L coincides with the probability of obtaining L from a sample of the nested CLE(4) in \Omega requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble.

We continue the study of the rational Picard group of the moduli space of Hitchin spectral covers started in Korotkin and Zograf (J Math Phys 59(9):091412, 2018). In the first part of the paper we expand the “boundary”, “Maxwell stratum” and “caustic” divisors introduced in Korotkin and Zograf (2018) via the set of standard generators of the rational Picard group. This generalizes the result of Korotkin and Zograf (2018), where the expansion of the full discriminant divisor (which is a linear combination of the classes mentioned above) was obtained. In the second part of the paper we derive a formula that relates two Hodge classes in the rational Picard group of the moduli space of Hitchin spectral covers.

We continue the study of the rational Picard group of the moduli space of Hitchin's spectral covers started in P. Zograf's and D. Korotkin's work [11]. In the first part of the paper we expand the ``boundary'', ``Maxwell stratum'' and ``caustic'' divisors introduced in [11] via the set of standard generators of the rational Picard group. This generalizes the result of [11], where the expansion of the full discriminant divisor (which is a linear combination of the classes mentioned above) was obtained. In the second part of the paper we derive a formula that relates two Hodge classes in the rational Picard group of the moduli space of Hitchin's spectral covers.

Building upon recent results of Dub\'edat (see arXiv:1403.6076) on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations $\Omega^\delta$ to a simply connected domain $\Omega\subset\mathbb C$ we prove the convergence of probabilities of cylindrical events for the \emph{double-dimer loop ensembles} on $\Omega^\delta$ as $\delta\to 0$. More precisely, let $\lambda_1,\dots,\lambda_n\in\Omega$ and L be a macroscopic lamination on $\Omega\setminus\{\lambda_1,\dots,\lambda_n\}$, i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities $P_L^\delta$ that one obtains L after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on $\Omega^\delta$ converge to a conformally invariant limit $P_L$ as $\delta \to 0$, for each L. Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety $\mathrm{Hom}(\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb C))$ and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do \emph{not} use any RSW-type arguments for double-dimers. The limits $P_L$ of the probabilities $P_L^\delta$ are defined as coefficients of the isomonodormic tau-function studied by Dub\'edat with respect to the Fock--Goncharov lamination basis on the representation variety. The fact that $P_L$ coincides with the probability to obtain L from a sample of the nested CLE(4) in $\Omega$ requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble.

In this paper we introduce a divisor in the moduli space of odd spin curves parametrizing spin curves such that some meromorphic function naturally associated with the spin structure has less number of branch points then maximal. We express the class of this divisor in the rational Picard group via the set standards generators.

The goal of the paper is to give an analytic proof of the formula of G. Farkas for the divisor class of spinors with multiple zeros in the moduli space of odd spin curves. We make use of the technique developed by Korotkin and Zograf that is based on properties of the Bergman tau function. We also show how the Farkas formula for the {\it theta-null} in the rational Picard group of the moduli space of even spin curves can be derived from classical theory of theta functions.