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## On decompositions of a cyclic permutation into a product of a given number of permutations

The investigation of decompositions of a permutation into a product of permutations satisfying certain conditions plays a key role in the study of meromorphic functions or, equivalently, branched coverings of the 2-sphere; it goes back to A. Hurwitz’ work in the late nineteenth century. In 2000 M. Bousquet-Melou and G. Schaeffer obtained an elegant formula for the number of decompositions of a permutation into a product of a given number of permutations corresponding to coverings of genus 0. Their formula has not been generalized to coverings of the sphere by surfaces of higher genera so far. This paper contains a new proof of the Bousquet-Melou-Schaeffer formula for the case of decompositions of a cyclic permutation, which, hopefully, can be generalized to positive genera. © 2015, Springer Science+Business Media New York.

I show that Hurwitz numbers may be generated by certain correlation functions which appear in quantum chaos.

We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues are expressed through the characters of symmetric groups. These operators generate differential equations for partition functions of Hurwitz numbers.

We construct partition functions that are tau-functions of integrable hierarchies.

We define cut-and-join operators in Hurwitz theory for merging two branch points of an arbitrary type. These operators have two alternative descriptions: (1) the GL characters are their eigenfunctions and the symmetric group characters are their eigenvalues; (2) they can be represented as W-type differential operators (in particular, acting on the time variables in the Hurwitz–Kontsevich τ -function). The operators have the simplest form when expressed in terms of the Miwa variables. They form an important commutative associative algebra, a universal Hurwitz algebra, generalizing all group algebra centers of particular symmetric groups used to describe the universal Hurwitz numbers of particular orders. This algebra expresses arbitrary Hurwitz numbers as values of a distinguished linear form on the linear space of Young diagrams evaluated on the product of all diagrams characterizing particular ramification points of the branched covering.

This book collects papers based on the XXXVI Białowieża Workshop on Geometric Methods in Physics, 2017. The Workshop, which attracts a community of experts active at the crossroads of mathematics and physics, represents a major annual event in the field. Based on presentations given at the Workshop, the papers gathered here are previously unpublished, at the cutting edge of current research, and primarily grounded in geometry and analysis, with applications to classical and quantum physics. In addition, a Special Session was dedicated to S. Twareque Ali, a distinguished mathematical physicist at Concordia University, Montreal, who passed away in January 2016.

For the past six years, the Białowieża Workshops have been complemented by a School on Geometry and Physics, comprising a series of advanced lectures for graduate students and early-career researchers. The extended abstracts of this year’s lecture series are also included here. The unique character of the Workshop-and-School series is due in part to the venue: a famous historical, cultural and environmental site in the Białowieża forest, a UNESCO World Heritage Centre in eastern Poland. Lectures are given in the Nature and Forest Museum, and local traditions are interwoven with the scientific activities.