We consider a random planar map Mn which is uniformly distributed over the class of all rooted q-angulations with n faces. We let mn be the vertex set of Mn, which is equipped with the graph distance dgr. Both when q≥4 is an even integer and when q=3, there exists a positive constant cq such that the rescaled metric spaces (mn,cqn−1/4dgr) converge in distribution in the Gromov–Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.

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Uniqueness and universality of the Brownian map

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.

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Complete set of cut-and-join operators in the Hurwitz-Kontsevich theory

https://math.temple.edu/~dfuter/research/Jones-ribbon-graph.pdf

The Jones polynomial and graphs on surfaces

We review recent developments in the theory of brane tilings and four-dimensional = 1 supersymmetric quiver gauge theories. This review consists of two parts. In part I, we describe foundations of brane tilings, emphasizing the physical interpretation of brane tilings as fivebrane systems. In part II, we discuss application of brane tilings to AdS/CFT correspondence and homological mirror symmetry. More topics, such as orientifold of brane tilings, phenomenological model building, similarities with BPS solitons in supersymmetric gauge theories, are also briefly discussed. This paper is a revised version of the author's master's thesis submitted to Department of Physics, Faculty of Science, the University of Tokyo on January 2008, and is based on his several papers and some works in progress [1–7].

Brane tilings and their applications

We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the 2-sphere.

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An algebro-geometric proof of Witten's conjecture

0 Introduction: What is This Book About.- 1 Constellations, Coverings, and Maps.- 2 Dessins d'Enfants.- 3 Introduction to the Matrix Integrals Method.- 4 Geometry of Moduli Spaces of Complex Curves.- 5 Meromorphic Functions and Embedded Graphs.- 6 Algebraic Structures Associated with Embedded Graphs.- A.1 Representation Theory of Finite Groups.- A.1.1 Irreducible Representations and Characters.- A.1.2 Examples.- A.1.3 Frobenius's Formula.- A.2 Applications.- A.2.2 Examples.- A.2.3 First Application: Enumeration of Polygon Gluings.- A.2.4 Second Application: the Goulden-Jackson Formula.- A.2.5 Third Application: "Mirror Symmetry" in Dimension One.- References.

Graphs on Surfaces and Their Applications

The connections recently established between combinatorial bicolored plane trees and Shabat polynomials show that the world of plane trees is incredibly rich with different mathematical structures. In this article we use Shabat polynomials to introduce a new operation, that of a composition, for combinatorial bicolored plane trees. The composition may be considered as a generalized symmetry.

Composition of Plane Trees

One of the consequences of the classification of finite simple groups is the fact that non-rigid polynomials (those with more than two finite critical values), considered as branched coverings of the sphere, have exactly three exceptional monodromy groups (one in degree 7, one in degree 13 and one in degree 15). By exceptional here we mean primitive and not equal to S_n or A_n, where n is the degree. Motivated by the problem of the topological classification of polynomials, a problem that goes back to 19th century researchers, we discuss several techniques for investigating orbits of braid groups on ``cacti'' (ordered sets of monodromy permutations). Applying these techniques, we provide a complete topological classification for the three exceptional cases mentioned above.

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Orbits of braid groups on cacti

We consider the usual model of hypermaps or, equivalently, bipartite maps, represented by pairs of permutations that act transitively on a set of edges E. The specific feature of our construction is the fact that the elements of E are themselves (or are labelled by) rather complicated combinatorial objects, namely, the 4-constellations, while the permutations defining the hypermap originate from an action of the Hurwitz braid group on these 4-constellations.The motivation for the whole construction is the combinatorial representation of the parameter space of the ramified coverings of the Riemann sphere having four ramification points.

/pdf/megamaps-construction-and-examples-2e7uo32hzc.pdf

Megamaps: Construction and Examples

Let R, S ∈ ℂ[x] be two coprime polynomials of the same degree with prescribed multiplicities of their roots. A classical problem of number theory actively studied during the last half-century is, What could be the minimum degree of the difference T = R − S? The theory of dessins d’enfants implies that such a minimum is attained if and only if the rational function f = R/T is a Belyi function for a bicolored plane map all of whose faces except the outer one are of degree 1. Such maps are called weighted trees, since they can be conveniently represented by plane trees whose edges are endowed with positive integral weights.

Alexander K. Zvonkin

Papers

Graphs on Surfaces and Their Applications

Composition of Plane Trees

Orbits of braid groups on cacti

Megamaps: Construction and Examples

Weighted Trees with Primitive Edge Rotation Groups