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Graphs on Surfaces and Their Applications
TLDR
In this article, the authors introduce the Matrix Integrals method and the Goulden-Jackson Formula for computing polygonal gluings, and apply it to the problem of "Mirror Symmetry in Dimension One".Abstract:
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.read more
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Uniqueness and universality of the Brownian map
TL;DR: In this article, a random planar map Mn which is uniformly distributed over the class of all rooted q-angulations with n faces is considered, where mn is the vertex set of Mn, which is equipped with the graph distance dgr.
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Complete set of cut-and-join operators in the Hurwitz-Kontsevich theory
TL;DR: Cut-and-join operators in Hurwitz theory for merging two branch points of an arbitrary type have been studied in this paper, where they can be represented as W-type differential operators acting on the time variables in the Hurwitz-Kontsevich τ-function.
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The Jones polynomial and graphs on surfaces
TL;DR: It is shown that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynometric of a certain oriented ribbon graph associated to a link projection.
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Brane tilings and their applications
TL;DR: A review of recent developments in the theory of brane tilings and four-dimensional = 1 supersymmetric quiver gauge theories can be found in this article, where the authors describe the foundations of the theory and apply it to AdS/CFT correspondence and homological mirror symmetry.
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An algebro-geometric proof of Witten's conjecture
Maxim Kazarian,Sergei K. Lando +1 more
TL;DR: In this paper, a new proof of Witten's conjecture is presented 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.