3/3 Mathematical Physics of Hurwitz numbers
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Metadata
Formal Metadata
Title  3/3 Mathematical Physics of Hurwitz numbers 
Title of Series  Physique mathematique des nombres Hurwitz pour debutants 
Number of Parts  6 
Author 
Kazarian, Maxim

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/17008 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2014 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  Hurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge number of relations represented in the form of partial differential equations for their generating function. This includes equations of the KP hierarchy, Virasorotype constraints, ChekhovEynardOrantintype recursion and others. Only a few of these relations can be derived from elementary combinatorics of permutations. All other relations follow from a deep relationship of Hurwitz numbers with moduli spaces of curves, GromovWitten invariants, matrix models, integrable systems and other domains of mathematics which are often referred to as `mathematical physics'. When discussing Hurwitz numbers in the talks, we consider them, thereby, as a sufficiently elementary but highly nontrivial model of all mentioned theories where all computations can be fulfilled completely, and all formulated relations can be checked explicitly in computer experiments. 