The study of Hurwitz numbers, which enumerate branched coverings of the Riemann sphere, is classical, going back to the pioneering work of Hurwitz in the 1880’s. There is an equivalent combinatorial problem, related by monodromy that was developed by Frobenius in his pioneering work on character theory, consisting of enumeration of factorizations of elements of the symmetric group. In 2000, Okounkov and Pandharipande began their program relating Hurwitz numbers to other combinatorial/topological invariants associated to Riemann surfaces, such as as Gromov-Witten and Donaldson-Thomas invariants. This has since been further developed by others to include, e.g., Hodge invariants and relations to knot invariants. A key result of Okounkov and Pandharipande was to express the generating functions for special classes of Hurwitz numbers, e.g., including only simple branching, plus one, or two other branch points, as special types of Tau functions of integrable hierarchies such as Sato's KP hierarchy and Takasaki-Takebe’s 2D Toda lattice hierarchy, together with associated semi-infinite wedge product representations. The differential/algebraic equations satisfied by such generating functions provide a new perspective, implying deep interrelations between these various types of enumerative invariants. In more recent work, these ideas have been extended to include generating functions for a very wide class of branched coverings, with suitable combinatorial interpretations, including broad class of weighted enumerations that select amongst infinite parametric families of weights. These make use not only of the six standard bases for the ring of symmetric functions, such as Schur functions, and monomomial sum symmetric functions, but also their “quantum” deformations, involving the pair of deformation parameters (q,t) appearing the in theory of Macdonald polynomials. The general theory of weighted Hurwitz numbers, together with various applications and examples coming from Random Matrix theory and enumerative geometry will be explained in a simple, unified way, based on special elements of and bases for the center of the symmetric group algebra, and the characteristic map to the ring of symmetric polynomials. The simplest quantum case provides a relation between special weighted enumerations of branched coverings and the statistical nechanics of Bose-Eintein gases. Various other specializations, to such bases as: Hall-Littlewood, Jack, q-Whittaker, dual q-Whttaker as well as certain special classical weightings have further applications, in physics, geometry, group theory and combinatorics.
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