Hurwitz numbers

E904001

algebraic invariant combinatorial invariant enumerative invariant object of Gromov–Witten theory object of algebraic geometry object of enumerative geometry object of intersection theory object of mathematical physics object of moduli theory topological invariant

Hurwitz numbers are algebraic invariants that count branched coverings of the Riemann sphere (or other curves) with specified ramification data, playing a key role in enumerative geometry and mathematical physics.

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Statements (52)

Predicate	Object
instanceOf	algebraic invariant ⓘ combinatorial invariant ⓘ enumerative invariant ⓘ object of Gromov–Witten theory ⓘ object of algebraic geometry ⓘ object of enumerative geometry ⓘ object of intersection theory ⓘ object of mathematical physics ⓘ object of moduli theory ⓘ topological invariant ⓘ
counts	branched coverings of algebraic curves ⓘ branched coverings of the Riemann sphere ⓘ holomorphic maps between Riemann surfaces with prescribed ramification ⓘ
dependsOn	degree of the covering map ⓘ genus of the source curve ⓘ genus of the target curve ⓘ number of branch points ⓘ ramification profile over branch points ⓘ
hasVariant	ELSV-type Hurwitz numbers ⓘ double Hurwitz numbers ⓘ monotone Hurwitz numbers NERFINISHED ⓘ orbifold Hurwitz numbers ⓘ quantum Hurwitz numbers ⓘ simple Hurwitz numbers ⓘ spin Hurwitz numbers ⓘ tropical Hurwitz numbers ⓘ
namedAfter	Adolf Hurwitz NERFINISHED ⓘ
relatedTo	2D topological gravity ⓘ Belyi maps NERFINISHED ⓘ ELSV formula NERFINISHED ⓘ Gromov–Witten invariants NERFINISHED ⓘ Hodge integrals NERFINISHED ⓘ KP hierarchy NERFINISHED ⓘ Riemann–Hurwitz formula NERFINISHED ⓘ Toda hierarchy NERFINISHED ⓘ Young diagrams NERFINISHED ⓘ conjugacy classes in symmetric groups ⓘ dessins d’enfants NERFINISHED ⓘ integrable hierarchies ⓘ intersection theory on moduli of curves ⓘ matrix models ⓘ moduli space of branched covers ⓘ moduli space of curves ⓘ random partitions ⓘ string theory ⓘ symmetric group representations ⓘ topological recursion ⓘ tropical geometry ⓘ
usedIn	algebraic geometry ⓘ combinatorics of permutations ⓘ enumerative geometry ⓘ mathematical physics ⓘ

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Full triples — surface form annotated when it differs from this entity's canonical label.

Hurwitz space → relatedTo → Hurwitz numbers ⓘ