Combinatorics of Graphs and Topological Recursion

Student: Tsepelev Vladimir

Educational Programme: Mathematics (Master)

Final Grade: 7

Year of Graduation: 2018

Spectral curve topological recursion associates an infinite family of multi-differentials to a spectral curve. It turns out that under the proper choice of spectral curve these multi-differentials become generating functions for many well-known objects in mathematics, like Gromov-Witten invariants, knot polynomials (Jones, HOMFLY) and Hurwitz numbers. This thesis provides an overview of general topological recursion technique and examples of applications to selected problems in enumerative geometry and graph theory. The original research results include proof of the particular case of the Bouchard-Marino conjecture and derivation of a necessary condition for topological recursion applicability to numerical sequence through condition on its subsequence corresponding to genus zero and Young diagram with two rows.

Full text (added June 1, 2018)

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