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Lern- und Qualifikationsziele
The topological recursion is a universal recursion, whose structure resembles the recursive construction of surfaces of any topology by cutting and pasting, and governs diverse problems in mathematics and physics. The first part of the course will introduce the recursion, discuss its algebraic meaning from two perspectives (quantization in affine symplectic spaces, geometry of Riemann surfaces) and study its essential properties. The second part will show how it can be applied to asymptotic expansion in random matrix theory, to the enumeration of branched covers of the sphere (Hurwitz theory) and to intersection theory on the moduli space of complex curves.
Voraussetzungen
Prerequisites are a good command of complex analysis (as well as differential geometry for Part V). Prior knowledge in random matrix theory and on the moduli space of curves is not necessary.
Gliederung / Themen / Inhalte
I. Airy structures, partition function, quantization of quadratic Lagrangian in affine symplectic spaces and its relation to the topology of surfaces, properties. Basic examples and constructions from the Virasoro algebra.
II. Spectral curves and topological recursion. Abstract loop equations and their solution by the topological recursion. Form cycle duality, variation of initial data and special geometry, holomorphic anomaly equation. Prepotential.
III. Formal and convergent matrix integrals. Schwinger-Dyson equations. The large size limit and the spectral curve. Existence of all-order asymptotic expansion in the size. Relation to loop equations and solution by the topological recursion.
IV. Hurwitz theory: branched covers of surfaces, representation of the symmetric group, enumeration problem. Cut and join equations, relation to loop equations, solution by the topological recursion. Glimpse into mirror symmetry.
V. The moduli space of curves, tautological classes. psi-classes. Combinatorial model of the moduli space and its geometry. Mirzakhani-McShane type-identities and topological recursion. Witten-Kontsevich theorem. General representation of the output of topological recursion via intersection theory. Application: ELSV formula for Hurwitz numbers. Cohomological field theories, TQFTs, Givental group action. Teleman's classification.
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