Quantization of moduli spaces and the enumerative information it contains is a very common theme in modern mathematical physics, which can be approached by several different tools: algebraic geometry, mirror symmetry, quantum algebra, hyperbolic geometry, representation theory, integrable systems. The enumerative content in questions concerns ”counting surfaces” and therefore relates to the geometry of the moduli space of curves and their cousins (spin, open, super, etc.). The purpose of this summer school is to present a selection of recent developments in this area with a broad range of applications, which by revisiting classical problems and geometric constructions with new ideas offer transversal views on these problems. A common theme of all lectures will be that geometry – of surfaces and of moduli spaces attached to them – enlightens the algebraic structures that often permit computations in various manifestations of topological field theories.

The courses aim at PhD/postdoc level. All researchers are welcome.


  • Geometric recursion
    Gaëtan Borot, Humboldt-Universität zu Berlin

  • Double ramification hierarchies
    Alexandr Buryak, Higher School of Economics, and Paolo Rossi, Universita degli Studi di Padova

  • Quantised Painlevé monodromy manifold, Sklyanin and Calabi-Yau algebras
    Marta Mazzocco, Birmingham University

  • Spin structures and cohomology classes on the moduli space of curves
    Paul Norbury, University of Melbourne

  • Geometric realisations of Bethe equations
    Anton Zeitlin, Lousiana State University

  • School directors

    Gaëtan Borot, Paul Norbury, Paolo Rossi


    Centro Internazionale Matematico Estivo, University degli Studi di Padova, Research grant BIRD190395, Humboldt Universität zu Berlin, ERC ReNew Quantum, ARC discovery grant DP180103891, University of Melbourne, National Group for Algebraic and Geometric Structures, and their Applications (GNSAGA – INDAM).