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Name of the project: Cluster algebras and moduli spaces of flat and holomorphic connections

Goals and objectives

The main objective of this project is the expansion of methods applicable for the computation of the canonical characteristic classes in moduli spaces of complex curves over spaces of flat connections. The modern approach to the computation of integrals of canonical classes is based on topological recursion that allows to recurrently express integrals of characteristic classes. The initial state of the proposed project is the formation of a cluster theory for moduli spaces of curves and the study of their applications in the computation of integrals of canonical classes.

The practical value of the study
Projected research results: 

  • To study the degeneracy of the so-called  «‎shear» coordinates and Weil-Petersson forms on the boundary strata of the Teichmüller space;
  • To generalise degeneration formulas for other cluster manifolds, for example, for double Bruhat cells;
  • To generalise topological recursion methods for boundary strata of general cluster manifolds by studying Poisson bracket degeneracy;
  • To study the relation between tropical cluster varieties and varieties of tropical curve;
  • To find a geometric representation for of Belavin-Drinfeld brackets and to determine the corresponding connectivity spaces;
  • To generalise real Hurwitz numbers introduced by Zvonkin and Itenberg for polynomial maps to rational maps.

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