Billiards and Surfaces à la Teichmüller and Riemann, Online

organized by Simion Filip, Carlos Matheus, Curtis McMullen and Martin Möller

14 June 2021
Time: 11am (PDT) / 2pm (EDT) / 8pm (CEST)

Maxim Kontsevich (IHÉS)

Title: Wall-crossing for abelian differentials

Abstract: For an abelian differential on a complex curve one can count saddle connections in all possible relative homology classes. These numbers jump when one crosses a wall in the moduli space of abelian differentials. I will show that the jumping formula is a particular case of the general wall-crossing formalism of Y.Soibelman and myself. The corresponding graded Lie algebra is the algebra of matrices over the ring of Laurent polynomials in several variables. The wall-crossing structure is explicitly calculable, and is determined by a finite collection of invertible matrices over the field of rational functions. The whole story generalizes from curves to higher-dimensional complex algebraic varieties.

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