Stokes structures and moduli spaces related to Markov triples and to Hurwitz spaces

• Applicant: Professor Dr. Claus Hertling
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 494849004

Families of holomorphic functions with isolated singularities come equipped with a rich geometry, with transcendental as well as combinatorial data. They are studied within algebraic geometry, but the involved structures are related to many other areas. Their Milnor lattices are Z-lattices with many additional structures, a monodromy operator, several bilinear forms, several monodromy groups, a set of vanishing cycles, and a set of distinguished bases with a braid group action on it.These distinguished bases are related to the Stokes structures of the Fourier-Laplace transformation of the Gauss-Manin connection of the holomorphic family of functions. Stokes regions, one for each distinguished basis (up to signs), glue to an abstract moduli space, which has to be compared with other constructions of moduli spaces.In the project, these data shall be studied in two situations. One is the family of Laurent polynomials which are the mirror partner of the quantum cohomology of the complex projective plane. Here the distinguished bases and the Stokes matrices are closely related to the Markov triples. The project will hopefully shed new light on the rich structures around the Markov triples and especially on the uniqueness conjecture for them.The other situation are the families of functions behind the Hurwitz spaces. The Hurwitz spaces and especially the induced Hurwitz numbers have been studied intensily. But the point of view from distinguished bases and from glueing Stokes regions to moduli spaces has not been taken. These moduli spaces are certain coverings of the Hurwitz spaces with properties which shall be explored.

• DFG Programme: Research Grants