Konstantin Jakob (Darmstadt): Algebraic Loops and Stokes Braids

The irregular class of a rank n differential equation with an irregular singularity may be viewed as a polynomial map to the configuration space of n points. Such irregular classes feature prominently in Boalch’s construction of wild character varieties, which are Betti moduli spaces for irregular connections. As the local coordinate winds around the singularity, the irregular class describes the motion of eigenvalues of leading terms; this motion determines a braid, called the Stokes braid.

I will report on joint work with Masoud Kamgarpour and Ian Le in which we classify braids arising from such algebraic loops in terms of valuation data. I will explain why tropical geometry provides the natural language for this question, and, if time permits, outline relations to several phenomena where the geometry or cohomology of spaces such as braid varieties is expected to depend only on tropical data.

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