Connectivity and simple connectivity of spaces of differentials

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The geometry of moduli spaces of differentials on curves is connected to a number of seemingly distant parts of mathematics, notably to spaces of stability conditions and to torsion point questions on hyperelliptic curves. Our goal is to investigate connectivity questions for strata of these moduli spaces and leaves of certain foliation given by imposing conditions on periods as well as questions about simple connectivity of the covering spaces given by adding Teichmüller markings.