Abstract: The multiplicative Deligne-Simpson problem (DSP) asks the following question: given an $n$-tuple of conjugacy classes of matrices, can we choose $n$ matrices from these classes such that their product is the identity and they have no common invariant subspace? Another way to formulate the DSP is to ask for a criterion for the existence of irreducible local systems on the punctured sphere with prescribed monodromy data. Many works have been done in this direction using various methods by Simpson, Katz, Kostov, Crawley-Boevey, and Shaw. In this talk, I will present an alternative approach to the DSP by establishing a relative spectral correspondence for parabolic Higgs bundles. This is joint work with Sukjoo Lee.
