I'll start by introducing the notions of logarithmic lambda-connections, parabolic structure, stability, moduli space, and Hilbert scheme of points on surfaces.

Then I'll go on to discuss apparent singularities of lambda-connections, elementary transformations and opers. In the second part of the talk I will explain joint work with Masa-Hiko Saito on a birational map between the moduli space of stable parabolic logarithmic Higgs bundles on a smooth projective curve C and the Hilbert scheme of points on a ruled surface over C. I end the talk by discussing the indeterminacy locus of this birational map.

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