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This isn't an excessively deep answer, but you can express a type system based on STLC with prenex polymorphism as a Pure Type System in a quite simple way, using sorts $*_{\mathrm{mono}}$, $*_{\mathrm{poly}}$ and $\square$ along with the axioms $$*_{\mathrm{mono}}, *_{\mathrm{poly}}\ :\ \square$$ and the rules (*_{\mathrm{mono}},*_{\mathrm{mono}},*_{\...
Apart from what's already written in the slides you linked to, let me describe one possible approach. For studying type inference semantically we need a model in which a term can have many types, or none. This naturally leads to Curry-style typing, i.e., we think of $t : A$ as a relation where both the term $t$ and the type $A$ are meaningful by themselves. (...