• Basic differential topology, basic Morse theory
• Category theory and standard theorems (Yoneda lemma, adjoint functor theorem, etc.)
• Simplicial sets
• Basic homotopy theory
• Some models of $(\infty,n)$-categories, and their equivalences

It would be amazing if we could formalise some version of the cobordism hypothesis, which connects Morse theory and $(\infty,n)$-category theory. At least the statement should be feasible.