Mirror symmetry for chain type polynomials

I will start by explaining Takahashi's homological mirror symmetry (HMS) conjecture regarding invertible polynomials, which is an open string reinterpretation of Berglund-Hubsch-Henningson mirror symmetry. In joint work with A. Polishchuk, we resolved this HMS conjecture in the chain type case up to rigorous proofs of general statements about Fukaya-Seidel categories. Our proof goes by showing that the categories in both sides are obtained from the category Vect(k) by applying a recursion. I will explain this recursion categorically and sketch the argument for why it is satisfied on the A-side assuming the aforementioned foundational results. If time permits, I will also mention what goes into the proof in the B-side.