Homological mirror symmetry for invertible polynomials in two variables
The starting point for homological mirror symmetry for invertible polynomials is an n x n invertible matrix with non-negative integer entries. To such a matrix, as well as to its transpose, one can associate polynomials. These polynomials are called invertible if they are weighted homogeneous, and both define isolated singularities at the origin. Homological mirror symmetry for invertible polynomials is a series of conjectures which posits the equivalence of the different flavours of Fukaya category associated to the Lefschetz fibration defined by one polynomial with various flavours of graded matrix factorisations defined by the transpose polynomial. Particular to the case of two variables is the fact that the partially wrapped Fukaya category of a Milnor fibre corresponds to the derived category of modules of a gentle algebra, and so HMS for invertible polynomials in two variables allows one to study the latter category geometrically. In this talk I will explain my recent work, part of which was done jointly with Jack Smith.