Moduli problems often manifest a remarkable symmetry now known as a shifted symplectic/Lagrangian structure. They are crucial because it provides the right framework for defining and counting objects in these spaces.

In the first part of the talk. I will explain why such symmetries arise in a natural way. Then I will present a construction of a homomorphism from the K-group of matrix factorisations of the critical locus to the K-group of the Lagrangian. The key step is the construction of a specialisation functor for categories of matrix factorisations along the deformation to the normal cone. This is joint work with Jeongseok Oh.