Lecture 1, Maximal Cohen-Macaulay modules over complete intersection rings.
Abstract: We explain the structure theory of maximal Cohen-Macaulay modules over the rings in the title and show how that structure can be revealed using the representation theory of the differential graded algebra provided by the Koszul complex on the defining equations.
We will discuss applications and remaining challenges.
Lecture 2, The Orlov octahedron on a graded Gorenstein algebra.
Abstract: One of the deepest results in the theory of maximal Cohen-Macaulay modules surely is Orlov's theorem from 2005 that relates the stable category of graded such modules over a not necessarily commutative graded Gorenstein ring A to the derived category of coherent sheaves on the underlying (virtual) projective scheme. We explain how this result defines for any graded A-module an octahedron of complexes of such modules that connects projective geometry and maximal Cohen-Macaulay approximations.
Time permitting we will discuss in some detail the case of graded matrix factorizations, or graded maximal Cohen-Macaulay modules, over (quasi-)homogeneous hypersurface rings.