Calabi-Yau threefolds in codimension 4

We study Calabi-Yau threefolds embedded in P^{^7} as arithmetically Cohen-Macaulay (aCM) subschemes. Using the theory of bilinkage and unprojection, we obtain a list of aCM Calabi-Yau threefolds, and present evidence which suggests our list forms a complete classification. We also have explicit descriptions for most of the threefolds in our list, and we can also understand various other properties such as their Kähler cones. This has important applications to the commutative algebra relating to manifolds embedded in codimension 4, and to the problem of finding mirror partner candidates for each Calabi-Yau in our list. This is joint work in progress with L. Golebiowski, G. Kapustka and M. Kapustka.