**Title**

Unique low rank completability of partially filled matrices

**Abstract**

I will consider the matrix completion problem - we are given a partially filled matrix and want to add entries in such a way that the resulting matrix has low rank. More precisely, I will assume that such a completion exists and ask whether it is unique. I will also consider the variants when the completed matrix should have the additional properties that it is a gram matrix or is skew-symmetric.

I will describe how techniques from rigidity theory can be applied to help analyse these problems. In particular, how the unique completability problem for a generic partially filled matrix can be converted to that of determining the rank function of a matroid defined on the edge set of a bipartite graph.