**Equiangular lines, mutually unbiased bases, and difference sets** - Aidan Roy

In 1975, Delsarte, Goethals and Seidel found an upper bound
for the size of a "system of complex lines": a set of vectors whose
scalar products have only a small number of distinct absolute values.
Recently, applications in quantum tomography have renewed interest in
constructing systems of maximal size. We consider two particular
problems: equiangular lines, which are sets in which only one
absolute value occurs; and mutually unbiased bases, in which two
values occur and the lines are partitioned into orthonormal bases.
For real vector spaces, these systems are closely related to regular
two-graphs and Hadamard matrices. However, the complex versions are
not nearly as well understood.
In this talk, we will see that combinatorial structures such as
difference sets and Cayley graphs can also be used for constructions
in complex case. In particular, we use difference sets in abelian
groups to produce equiangular lines, and we use relative difference
sets to produce maximal sets of mutually unbiased bases.
This talk is based on joint work with Chris Godsil.