Constructions of Complex Equiangular Lines

A set of unit vectors in $\C^d$ represents equiangular lines if the magnitudes of the inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is $d^2$, and it is conjectured that sets of this maximum size exist in $\C^d$ for every $d\geq 2$. In this thesis we describe evidence supporting the conjecture. We explain the fiducial vector method of construction, outlining the methods of approach and combining the viewpoints of several authors responsible for known examples of maximum-sized sets of equiangular lines. We also identify several milestone publications and note specific examples of maximum-sized sets of equiangular lines which we believe to be key in eventually resolving the conjecture.Furthermore we give two new maximum-sized sets of equiangular lines in dimension 8; one set is simpler than all previously known examples and the other illuminates previously unrecognized underlying structure through connections with other dimensions. Using these sets of lines we are able to demonstrate some important connections between equiangular lines and other combinatorial objects, including Hadamard matrices, mutually unbiased bases and relative difference sets.