The ACO Seminar (2018–2019)

The problem of finding the maximum number of equiangular lines in d-dimensional Euclidean space has been studied extensively over the past 80 years, in part because it is a fundamental problem in discrete geometry, but also because of its applications in signal processing, communications and coding theory. The absolute upper bound on the number of equiangular lines that can be found $\mathbb{R}^d$ is d(d+1)/2. However, examples of sets of lines that saturate this bound are only known to exist in dimensions d=2,3,7 or 23, and it is an open question whether this bound is achieved in any other dimension. By considering the additional property of incoherence, we prove that there exists a set of equiangular lines that saturates the absolute bound and the incoherence bound if and only if d=2,3,7 or 23.

For a given angle $\kappa$, there exists a relative upper bound on the number of equiangular lines in $\mathbb{R}^d$ with common angle $\kappa$. We show that classifying sets of lines that saturate this bound along with the incoherence bound is equivalent to classifying certain quasi-symmetric designs, which are combinatorial designs with two block intersection numbers. We will also show how such sets of lines are related to the integer points of an elliptic surface.

Interestingly, given a further natural assumption, we can classify the known sets of lines that saturate the relative and incoherence bounds. This family comprises of the known sets of lines that saturate the absolute bound along with the the maximal set of 16 equiangular lines found in $\mathbb{R}^6$. There are infinitely many known sets of lines that saturate the relative bound, so this result is surprising. To understand these sets of lines further, we identify the E8 lattice with the projection onto an 8-dimensional subspace of a sub-lattice of the Leech lattice defined by 276 equiangular lines in $\mathbb{R}^{23}$. This allows us to identify many of the maximal sets of of equiangular lines in small dimensions with subsets of the 276 equiangular lines in $\mathbb{R}^{23}$.

Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220.