Title: Connections between complex lines and distance-regular graphs
| Speaker: | Gabriel Coutinho |
| Affiliation: | University of Waterloo |
| Room: | MC 5501 |
Abstract: A set of lines in a vector space is called equiangular if the angle between any two lines is the same. Consider the following apparently simple problem: what is the maximum number of equiangular lines in a vector space of dimension $d$? For example, in $\mathbb{R}^2$, it is obvious that the best one can do is to pick the three lines corresponding to the medians of an equilateral triangle. Our intuition begins to fail when we look to complex vector spaces. For example, there are four equiangular lines in $\mathbb{C}^2$ and nine in $\mathbb{C}^3$.
As it turns out, sets of equiangular lines are intimately related to certain antipodal distance-regular graphs of diameter three. In a recent work joint with Godsil, Shirazi and Zhan, we were able to deepen the understanding of this connection, but there are still many important open questions about the topic. This talk will be about some of the history, the old results, and our new contributions. It will require no prior knowledge about any of these topics.