The state of the art on equiangular lines

In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in $R^r$ with angle arccos(α) and gave a good partial answer in the regime $r ≤ 1/α^2 − 2$. At the other extreme where r is at least exponential in $1/α^2$, recent breakthroughs have led to an almost complete solution. In this talk, we will describe our recent progress on this problem, partly in joint work with Matija Bucić. In particular, we obtain upper bounds that unify and significantly extend or improve essentially all previously known results, thereby bridging the gap between the aforementioned regimes and determining the answer up to a factor of 2.
Roughly speaking, our approach relies on new lower bounds on the second eigenvalue of a graph as well as new upper bounds on its multiplicity. In particular, we obtain the first extension of the Alon–Boppana theorem to dense graphs, with equality for strongly regular graphs corresponding to families of $r+1$ choose 2 equiangular lines in $R^r$. Time permitting, we will also mention some results in the complex setting.