Abstract:[NOTE THE UNUSUAL TIME] In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in a R^r with angle arccos(alpha) and gave a partial answer in the regime r <= 1/alpha^2 - 2. At the other extreme where r is at least exponential in 1/alpha, recent breakthroughs have led to an almost complete resolution of this problem. In this talk, we introduce a new method for obtaining upper bounds which unifies and improves upon all previous approaches, thereby yielding bounds which bridge the gap between the aforementioned regimes and are best possible either exactly or up to a small multiplicative constant. Our approach is based on orthogonal projection of matrices with respect to the Frobenius inner product and it also yields the first extension of the Alon--Boppana theorem to dense graphs, with equality for strongly regular graphs corresponding to r(r+1)/2 equiangular lines in R^r. Applications of our method in the complex setting will be discussed as well.