  A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between
each pair of lines is the same. We address the question of determining the
maximum size of equiangular line sets in $\mathbb{R}^n$, using semidefinite
programming to improve the upper bounds on this quantity. Improvements are
obtained in dimensions $24 \leq n \leq 136$. In particular, we show that the
maximum number of equiangular lines in $\mathbb{R}^n$ is $276$ for all $24 \leq
n \leq 41$ and is 344 for $n=43.$ This provides a partial resolution of the
conjecture set forth by Lemmens and Seidel (1973).
