A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the last 70 years. Answering a question of Lemmens and Seidel from 1973, in this talk we show that for every fixed angle $\theta$ and sufficiently large $n$ there are at most $2n-2$ lines in $\mathbb{R}^n$ with common angle $\theta$. Moreover, this is achievable only when $\theta =\arccos\frac{1}{3}$. Various extensions of this result to the more general settings of lines with $k$ fixed angles and of spherical codes will be discussed as well.

Joint work with I. Balla, F. Drexler and P. Keevash.

*Benny Sudakov received his PhD from Tel Aviv University in 1999. He had appointments in Princeton University, the Institute for Advanced Studies and in UCLA, and is currently professor of mathematics in ETH, Zurich. He is the recipient of a Sloan Fellowship, NSF CAREER Award, Humboldt Research Award, is AMS Fellow and was invited speaker at the 2010 International Congress of Mathematicians. His main research interests are combinatorics and its applications to other areas of mathematics and computer science.*