  We give lowed bounds on the number of periodic trajectories in strictly
convex smooth billiards in $\R^{m+1}$ for $m\ge 3$. For plane billiards (when
m=1) such bounds were obtained by G. Birkhoff in the 1920's. Our proof is based
on topological methods of calculus of variations - equivariant Morse and
Lusternik - Schirelman theories. We compute the equivariant cohomology ring of
the cyclic configuration space of the sphere $S^m$, i.e., the space of n-tuples
of points $(x_1, ..., x_n)$, where $x_i\in S^m$ and $x_i\ne x_{i+1}$ for i=1,2,
..., n.
