Abstract: In 1959, B.Neumann introduced outer billiards in the plane, a
discrete-time dynamical system defined in the exterior of an oriented
plane oval. The outer billiard maps a point z to another point z' if the
line zz' is tangent to the curve at the midpoint Q=(z+z')/2. This map
preserves area, and, using KAM theory, J.Moser showed that if the curve
is smooth enough, the orbits of this billiard do not escape to infinity.
Outer symplectic billiards generalize outer billiards to
higher-dimensional symplectic spaces. Using a variational approach, we
can establish the existence of odd-periodic orbits. Interestingly,
however, we cannot guarantee the existence of even-periodic orbits. We
will also discuss the behavior of this correspondence when the "table"
is a curve or a Lagrangian submanifold. This is joint work with P.Albers
and S.Tabachnikov.