We will resume next week as usual.

Given the action of a compact Lie group on a smooth manifold, it is possible to define a class of (pseudo-)differential operators known as the transversally elliptic operators. The principal symbol of such an operator is only guaranteed to be invertible for nonzero covectors that are conormal to the group orbits. I will give the precise definition of these operators, and explain why they have a well-defined equivariant index, in spite of the fact that in general, they are not Fredholm operators.

Mirror symmetry proposes a rather general correspondence of algebraic structures in complex and symplectic geometry, and illustrates the interdependence of mathematics and theoretical physics. I will review selected aspects of the story, from the origins in Calabi-Yau compactifications of the superstring to recent calculations relating to arithmetic of algebraic cycles.