In this talk, we will investigate the symplectomorphism groups of the simplest open manifolds with both convex and concave ends, namely the symplectizations $sL(n,1)$ of Lens spaces $L(n,1)$. We will see that the compactly supported symplectomorphism group $Symp_c(sL(n,1))$ is homotopy equivalent to a loop space. As a corollary, we will show that the space of Lagrangian $RP^2$ in the cotangent bunble $T^*\RR P^2$ is weakly contractible. (Part of a joint work with R. Hind and W. Wu.)

We will discuss the basic formalism of Hamiltonian mechanics and of its generalization Nambu-mechanics. Notions from symplectic geometry will allow us to lay out this formalism in a coordinate independent way and will lead to the definition of Poisson manifolds, which serve as phase spaces for Hamiltonian mechanics. We will prove Liouville’s theorem for Hamiltonian mechanics which states that the volume of any region in phase space is preserved under the phase flow (time evolution) and we will see in a special case that the Liouville theorem itself is preserved in the generalization from Hamiltonian to Nambu-mechanics. Finally we will introduce the notion of a Nambu-Poisson manifold (Phase space for Nambu-mechanics) a natural generalization of the notion of a Poisson manifold. We will see for example that the theorem from Hamiltonian mechanics that the bracket of two integrals of motion is again an integral of motion holds also for Nambu’s dynamics by the very definition of Nambu-Poisson manifold. We will end with many examples; in particular, multi-symplectic manifolds will be introduced.

I shall survey the main steps in the heat equation proof of the Atiyah-Singer index theorem for Dirac operators on spin manifolds.