Abstract: Let $M$ be a $2n$ dimensional manifold with a symplectic form $\omega$. This symplectic form determines a Lie subgroup, $Symp_{\omega}(M)$, of $Diff(M)$ called the symplectomorphism group. There is yet another subgroup of interest called the Hamiltonian group. $Ham_{\omega}(M)$ is an infinite dimensional Lie group yet it has some properties of compact finite dimensional Lie groups. The presence of finite dimensional tori $T^k$, k≤n, inside $Ham_{\omega}(M)$ determines $M$ completely when $n = 2, 4$. I will sketch the proof of the case $n = 4$, a result of my supervisor and his colleagues.

Abstract: In the present talk I describe a (conjectural) relation between moduli spaces of (framed) sheaves on some two-dimensional root toric stacks and Nakajima quiver varieties of type the affine Dynkin diagram $\hat{A}_{n}$. If time permits, I will discuss an application of this relation to representation theory of Kac-Moody algebras (and vertex algebras).