If $S$ is a symplectic, then $S^{[n]}$ is naturally symplectic. Given an anti-symplectic involution of $S$, there is an induced involution on $S^{[n]}$ whose fixed point locus is a smooth Lagrangian submanifold. In this talk I explain how to calculate its cohomology and mixed Hodge structure. For the special case $S = \mathbb{C}^2$, this is done using a Morse theory argument borrowed from Ellingsrud-Stromme. For the general case, we adapt an approach due to Gottsche-Soergel involving perverse sheaves.

I will outline the origin of the Poisson structure. I will also describe how to define an almost complex structure on the symplectic leaves, in some situations.

Some parts of the material presented are joint with Indranil Biswas, Jacques Hurtubise and Sean Lawton. Other parts are joint with Yukai Zhang.

In 2003, Hassett constructed a weighted variant of $\overline{M_{0,n}}(\mathbb{R})$: For each of the $n$ labels, we assign a weight between 0 and 1; points can coincide if the sum of their weights does not exceed one. We seek combinatorial presentations for the fundamental groups of Hassett spaces with certain restrictions on the weights. In particular, we express the Hassett space as a blow-down of $\overline{M_{0,n}}$ and modify the cactus group to produce an analogous short exact sequence. The relations of this modified cactus group involves extensions to the braid relations in $S_n$. To establish the sufficiency of such relations, we consider a certain cell decomposition of these Hassett spaces, which are indexed by ordered planar trees.