Following A. Beauville, a complex algebraic variety $X$ is said to be symplectic if it admits a holomorphic symplectic form $\omega$ on its smooth locus such that, for every resolution $\pi: Y \to X$, $\pi^*\omega$ extends to a holomorphic $2$-form on $Y$. When this extension is actually non-degenerate (a de facto symplectic form) on $Y$, we call $\pi$ a symplectic (or crepant) resolution.

Let $G$ be a complex reductive group and $A$ an abelian variety of dimension $d$. The aim of this talk is to show that all moduli spaces of $G$-Higgs bundles over $A$ are symplectic varieties, and that, for $G=\mathrm{GL}(n,\mathbb C)$, the canonical Hilbert-Chow morphism is a symplectic resolution if and only if $d=1$.

Moreover, using a little representation theory, we can obtain explicit expressions for the Poincaré polynomials of all Hilbert-Chow resolutions (either $d=1$, all $n$; or $n=1,2,3$ and all $d$). This is joint work with I. Biswas and A. Nozad.

The question whether a symplectic manifold embeds into another is central in symplectic topology. Since Gromov nonsqueezing theorem, it is known that this is a different problem from volume preserving embedding. There are several nice results about symplectic embeddings between open subsets of $\mathbb R^{2n}$ showing that even for those examples the question can be completely nontrivial. The problem is substantially more well understood when the manifolds are toric domains and have dimension $4$, mostly because of obstructions coming from embedded contact homology (ECH). In this talk we are going to discuss symplectic embedding problems in which the target manifold is the disk cotangent bundle of a two-dimensional sphere, i.e., the set consisting of the covectors with norm less than $1$ over a Riemannian sphere. We shall talk about some tools such as ECH capacities and action angle coordinates. Much of this talk is based on joint works with Vinicius Ramos and Alejandro Vicente.

Studying metrics with special curvature properties on compact Kähler manifolds is a fundamental problem in Kähler geometry. In this talk, I will focus on the existence and uniqueness of singular Kähler-Einstein metrics whose singular behavior is prescribed. These results are based on a series of joint works with T. Darvas and C. Lu.