This talk will be an overview on Higgs bundles moduli spaces and on the geometry of the Hitchin system, focusing on its connections to mirror symmetry. I will focus on the existence of certain special subvarieties of these moduli spaces equipped with special sheaves, called branes. These are conjecturally exchanged under mirror symmetry, thus being so-called dual branes. One type of such branes is constructed from hyperholomorphic subvarieties and their conjectural dual arise from Lagrangian subvarieties.

In this talk we consider moduli spaces of flat Lie algebroid connections on a Riemann surface. These types of moduli spaces constitute a simultaneous generalization of several classes of moduli spaces which are broadly used in differential geometry, algebraic geometry, and mathematical physics, such as moduli spaces of Higgs bundles, twisted Higgs bundles, flat connections, and logarithmic or meromorphic connections.

Moduli spaces of tropical objects can often be obtained as tropicalization of suitable compactifications of algebro-geometric objects.

We describe the geometry and the topology of generalized polygon spaces with “edges” in projective spaces, and fixed "edge lengths".
These spaces are symplectic quotients of degenerate co-adjoint orbits on Lie algebras, satisfying a "closing condition". They can also be viewed as moduli spaces of quiver representations of a star-shaped quiver, and as spaces of parabolic bundles over the Riemann sphere. Concentrating on the su(n) case, and using wall-crossing methods (also called flips), we obtain a recursive formula for their Poincaré polynomials.

In this talk I will explain how virtual resolutions are useful to study the geometrical properties of subvarieties of products of projective spaces. Attention will be focused on sets of points in a product of two projective spaces.
We use this insight to compute implicit equations of parametrized surfaces that come up in Geometric Modeling. All definitions and concepts will be presented from scratch and many examples will be presented.

As is well known, a Hermitian metric on a complex manifold is called SKT (strong K\”ahler with torsion) if the Bismut torsion form $H$ is such that $dH=0$. In this talk, as a conformal generalisation of the SKT condition, we will introduce a new type of Hermitian structure, called \emph{locally conformally SKT}. Precisely, a Hermitian structure $(J,g)$ is said to be locally conformally SKT if there exists a closed 1-form $\alpha$ such that $dH = \alpha \wedge H$. We will discuss the existence of such structures on Lie groups and their compact quotients by lattices.

For some classes of hyperbolic surfaces, all locally finite ergodic measures  invariant under the horocycle flow are described. See the works of Omri Sarig and those of Lindenstrauss-Landesberg in that sense. This talk focuses on the study of another class of hyperbolic surfaces. Precisely, we give an analytical construction of a family of surfaces of infinite type whose corresponding horocycle flow is conservative but not ergodic with respect to the Liouville measure.

Compact Hyperkaeler manifolds play a central role in complex algebraic geometry, as they arise as one of the building blocks of compact Kaehler manifolds with trivial canonical class. I will briefly recall the (very rich) theory of compact Hyperkaehler manifolds, hence I will pass to their analogue in the singular setting, where several different definitions are possible and convenient for different purposes.

In 1975 Narasimhan and Ramanan studied the action of a line bundle $L$ of finite order on the moduli space of vector bundles of fixed rank and degree via tensorization. They proved that fixed points are pushforwards of vector bundles of lower degree over an Étale cover of $X$ determined by $L$. We extend this study to $G$-Higgs bundles for $G$ reductive and consider the more general action of a finite group $\Gamma$. The action of an element in $\Gamma$ involves "tensorization by a line bundle", extension of structure group by an automorphism of $G$ and rescaling the Higgs field.

Abstract. In this talk we will focus on semicomplete Halphen vector fields.
It is shown that rational semicomplete Halphen vector fields are in ``one-to-one'' correspondence with singular uniformizable projective structures on compact Riemann surfaces.
In turn, many examples of the projective structures in question can be obtained by means of Teichmüller space techniques or, more precisely, from Bers simultaneous uniformization theorem.
The proof of this correspondence is the main topic for this discussion.