In a work in progress with Gabriel Calsamiglia (UFF, Niteroi, Brasil) and  Titouan Serandour (Univ Rennes/ENS Lyon) we investigate a partial compactificat

The Minimal Model Program (MMP) is a far reaching conjecture in birational geometry which aims at constructing a good representative (minimal model) of any given complex projective variety W. When such a model exists it might not be unique and so it becomes natural to study the relations between them. In the case when W is covered by rational curves, its minimal model is a Mori fibre space, that is, a fibration whose generic fibre is positively curved, and its uniqueness is encoded in the notion of birational rigidity.

We recall the construction of a fundamental differential system of Riemannian geometry and survey on some of its applications. First, on the determination of the Euler characteristic of hypersurfaces in space forms. Secondly, the generalization of Hsiung-Minkowski identities concerning total mean curvatures of hypersurfaces. And thirdly, we discuss the study of the volume of unit vector fields on a Riemannian manifold via calibrations induced from the differential system.

Consider a holomorphic foliation F on a complex manifold. In the 1970s,  Baum-Bott constructed residue classes associated with each singular component of F. These classes satisfy an index theorem, which computes characteristic classes of F.

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.