The birational classification of projective manifolds was developed
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A Calabi-Yau orbifold is locally modeled on C^n/G with G a finite subgroup of SU(n). If the singularity is isolated, then the crepant resolution (if it exists) is an ALE manifold, for which index-type results are well known. However, most of the time the singularity is not isolated, and for the...

We will introduce Abelian and non-Abelian gerbes, from the point of view of Lie groupoids, and connections on those.

The moduli space $Hom(F,G)/G$ of completely reducible representations of a
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We give a notion of stability on the infinite Grassmannian Gr(k((z))^{\oplus r}) by the action of the group Sl(r,k[[z]]) coherent with the classical stability on the moduli space of vector bundles over a curve. We prove the existence of a geometric quotient, give a numerical criterium and the...

A double (Lie) groupoid is a groupoid in the category of (Lie) groupoids. A standard example is the fundamental double Lie groupoid of a Lie groupoid. I will describe a functor from the category of double Lie groupoids to the category of simplicial manifolds. In general, the simplicial...

A very long-standing problem in Algebraic Geometry is to determine
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Unprojection theory aims to analyze (and construct) complicated
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Mikhail Gromov introduced hyperbolic groups in 1987 through geometric properties of Cayley graphs, viewed as (geodesic) metric spaces. This class of groups captured very quickly the attention and the heart of (geometric)
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