A tangle T is a pair formed by a ball and collection of properly embedded disjoint arcs in B. Tangles are useful for decomposing links and have presence in applications in science. In this talk we will discuss the case where the unknot and split links have decompositions with a given tangle T as factor. As an application we obtain a complete classification for tangles up to seven crossings with such properties. 

This is joint work with António Salgueiro.

We are interested in equivariant embeddings of homogeneous spaces. Given the standard action
of C^2 on itself, we look for complex surfaces M acted upon by C^2 so as M should include an
invariant copy of C^2 where the restriction of the (extended) action recovers C^2
acting on itself. In particular, for a given equivariant C^2-embedding,  we shall provide a local classification of the action  by means of a description of the corresponding local transformation group.

This is joint work with Julio Rebelo, Helena Reis and Ana Cristina Ferreira.

 Sasaki manifolds are the odd dimensional analogues of Kähler manifolds. In this talk, I will discuss the equivariant index of the horizontal Dolbeault complex on toric Sasaki manifolds. We show that the index localises to certain closed Reeb orbits and can be expressed as a sum over lattice points of the moment cone. This kind of index problems have appeared recently in the evaluation of partition functions of certain supersymmetric gauge theories. 

On a symplectic manifold $M$ with a hamiltonian action by a group $K$, it is natural to perform geometric quantization with respect to $K$-invariant polarizations. We first consider the case of the cotangent bundle $T^\ast K$ (based on arXiv:2301.10853), where parallel transport along the geodesics yields a realization of the Peter--Weyl Theorem.

In the second part, we dicuss the universal role of construction for the cotangent bundle plays in the parametrization of invariant Mabuchi geodesic rays for general hamiltonian $K$-manifolds.

After reviewing the formalism of Simpson's shapes for the non-abelian Hodge theory associated to a smooth projective curve X, we will describe the construction of a certain class of functors between the derived categories of the Dolbeault, De Rahm and Betti shapes of X into the derived categories of the corresponding moduli stacks. We shall describe as well some of their main properties, in particular their interaction with the so-called Wilson operators.

This is a report on the thesis of Robert Hanson.

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.