Geometry and Topology

Riemann-Hilbert correspondence for classical and for twistor D-modules

Speaker: 

Teresa Monteiro Fernandes

Date: 

Friday, 18 November, 2016 - 15:30

Venue: 

Room 1.22

In this talk I will give an overview of the main concepts and results in D-module theory, and then switch to the notion of relative D-module. I will explain the main motivation for the study of holonomic relative modules given by Mochizuki's notion of mixed twistor D-module. We will explain the Riemann-Hilbert correspondence in this case as a joint work with Claude Sabbah.

Computing lines in smooth cubic hypersurfaces and application to the irrationality problem

Speaker: 

Xavier Roulleau

Date: 

Friday, 11 November, 2016 - 11:30

Venue: 

Room 0.30

A smooth cubic hypersurface X of dimension >1 is unirational. The variety of lines F(X) on these hypersurfaces is an essential tool to understand the geometry of X. In dimension 3, the study of F(X) enables to prove that X is always irrational.

In this talk we study the zeta function of F(X) and we obtain a simplified proof of the irrationality of a dense set of smooth cubic threefold. This is a joint work with D. Markouchevitch.

Complexified Hamiltonian symplectomorphisms and solutions of the homegeneous complex Monge-Ampere equation

Speaker: 

José Mourão

Date: 

Friday, 23 September, 2016 - 14:30

Venue: 

Room 1.22

The geodesics for the Mabuchi metric on the space of Kaehler metrics on a manifold correspond to solutions of the homogeneous complex Monge-Ampere (HCMA) equation. We will describe a method for reducing the Cauchy problem for the HCMA equation with analytic initial data to finding a related Hamiltonian flow followed by a "complexification". Examples and applications will be discussed. 

Work in collaboration with J.P. Nunes and T. Reis

Stability conditions for (G,h)-constellations.

Speaker: 

Alfonso Zamora

Date: 

Friday, 1 July, 2016 - 14:45

Venue: 

1.22

Given a reductive group G and an affine G-scheme X, constellations are G-equivariant sheaves over X such that their module of global sections has finite multiplicities. Prescribing these multiplicities by a function h, and imposing a stability condition $\theta$ there is a moduli space for $\theta$-stable constellations constructed by Becker and Terpereau, using Geometric Invariant Theory. This construction depends on a finite subset D of the set of irreducible representations of G.

Nielsen-Olesen cosmic strings, the Einstein-Bogomol'nyi equations, and algebraic geometry

Speaker: 

Luis Álvarez-Cónsul

Date: 

Friday, 1 July, 2016 - 13:30

Venue: 

1.22

Y. Yang observed 20 years ago that the 2-sphere is the only compact orientable surface admitting solutions of the Einstein-Bogomol'nyi equations, coupling vortices with gravity, and obtained sufficient conditions for the existence of cosmic strings in this situation. In this talk, we will give an algebro-geometric interpretation of Yang's conditions, and explain why they are in fact necessary for the existence of solutions.

The curvature veronese of a 3-manifold immersed in Euclidean space

Speaker: 

M. Carmen Romero Fuster

Date: 

Friday, 3 June, 2016 - 14:30

Venue: 

Room 1.22

The concept of curvature ellipse at a point of a surface immersed in 4-space has been known since a long time ago [2] and it has proven to be a useful tool in the study of the geometrical properties from both, the local and global viewpoint [1, 3, 4]. Its natural generalization to higher dimensional manifolds is given by the image of a convenient linear projection of a Veronese submanifold of order 2 in the normal space of the manifold at each point [3]. We call it the curvature locus or curvature veronese.

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