Posters

Dirac structures are a geometric object generalizing symplectic and Poisson structures. From a physics viewpoint, they describe mechanical systems with both symmetries and constraints. Geometrically, they are given by subbundles of a vector bundle with additional structure, called a Courant algebroid. A pair of complementary Dirac structures in a Courant algebroid decomposes it as the double of a Lie bialgebroid. Our goal is to describe, in a given Courant algebroid, the obstruction to the existence of a Dirac complement for a given Dirac structure. Our main result provides an algebraic obstruction in terms of curved differential graded Lie algebras to the existence of a Dirac complement.

I will present how the existence of a certain class of Killing spinors, which arise in the study of special holonomy in generalized geometry, determines embeddings of canonical supersymmetric vertex algebras in the space of sections of a sheaf of vertex algebras, called the chiral de Rham complex. The Killing spinor equations considered are part of an approach to special holonomy based on Courant algebroids in generalized geometry and are inspired by heterotic supergravity.
More specifically, I will consider embeddings of the N=2 superconformal vertex algebra in two different set-ups, namely in the superaffinization of a quadratic Lie algebra, satisfying appropriate algebraic conditions, and in the space of sections of the chiral de Rham complex of a Courant algebroid appearing in supergravity.
This poster is based on my PhD thesis, and joint works with Luis Álvarez-Cónsul and Mario Garcia-Fernandez.

We construct steady states of the Navier-Stokes equations with a prescribed return map to an embedded disk on a class of 3-manifolds. The techniques used include a cosymplectic embedding theorem and a correspondence between cosymplectic structures and nowhere vanishing harmonic 1-forms.

We introduce symmetric Poisson structures as pairs consisting of a symmetric bivector field and a torsion-free connection satisfying an integrability condition analogous to that in usual Poisson geometry. Equivalent conditions in Poisson geometry have inequivalent analogues in symmetric Poisson geometry and we distinguish between symmetric and strong symmetric Poisson structures. Geometrically, symmetric Poisson structures correspond to locally geodesically invariant distributions together with a characteristic metric, whereas strong symmetric Poisson structures correspond to totally geodesic foliations together with a leaf metric and a leaf connection. We present several classes of examples of such structures. In particular, we show that linear symmetric Poisson structures are in correspondence with Jacobi-Jordan algebras, whereas strong symmetric correspond to those that are moreover associative.

Logarithmic and b-tangent bundles provide a versatile framework for addressing singularities in geometry. Introduced by Deligne and Melrose, these modified bundles resolve singularities by reframing singular vector fields as well-behaved sections of these singular bundles. This approach has gained significant attention in symplectic geometry, particularly through its applications to the study of Poisson manifolds that are symplectic away from a hypersurface (b^m-symplectic forms). We investigate the conditions under which these singular tangent bundles are isomorphic to the tangent bundle, analyzing in detail the case of spheres. Furthermore, we establish a Poincaré–Hopf theorem for the b^m-tangent bundle, offering new insights into the interplay between singular structures and topological invariants. This is joint work with Eva Miranda and based on arXiv:2502.18602.