In the talk, we study immersions between cell complexes using inverse monoids. By an immersion f : D -> C between cell complexes, we mean a continous map which is a local homeomorphism onto its image, and we further suppose that commutes with the characteristic maps of the cell complexes. We describe immersions between finite-dimensional connected Delta-complexes by replacing the fundamental group of the base space by an appropriate inverse monoid.

# Seminars

In this talk, we will review the famous Hopf problem, which dates back to 1948, of whether there is a complex structure on the round 6-sphere. Although some attempted answers have been advanced, both in the negative and in the positive directions, this problem is still open. We will review some of the work and partial results that appear in the literature and take a closer look at almost complex structures compatible with the metrics of constant sectional curvature.

Two groups are called commensurable if they have isomorphic subgroups of finite index. In the first part of the talk I will discuss our results with Montse Casals-Ruiz and Ilya Kazachkov on the commensurability classification of right-angled Artin groups (RAAGs) defined by trees. In the second part of the talk I will mention some algorithmic properties of RAAGs and discuss our results with Jordi Delgado and Enric Ventura on the subgroup intersection problem for Droms RAAGs.

The so-called Hopf-zero singularity consists in a vector field in $R^3$ having the origin as a critical point, with a zero eigenvalue and a pair of conjugate purely imaginary eigenvalues.

Let G be the Grassmannian of lines in P3 embedded in P5 as the Plücker quadric Q. The intersection of Q with a second hypersurface of degree d is what is called a complex of lines of degree d. When we consider the intersection of Q with a second quadratic hypersurface in P5, P, we have a quadratic complex. Let X = Q ∩ P be a quadratic complex that, in this talk, we assume to be non-singular, meaning X is non-singular.

The subject of this talk will be Hopf algebras and their dual theory. We will mostly focus on a particular class of Hopf algebras: noetherian Hopf algebras that are finitely-generated modules over some commutative normal Hopf subalgebra. Some properties and examples of these Hopf algebras will be mentioned. Furthermore, we will see some results on the dual of this class of Hopf algebras, some of its properties, decompositions e maybe some interesting Hopf subalgebras.

Right-angled Artin groups arise naturally in different branches of mathematics and computer science. In this talk we will introduce the class of right-angled Artin groups and discuss when they are algebraically, geometrically and logically similar, or, more formally, when they are commensurable, quasi-isometric and universally equivalent.

The pseudowords in a finitely generated free profinite aperiodic semigroup

are faithfully represented by labeled linear orders induced by the factorizations of the pseudowords.

We address the problem of knowing which labeled linear orders are in the image of this representation (This is joint work with Jorge Almeida, José Carlos Costa and Marc Zeitoun).

Spin foam models are a "state-sum" approach to loop quantum gravity which aims to facilitate the description of its dynamics, an open problem of the parent framework. Since these models' relation to classical Einstein gravity is not explicit, it becomes necessary to study their asymptotics - the classical theory should be obtained in a limit where quantum effects are negligible, taken to be the limit of large triangle areas in a triangulated manifold with boundary.

A smooth (real analytic or complex) manifold is said to be foliated when it is partitioned into immersed and connected sub-manifolds. It appears that the same foliation can be induced by totally different sets of vector fields. Thus, we turn our attention to the sheaves of vector fields that induce an integrable distribution. We have noticed that if this sheaf is resolved by a graded vector bundle E (at the level of sections), one can lift the Lie bracket of vector fields into a Lie ∞-algebroid structure on E.