II Spanish Young Topologists Meeting

In a recent paper with A. Minasyan and A. Sisto, we have described the commensurating endomorphisms of a cylindrical hyperbolic groups.
Combining this result with the conjugacy separability of fundamental groups of compact 3-manifolds, proved by Hamilton, Wilton and Zalesskii based on the deep work of Wise and Agol, we are able to conclude that the outer automorphism group of the fundamental group of a compact 3-manifold is residually finite. I will give an overview of all this results.

The Conley index of an isolated invariant set is a powerful tool in the study of flows defined on locally compact metric spaces. It is obtained as the homotopy type of the pointed space $(N/L, [L])$, where $(N, L)$ is an index pair of an isolated invariant set K, this is a pair where $N$ is a suitable isolating neighbourhood of $K$ and $L \subset N$ is the set through which the flow leaves $N$. In this talk we will introduce a new way of obtaining the Conley index of a plane continua without using index pairs. For this purpose we will make use of shape theoretical tools, particularly we will make use of the fact that the shape of a plane continua is determined by the number of connected components of his complementary. In addition we will point out that this method give us a complete classification of the spaces which could be the Conley index of some isolated invariant plane continua and, in some cases, we can infer the shape of our continua in terms of his Conley index. This results were obtained in collaboration with José M. R. Sanjurjo.

Classical rational homotopy theory relies in the equivalence of the usual homotopy category of simply connected, or more generally, nilpotent rational CW-complexes with the homotopy categories of commutative differential graded algebras and differential graded Lie algebras concentrated, roughly speaking, in positive degrees. Here, we show that considering the unbounded situation is also of interest when studying the rational homotopy type of non-connected spaces.

We describe characteristic classes of families of surfaces in fixed background manifolds and explain the problems that arise when the background manifold has no boundary. This is joint work with Oscar Randal-Williams.

The geometry of curves in the 3-sphere $\mathbb{S}^3$ is neatly described using the algebra of quaternions. The Hopf fibration provides a geometric realization of the space of contact elements in the 2-sphere $\mathbb{S}^2$. The group of contact symmetries is a rich object. We study the homotopic non-triviality of families of transformations preserving the contact structure. This is proven via an explicit description of the monodromy of a fibration over the sphere $\mathbb{S}^2$, the quaternionic symmetries and the description of $\mbox{Diff}(\mathbb{S}^n)$. The required preliminaries on differential and algebraic topology will be introduced. Further applications of algebraic topology to the group of symplectic and contact diffeomorphisms will be discussed.

It is known that 2-dimensional open Topological Conformal Field Theories (TCFT) are equivalent to cyclic $A_\infty$-algebras, whose (co)homology is computed by Hochschild (co)homology. In his thesis, C. Braun extends this known result to the context of unoriented TCFT and states that 2-dimensional open unoriented TCFT are equivalent to cyclic $A_\infty$-algebras equipped with an involution, and computes its cohomology by using an involutive version of Hochschild cohomology.
In this talk we will deal with the concepts of Hochschild (co)homology for associative and $A_\infty$-algebras and its involutive versions and will present a derived definition for involutive Hochschild (co)homology of involutive $A_\infty$-algebras.

The catastrophe theory was originated with the work of René Thom in the 60s, based on some previous work of Whitney, and now it can be considered as a branch of singularity theory in geometry. Roughly speaking, this theory deals with the sudden changes in qualitative behaviour on some parametric systems.
In this talk I will make a quick introduction of this theory, focusing on the example of the Zeeman's Catastrophe Machine. This device was developed by Christopher Zeeman in the 70s and it shows some "pathological" behaviours which somehow lead to the catastrophe theory. With this example in mind, I will introduce the main concepts of the theory and finish with the Thom's classification Theorem of elementary catasthrophes.

Let $f: (\mathbb{R}^{2}, 0) \rightarrow (\mathbb{R}^{2}, 0)$ be a finitely determined map germ. The link of $f$ is obtained by taking a small enough representative $f: U \subset \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ and the intersection of its image with a small enough sphere $\mathbb{S}_{\epsilon}^{1}$ centered at the origin in $\mathbb{R}^2$. We will describe the topology of $f$ in terms of the Gauss word associated to its link.
As an application of these techniques we will give a wide classification of corank 1 map germs and we will classify corank 2 map germs in some particular cases.

A common method in Morse theory is to embed a manifold in a suitable Euclidean space and study the critical set of some height function. We consider height functions on symmetric spaces $M \cong G/K$ embedded in the associated matrix Lie group $G$.
For symmetric spaces, there is no systematic characterization of Morse functions. In fact, the results for Lie groups led in the past to study only real diagonal matrices. As a consequence, the behaviour of the height functions on symmetric spaces might seem much more regular than what it actually is. The situation is much more interesting indeed. We shall show that it is possible to compute the critical set of the restricted function $h^M_X$ in terms of the matrix $\hat X = X^* + \sigma(X)$, where $\sigma$ is the involutive automorphism of $G$ defining the symmetric space. Anyway, we will be able to prove a result of reduction of an arbitrary height function $h^M_X$ to the diagonal case.

El objetivo de esta memoria es presentar el concepto de grafo aleatorio según D.Aldous en el contexto degrafos de Cayleyde grupos finitamente generados. Para ello se destacará ladescripción de dos ejemplos básicos: la envoltura de un grafo repetitivo y la percolación de Bernoulli. De hecho, el propósito final es demostrar que estos dos tiposde grafos aleatorios se distinguen gracias a una propiedad importante, llamadatolerancia a la inserción.Este resultado muestra que las técnicas propias de la teoría de percolaciónusadas por D. Aldous, R. Lyons, I. Benjamini y N. Curien no son validas para todos los grafos aleatorios, lo que sugierela necesidad de emplear otras técnicas, como las propias de los sistemas dinámicos, en elestudio de los grafos aleatorios.