Coupled cell systems are dynamical systems which respect the structure of a network. In an homogeneous network the number of inputs directed to each cell is constant and it is called the network valency. In this talk, we focus on bifurcations in coupled cell systems with a condition associated to the network valency. Moreover, we address the lifting bifurcation problem that concerns whether all bifurcation branches are lifted from a small network to a bigger one. We present some results and examples about this problem.

# Seminars

V. Alexeev proved in 1994 that the set *S* of self-intersections of the canonical class of stable surfaces satisfies the descending chain condition, this is, any monotone sequence is increasing. (This set *S* is a subset of the positive rational numbers.) In particular *S* has a minimum, and it may have accumulation points. I will discuss what is known about *S*, certain new theorems on accumulation points, and open questions. This is a joint work with José Ignacio Yáñez.

Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of Boolean representable simplicial complexes. This turns out to be the direct sum of a complete matroid with a lift matroid of the complete biased graph over G. As is well known, the Dowling lattice defines the frame matroid over a similar biased graph.

Abstract. The National Security Agency (NSA) in August 2015 announced plans to transition to post-quantum algorithms “Currently, Suite B cryptographic algorithms are specified by the National Institute of Standards and Technology (NIST) and are used by NSA’s Information Assurance Directorate in solutions approved for protecting classified and unclassified National Security Systems (NSS).

For a finitely generated group, the Cayley graph is a metric space encoding the structure of the group. Gromov introduced the notion of a $\delta$-hyperbolic group, a finitely generated group with a negatively curved Cayley graph, that is, for any triangle in the graph with geodesic sides, each side is contained in the $\delta$-neighborhood of the union of the two other sides. Hyperbolic groups are "prevalent" among finitely generated groups.

We consider algorithmic problems for automaton semigroups and automaton groups of the freeness and finiteness kind. We first show that checking whether an automaton group has empty set of positive relations is undecidable. Moreover we prove that the emptyness of the set of positive relations is equivalent to the dynamical property of having all the orbital graphs centred at the non-singular points which are acyclic. We also settle the problem of checking the freeness for the semigroup defined by an automaton group by proving that such problem is undecidable.

Recall that a finite semigroup S is said to be inherently nonfinitely based (INFB) if S does not belong to any finitely based locally finite variety. In 1987, Mark Sapir proved that the 6-element Brandt monoid B_2^1 is INFB; later he gave an algorithmically efficient description of INFB semigroups. Sapir's description implies, in particular, that no finite J-trivial semigroup is INFB.

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.

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.