The notion of topological complexity of a space has been introduced by M. Farber in order to give a topological measure of the complexity of the motion planning problem in robotics. Surprisingly, the determination of this invariant for non-orientable surfaces has turned out to be difficult. A. Dranishnikov has recently established that the topological complexity of the non-orientable surfaces of genus at least 4 is maximal. In this talk, we will determine the topological complexity of the Klein bottle and extend Dranishnikov's result to all the non-orientable surfaces of genus at least 2.

# Seminars

Nilpotent semigroups in the sense of Mal'cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, MN, which has finite rank. The semigroup identities that define nilpotent semigroups, lead us to define strongly Mal'cev nilpotent semigroups. Finite strongly Mal'cev nilpotent semigroups constitute a non-finite rank pseudovariety, SMN. The pseudovariety SMN is strictly contained in the pseudovariety MN but all finite nilpotent groups are in SMN.

Using the Dirac–Higgs bundle, we consider a new class of space-filling (BBB)-branes on moduli spaces of Higgs bundles, given by a generalized Nahm transform of a stable Higgs bundle. We then use the Fourier–Mukai–Nahm transform to describe its dual brane, which is checked to be a (BAA)-brane supported on a complex Lagrangian multisection of the Hitchin fibration.

After recall classical results in order to place our work, we introduce the class of split regular BiHom-Lie algebras as the natural extension of the one of split Hom-Lie algebras and so of split Lie algebras. By making use of connection techniques, we focus our attention on the study of the structure of such algebras and, under certain conditions, the simplicity is characterized.

We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of D2- symmetric maps, for which we obtain a similar result for C1 homeomorphisms. Some applications to differential equations are also given.

This is joint work with B. Alarcón (UFF — Brasil) and S.B.S.D. Castro (CMUP) and is part of a project of using symmetries to obtain global stability result

We study bifurcations in area-preserving maps with homoclinic tangencies. We consider $C^r$-smooth maps ($r\geq 3$) having a saddle fixed point whose stable and unstable invariant manifolds have a quadratic or cubic tangency at the points of some homoclinic orbit and study bifurcations of periodic orbits near the homoclinic tangencies in closed area-preserving maps. In the case of a quadratic homoclinic tangency we prove the existence of cascades of generic elliptic periodic points for one and two parameter unfoldings.

We study the complex dynamics in analytic area-preserving maps in a neighbourhood of a resonant elliptic fixed point. We assume that the resonance is weak, i.e., the linear part is a rotation of an angle 2\pi where n\geq5. Normal form theory suggests that there is a flower with n petals which consists of points bi-asymptotic to the fixed point.

We show that the flower splits: there are parabolic stable and unstable complex manifolds and they do not intersect. We measure the splitting of the manifolds and relate it to the Stokes phenomenon.

A new scheme for proving pseudoidentities from a given set Σ of

pseudoidentities, which is clearly sound, is also shown to be complete

in many instances, such as when Σ defines a locally finite variety, a

pseudovariety of groups or, more generally, of completely simple

semigroups. Many further examples when the scheme is complete are

given when Σ defines a pseudovariety V which is σ-reducible for the

equation x = y, provided Σ is enough to prove a basis of identities

for the variety of σ-algebras generated by V. This gives ample

The notion of structural stability is one of the oldest concepts in dynamical systems. Structurally stable diffeomorphisms are conjugate to all nearby diffeomorphisms and are known to be strongly related to dynamics with persistence of the shadowing property.

Cluster maps are birational maps arising from mutation-periodic quivers. These quivers are associated to (mutation-periodic) cluster algebras, introduced in 2002 by Fomin and Zelevinsky.