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.

# Seminars

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.

The next session of the Research Seminar Program will take place in the Department of Mathematics of University of Porto in a room to be announced on 18th of October of 2017.

11h00 - Peter Lombaers: Integers and Ideals: There and Back Again.

Abstract: In number theory, when you try to solve an equation in a number field, it is often more convenient to work with ideals than with integers. This stems from the fact that ideals have unique factorization, but integers may not. I will explain the advantages and difficulties of this method using concrete examples.

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.

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.