We introduce the classical results of de Rham and Berger on the holonomy of a Riemannian manifold. We compare these to the situation of parallel skew-torsion, where we obtain Riemannian submersions from reducible holonomy. If time permits I will give an introduction to 3-(α, δ)-Sasaki manifolds and their submersion onto quaternionic Kahler manifolds.

The topological complexity and its higher versions are homotopy invariants which were introduced by Farber and Rudyak in order to give a topological measure of the complexity of the motion planning problem. We will discuss some properties of these invariants for closed manifolds with abelian fundamental group. In particular, we will give sufficient conditions for the (higher) topological complexity of such a manifold to be non-maximal. This is based on joint works with N. Cadavid, D. Cohen, J. González and S. Hughes.

Mirror symmetry is a somewhat mysterious phenomenon that relates the geometry of two distinct Calabi-Yau manifolds. In the realm of trying to understand this relationship several conjectures on the existence of so-called special Lagrangian submanifolds appeared. In this talk, I will report on joint work with Jason Lotay on which we prove versions of the Thomas and Thomas-Yau conjectures regarding the existence of these special Lagrangian submanifolds and the role of Lagrangian mean curvature flow as a way to find them. I will also report on some more recent work towards proving more recent conjectures due to Joyce.

Lie groupoids can encode geometric objects such as smooth actions, and foliations; deformations of Lie groupoids also relate to deformations of these objects. In this talk we’ll first see the deformation cohomology of a (real) Lie groupoid and mention relations to deformations the mentioned examples.

We will then see the cohomology controlling deformations of a complex Lie groupoid: it combines deformation cohomology of the groupoid structure and Kodaira-Spencer cohomology of the underlying complex manifold, via a double complex.

The talk is based on ongoing work with Luca Vitagliano.

A striking difference between Riemannian and pseudo-Riemannian metrics is that pseudo-Riemannian ones often fail to be geodesically complete even in the compact case. We will present some developments in the classification of Lie groups with all their left-invariant pseudo-Riemannian metrics complete. More concretely, we will discuss the specifics of geodesic completeness when the manifold in question is a Lie group and recall the Euler-Arnold theorem as well as the seminal work of Marsden for the compact (homogeneous) case. We will see how an interpretation in Riemannian terms of his techniques provided us with tools for characterising completeness even for general manifolds. As for Lie groups, we will show how a certain notion of "linear growth'' allowed us to establish large classes of Lie groups whose left-invariant metrics are all complete. Time permitting, we will also discuss the generalisation of the Euler-Arnold formalism to the holomorphic-Riemann setting and discuss the classification of geodesic completeness for 3-dimensional (non-unimodular) Lie groups.

This is a series of joint works with S. Chaib, A. Elshafei, H. Reis, M. Sánchez and A. Zeghib.