PhD Defense: Clarisson Rizzie Canlubo

Title: Non-commutative covering spaces and their symmetries

Abstract: The main goal of this thesis is to propose a notion analogous to covering spaces in classical geometry. This is motivated by the author's long term goal of defining the (étale) fundamental group of a non-commutative space and put forth a good notion of monodromy.

We will present a notion of a non-commutative covering space using Galois theory of Hopf algebroids. We will look at basic properties of classical covering spaces that generalize to the non-commutative framework. Afterwards, we will explore a series of examples. We will start with coverings of a point and central coverings of commutative spaces and see how these are closely tied up. Coupled Hopf algebras will be presented to give a general description of coverings of a point. We will give a complete description of the geometry of the central coverings of commutative spaces using the coverings of a point. A topologized version of Hopf categories will be defined and its corresponding Galois theory. Using this and basic concepts from algebraic geometry and spectral theory, we will give a full description of the general structure of non-central coverings. Examples of coverings of the rational and irrational non-commutative tori will also be studied. Using the non-commutative analogue of the hyperelliptic involution, we will show that unlike the classical case, the non-commutative sphere is a covering of the non-commutative torus. There is a purely non-commutative phenomenon happening to non-commutative coverings, namely, their symmetry is two-sided. We will explain this and relate it to bi-Galois theory. Using the OZ-transform, we will show that non-commutative covering spaces come in pairs. Several categories of covering spaces will be defined and studied. Appealing to Tannaka duality, we will explain how this lead to a notion of an étale fundamental group. Finally, in the last chapter we will discuss possible future projects.

Supervisor: Prof Ryszard Nest, Math, University of Copenhagen

Assessment committee:

• Assoc. Prof. Lars Halle (Chairman), MATH, University of Copenhagen
• Prof. Piotr Hajac, IMPAN, PL
• Prof. Ulrich Krähmer, TU, Dresden, DE