An overview of the course will be given at the beginning of the first class. The first three weeks will cover the background material. From the fourth week, each weekly class will consist of three parts:

• A 45min survey lecture.

• A 90min hands-on working session.

• A 30min student-directed working session.

Those interested in knowing about this subject are most welcome to attend only the first part of the class from the fourth week. Those interested in a deep understanding and working knowledge can attend the other parts. Those officially registered in the course are expected to attend and participate in most of the classes.

The representations of the fundamental group of a closed orientable surface can be understood by means of a manifold (a vector bundle over the surface) together with some extra structure (a flat connection on this bundle).

When a complex structure is chosen on the surface, the concept of a Higgs bundle over a Riemann surface arises naturally. The spaces parameterizing these objects, or moduli spaces, are related by the Hitchin-Kobayashi correspondence.

The rich geometric structure of the moduli space of Higgs bundles then becomes a powerful tool to study the topology of the moduli space of surface group representations, or character variety.

In this introductory course we will cover the following topics:

• The fundamental group of a surface and its representations.

• Basics on bundle theory.

• Local systems and flat connections.

• Higgs bundles on Riemann surfaces.

• Moduli spaces and the Hitchin-Kobayashi correspondence.

• The geometry of the moduli space of Higgs bundles and the topology of the character variety.

Upon completion of the course, students

• will achieve a working knowledge of concepts such as associated bundles, cross sections, connections, etc.

• will know the basics about the Hitchin-Kobayashi correspondence and the concepts appearing in it.

• will be acquainted with an example of the powerful interactions between diffrent fields, namely, geometry into topology and representation theory.

• will be able to independently read relevant literature on the topic.

Some basic definitions: surface, differentiable manifold, differential forms, complex structure, etc., although we will recall all of them. No previous knowledge on the fundamental group, vector bundles or connections will be assumed.