## Welcome

This course is meant to be a light but motivated trip from the study of surface group representations to the definition of a Higgs bundle, a rich geometric object whose applications reach subjects like mirror symmetry or Langlands duality. At the end of this trip we will see what Higgs bundles can tell us about the representations of the fundamental group of a surface. The course takes place on Sundays 9:15-12:00 at Room 155 of the Ziskind Building in the period 29th October 2017 - 28th January 2018. It gives 3.00 credit points in Mathematics and Computer Science and 2.00 credit points in Physical Sciences. The grade (pass/fail) is based on active participation in the class. If you want to know more, please, check the sections Syllabus, Assignments and Lecture Notes, or contact me by email. The official course information can be found at the Feinberg Graduate School. |

## Summary of the lectures

This is a summary of the 14 lectures:

6 Feb 2018. We talked about the Hitchin-Kobayashi correspondence and mentioned Morse theory and hyperKahler structures. We wrapped up the course and review the main things we have learnt.

28 Jan 2017. We defined Higgs bundle again and explained how we can understand this definition from the relation between connections and holomorphic structures on a complex vector bundle over a surface, by means of a choice of a hermitian metric. Finally, we mentioned the Hitchin-Kobayashi correspondence.

21 Jan 2017. We talked about the relation between flat connections and trivializations with locally constant transition maps. This helped us to understand better the relation between surface group representations and flat connections. We ended by writing the defintion of a Higgs bundle.

14 Jan 2017. We explained what it means for a fibre bundle to have constant transition maps in geometric terms, and mentioned how to recover a representation from a principal G-bundle with a flat connection.

7 Jan 2017. We introduced connections on fibre bundles, both as distributions and as projection maps. We defined the curvature of a connection and looked at what it means to be flat.

31 Dec 2017. We associated a principal G-bundle to any surface group representation, gave trivializations and computed its transition maps, which turned out to be constant.

17 Dec 2017. We looked at the action of G on the space of representations and saw a concrete example of the quotient not being Hausdorff . We reviewed the vector space as a space of derivations, looked at the differential of a map between manifolds, recalled the exponential of a Lie group, introduced the vertical subbundle of a fibre bundle and started to talk about connections.

10 Dec 2017. We started with the study of the space of representations of the fundamental group of a connected compact orientable surface into a Lie group G. We described the space as a real algebraic set when G is an algebraic set, whereas we get a real analytic set when G is just a Lie group. We described the natural actions that the space of representations carries and we look at the example of G being the real additive group.

3 Dec 2017. We presented the topology of the fundamental group coming from the compact-open topology of the space of maps. We mentioned that the fundamental group is not in general a topological group, but a quasitopological group. However, for our purposes, the fundamental group of a manifold is countable and has the discrete topology. Finally we defined the universal cover of a manifold.

26 Nov 2017. We gave an overview of the classification theorem of compact connected topological surfaces. We then moved to general topological spaces and talked about homotopy and retractions, introduced the fundamental group, and stated the Seifert-Van Kampen theorem. From the fundamental group of the circle, we computed the fundamental group of any compact connected orientable surface.

19 Nov 2017. We looked at the G-torsor structure of the fibres of a principal G-bundle. We defined sections and saw what they mean in vector and principal bundles. We looked at the constructions of the frame bundle and the associated vector bundle. Finally, we discussed about the tangent bundle.

12 Nov 2017. We proved that a fibre bundle is equivalently given by a cocycle of transition maps. We defined vector bundles, diffeomorphism between vector bundles and saw that the cylinder is not diffeomorphic to the Mobius band, but is diffeomorphic to the 2-twisted cylinder.

5 Nov 2017. We looked at some more examples, defined smooth maps between manifolds, and introduced the notion of a bundle, paying special attention to the transition maps.