Abstract: In this talk, I will highlight some differences between the moduli space of Higgs bundles (in the sense of Hitchin) on a curve of positive genus and the the moduli space of "twisted" Higgs bundles at genus 0. The Betti numbers of both spaces can be determined by a localization calculation, with respect to an $S^1$ action. This was exactly Hitchin's method for obtaining the Betti numbers of the rank-2 instance of the usual Higgs moduli space. The $S^1$ fixed points are what are called "holomorphic chains": these are similar to complexes of vector bundles, but the differential (the Higgs field itself) is nilpotent with order equal to the length of the complex. I will show how the localization calculation can be made very combinatorial in the genus 0 case. The appropriate language for organizing this data is that of quivers, which we use to represent (and construct) families of chains.

Abstract: In this introductory talk I will start with basic definitions and examples. After defining natural transformations I will prove the Yoneda lemma which is a fundamental result in category theory. It will indeed be a good opportunity for the beginners to befriend the theory and refresher for the others.