We will begin by discussing the relationship between divisors, line bundle and maps to projective space on a compact Riemann surface. This will motivate the main theorem of the talk, the Riemann-Roch theorem on a compact Riemann surface. Using a result of Chow, we show that this theorem implies that every compact Riemann surface comes from a projective algebraic curve.

In a 2002 paper, D.-C. Cisinski completely characterized the accessible model structures on a Grothendieck topos in which the cofibrations are the monomorphisms. All such model structures are Bousfield localizations of a "minimal model structure." I'll discuss some properties of these model structures and two extreme examples: model structures on presheaf topoi and the minimal model structure on the category of simplicial sets. This latter example sheds some light on the weak equivalences in Rezk's category of complete Segal spaces.

This is Shiing-Shen Chern's original work on Chern classes. In this talk, we will discuss three things: 1. Any complex vector bundle has a Hermitian structure. 2. Due to the Hermitian holomorphic structure, the formulas for the canonical connection and its curvature are very simple and easy to calculate. 3. More examples of Chern classes will be computed through Chern-Weil's theory.

The Maximum Principle in one complex variable implies that every holomorphic function on any compact complex manifold is constant. One can then ask the question: which non-compact complex manifolds have no non-constant holomorphic functions? In full generality this is difficult to answer. However, if one ask this of complex Lie groups, then one is considering Cousin groups (also called toroidal groups). It turns out that these groups play a central role in the structure theory of some non-compact homogeneous complex manifolds - a subtitle of the talk could be: "How I came to know and love Cousin groups".

In our talk we recall the notions of Lie groups, Lie algebras, and the exponential maps between them. We will show how to construct one particular Cousin group and prove all holomorphic functions on it are constant by using Liouville's theorem, the density of one set in another, and the Identity Principle. From this construction one sees what the structure of all Cousin groups must be and can then classify Abelian complex Lie groups (they are direct products of copies of $\mathbb C$ and $\mathbb C^*$ with Cousin groups). We also define and investigate properties of holomorphic reductions of complex Lie groups and complex homogeneous spaces (with examples, particularly in the nilpotent and solvable cases). In an analogous way one can get reductions relative to other analytic objects on our manifolds, e.g., bounded holomorphic functions and analytic hypersurfaces.

We will present some conditions for the existence of Kahler structures on (non-compact) manifolds that are homogeneous under the holomorphic actions of complex Lie groups, particularly when the groups are either solvable or reductive. This includes recent joint work with Karl Oeljeklaus and Christian Miebach of the Universite de Provence in Marseille, France.