Abstract: 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.

Abstract: 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.