Abstract: This talk focuses on Koszul duality of quadratic algebras. We will quickly review the basic definition and important properies of twisted morphism from a (differential graded) coalgebra to a (differential graded) algebra, equivalent characterization of Koszul morphisms, and the adjunction between cobar and bar constructions.Then we apply this to quadratic algebras, and some classical examples will be introduced

Abstract: In 1988, Delzant showed how to construct a toric manifold from a special type of polytope (now called a Delzant polytope). This result, combined with the Atiyah, Guillemin-Sternberg convexity theorem, establishes a 1-1 correspondence between symplectic toric manifolds and Delzant polytopes. A few years later, Lerman and Tolman did the same with toric orbifolds, thus establishing an analogous correspondence in this more general case, though the polytopes are now more general and come with weights attached to each facet. In this talk, we will focus on a very specific case of the Lerman-Tolman construction; that of constructing a weighted projective space starting with a weighted (non isosceles) triangle.