Aspects of Algebraic Geometry
May 11 - 22, 2015
Elizabeth Milićević, Haverford College - Grassmannians and Flag Varieties
Abstract: The Grassmannian is the set of all subspaces of a vector space which are the same size, and flags more generally are chains of subspaces. While these ideas clearly involve linear algebra, flags also turn out to be projective varieties which enjoy rich geometric, topological, and combinatorial structure. This course will focus on the topology of flag varieties, as well as important subvarieties which encode classical problems in enumerative geometry.
Wei Ho, University of Michigan - Algebraic Curves over Finite Fields
Abstract: In this course, we will study the arithmetic and geometry of algebraic curves, especially over finite fields. We will explore the remarkable properties of the zeta function for a curve over a finite field, which gives a close link between the arithmetic of the curve (such as the number of rational points) and its geometry (such as the genus).
Lucia Caporaso, Roma Tre University - Moduli Space of Curves
Abstract: As a set, the moduli space of curves is the set of isomorphism classes of (nonsingular, connected, projective) algebraic curves. It is endowed with a natural structure of algebraic variety which reflects many interesting properties of the curves themselves. The moduli space of curves and its compactifications play an important role in several areas of algebraic geometry, arithmetic, and other areas of mathematics. I will discuss their basic properties, sketch their construction, and explain some recent research directions involving them.
Claire Voisin, Ecole Polytechnique - Birational Invariants
Abstract: Rational and birational maps are very important in algebraic geometry, as there are very few true morphisms. I will discuss very classical questions around the characterization of varieties birational to projective space, and describe the many different methods and ideas which have been used classically or recently to study these questions.