Differential Geometry of Submanifolds of Projective Space: Rough Draft

• Introduction to the local differential geometry of submanifolds of projective space • Introduction to moving frames for projective geometry • How much must a submanifold X ⊂ PN resemble a given submanifold Z ⊂ PM infinitesimally before we can conclude X ≃ Z? • To what order must a line field on a submanifold X ⊂ PN have contact with X before we can conclude the lines are contained in X? • Applications to algebraic geometry • A new variant of the Hwang-Yamaguchi rigidity theorem • An exposition of the Hwang-Yamaguchi rigidity theorem in the language of moving frames. • Open questions and problems. Representation theory and algebraic geometry are natural tools for studying submanifolds of projective space. Recently there has also been progress the other way, using projective differential geometry to prove results in algebraic geometry and representation theory. These talks will focus on the basics of submanifolds of projective space, and give a few applications to algebraic geometry. For further applications to algebraic geometry the reader is invited to consult chapter 3 of [11] and the references therein. Due to constraints of time and space, applications to representation theory will not be given here, but the interested reader can consult [23] for an overview. Entertaining applications include new proofs of the classification of compact Hermitian symmetric spaces, and of complex simple Lie algebras, based on the geometry of rational homogeneous varieties (instead of root systems), see [22]. The applications are not limited to classical representation theory. There are applications to Deligne’s conjectured categorical generalization of the exceptional series [25], to Vogel’s proposed Universal Lie algebra [27], and to the study of the intermediate Lie algebra e7 1 2 [26].