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### On the projective geometry of homogeneous varieties

**Colloquium**

Speaker: | Joseph Landsberg, University of North Carolina |

Location: | 693 Kerr |

Start time: | Tue, Nov 3 1998, 2:10PM |

In this talk I will discuss joint work with L. Manivel, where we explore new relations between the representation theory of complex simple Lie groups and the algebraic and differential geometry of their homogeneous varieties. The talk will have two parts: Geometry via Lie theory and Lie theory via geometry.

First, I will describe the homogeneous varieties as subvarieties of projective spaces, and answer basic questions about them with the aid of Lie theory. In the execeptional cases I'll interpret the answers in terms of octonionic geometry. This work complements and advances that of Freudenthal and Tits, who study the exceptional homogeneous spaces in an abstract/axiomatic setting. In the second part, I will explain how to construct all Hermitian symmetric varieties and adjoint varieties by simple geometric algorithms begining with $Bbb CBbb P^1$. In particular, these algorithms provide the construction of all complex simple Lie algebras without reference to root systems.