Robin Graham: Parabolic Geometries: a tour via examples

Current Topics Seminar

Speaker	Robin Graham
Title	Parabolic Geometries: a tour via examples
Date	May 26 4:00pm PDL C-36

In the second half of the 1800's, two generalizations of classical Euclidean geometry were conceived. Klein gave a geometric interpretation of homogeneous spaces of Lie groups, and Riemann introduced Riemannian metrics. One point of view on much of Cartan's work in the early 1900's is that it combines Klein's and Riemann's perspectives in a new conception of geometry, in which geometric structures correspond to spaces modeled locally on homogeneous spaces. Much recent research has focused on the case of parabolic geometries, in which the model homogeneous space is of the form G/P, where G is a real semisimple Lie group and P is a parabolic subgroup. In this talk I will try to give something of the flavor of the subject through a discussion of some examples of parabolic geometries. In addition to the canonical example of conformal geometry, I will describe examples arising from the study of "geometry of differential equations", and a particularly interesting example which arises in many different settings in which G is the exceptional Lie group G₂. Note: this seminar will be held on Tuesday.