Differential Geometry Seminar: Karsten Grove (Notre Dame), 'Positive curvature, symmetry, and Tits geometry'

Event Date: 

Friday, March 8, 2013 -
3:00pm to 4:00pm

Event Location: 

  • 6635 South Hall

Event Contact: 

Lee Kennard

Email: kennard@math.ucsb.edu

Abstract:

During the past two decades investigations of positively curved manifolds in the presence of (large) isometry groups have flourished. In particular this has lead to a number of classification type results as well as to the discovery that the unit sphere bundle of the 4-sphere with an exotic structure has a metric with positive curvature (Dearricott and joint work with Verdiani and Ziller).

The new example was discovered via classification work on positively curved manifolds of cohomogeneity one, i.e., with one dimensional orbit space, or equivalently with orbits of codimension one (joint work with Wilking and Ziller).

Cohomogeneity one actions are examples of so-called polar actions. It turns out that polar actions of cohomogeneity at least two on positively curved manifolds are intimately related to the theory of buildings by Tits. This in part has lead to a complete classification (joint work with Fang and Thorbergsson).

In this talk, I will provide a general background for the investigations alluded to above, and emphasize the link to Tits geometry. All necessary concepts will be discussed.