For more information about this meeting, contact Dmitri Burago, Anatole Katok.

Title: | Tits Geometry and Positive Curvature |

Seminar: | Center for Dynamics and Geometry Seminars |

Speaker: | Karsten Grove, University of Notre Dame |

Abstract: |

There is a well known link between (maximal) irreducible polar representations
and isotropy representations of irreducible symmetric spaces provided by Dadok. Moreover,
the theory by Tits and Burns - Spatzier provides a link between irreducible symmetric spaces
of non-compact type of rank at least three and compact topological spherical irreducible buildings
of rank at least three.
In joint work with Fang and Thorbergsson we discover and exploit a rich structure of a (connected)
chamber system of finite (Coxeter) type "M" associated with any polar action of cohomogeneity at
least two on any simply connected (closed) positively curved manifold. Although this chamber system
is typically not a (Tits) geometry of type "M", we prove in all cases but one that its universal (Tits) cover
indeed is a building. We construct a topology on this universal cover making it into a compact
topological building in the sense of Burns and Spatzier.
Our work shows that the exception indeed provides a new example (also discovered by Lytchak) of a
Tits "C_3" geometry whose universal cover is not a building.
We use this structure to prove the following
Rigidity Theorem: Any polar action of cohomogeneity at least two on a simply connected positively
curved manifold is smoothly equivalent to a polar action on a rank one symmetric space.
The analysis and methods used in the reducible case (including the case of fixed points), the case of
cohomogeneity two, and the general irreducible case in cohomogeneity at least three are quite different
from one another. Throughout the local approach to buildings by Tits plays a significant role. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 10 / 24 / 2012 |

Time: | 03:35pm - 05:30pm |