For more information about this meeting, contact Mary Anne Raymond.

Title: | What is an L^2 Betti number? |

Seminar: | Slow Pitch Seminar |

Speaker: | John Roe |

Abstract: |

We all know the idea of the “dimension” of a vector space: 0,1,2,3,… or maybe infinite. Many interesting topological invariants can be defined as the dimensions of appropriate vector spaces. Among these are the “Betti numbers”, which are the dimensions of homology spaces, or intuitively the “number of holes” of various kinds in a topological space.
In the 1930s John von Neumann envisaged the possibility of a “continuous” geometry where the dimension of any object could be any positive real number. Examples of such continuous geometries can be realized in terms of operators on Hilbert space. When the Hilbert spaces involved are defined geometrically, one ends up with real-valued topological invariants.
These “L2 Betti numbers” were invented by Atiyah and Singer in the early 1970s. Their study involves a mixture of topology and analysis, and they have recently played a part in solving problems related to combinatorial group theory. I’ll try to explain how this story goes, without getting involved in too many technicalities. All are welcome. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 09 / 18 / 2007 |

Time: | 05:00pm - 06:00pm |