# Douglas Lectures

## Spring 2019

Date: | April 2, 2019 |

Time: | 4:00pm |

Location: | Blocker 117 |

Speaker: | Vaughan Jones, Vanderbilt University |

Title: | An introduction to subfactors |

Abstract: | Von Neumann algebras are closed *-algebras of operators on Hilbert space. A factor is a von Neumann algebra whose center is just the scalar multiples of the identity. Murray and von Neumann discovered novel factors in the 1930’s. Among which are the II_1 factors whose Hilbert spaces admit a dimension function. Thus a subfactor N in M has an index [M:N] given by the dimension of M as a Hilbert space over N. A finer invariant is the principal graph of a subfactor, a graph whose norm is the square root of the index. I will list all subfactors of index less than 4 together with their indices and principal graphs. |

Date: | April 4, 2019 |

Time: | 4:00pm |

Location: | Blocker 117 |

Speaker: | Vaughan Jones, Vanderbilt University |

Title: | What we know and do not know about subfactors |

Abstract: | Haagerup discovered the "first" subfactor whose index is bigger than 4-the index is $\frac{5+\sqrt {13}}{2}$. Inspired by Haagerup a group of people tried to extend the classification of subfactors as far beyond index 4 as possible. The current record is 5.25 but the effort has stalled to a certain extent because of a proliferation of potential principal graphs. I will describe the classification and some of the methods used in it. In fact there are other subfactors which fall outside the scope of the classification program which remain totally mysterious. |

Date: | April 5, 2019 |

Time: | 4:00pm |

Location: | Blocker 220 |

Speaker: | Vaughan Jones, Vanderbilt University |

Title: | Subfactors, quantum field theory, diffeomorphism groups and Thompson groups |

Abstract: | Doplicher Haag and Roberts showed how subfactors arise in quantum field theory. Their examples were not terribly interesting from the point of view of subfactors but that situation has dramatically reversed in the last 30 years with the consideration of low dimensional quantum field theories, especially with conformal symmetry. Indeed interesting subfactors can be constructed from Diff$(S^1)$ which appears, as a consequence of conformal symmetry, in one dimensional QFT. An attempt to construct these QFT’s directly from subfactors has given an interesting family of unitary representations of R. Thompson’s groups F and T. |