Math 9101 - Seminar in Analysis: Operator Algebras and Acylindrically Hyperbolic Groups - Vanderbilt University - Spring 2021

Syllabus |
Lecture notes (updated March 24th, 2021) |
Jesse Peterson's homepage |

All lectures will be held MWF 3:00-3:50pm over Zoom. Meeting ID: 941 4756 2219. Email the instructor for the password.

[AP] Anantharaman, Popa: An introduction to II_1 factors, preprint.

[P] Paulsen: Completely bounded maps and operator algebras, (2002), Cambridge University Press, Cambridge, UK.

[Os] Osin: Acylindrically hyperbolic groups, Transactions of the American Mathematical Society (2016), 368 (2): 851-888.

[Ba] Balasubramanya: Acylindrical group actions on quasi-trees, Algebr. Geom. Topol. Volume 17, Number 4 (2017), 2145-2176.

[BBF] Bestvina, Bromberg and Fujiwara: Bounded cohomology with coefficients in uniformly convex Banach spaces, Commentarii Mathematici Helvetici 91(2) (2016, 203-218.

[Oz] Ozawa: Weak amenability of hyperbolic groups, Groups Geom. Dyn. 2 (2008), 271-280.

[ER] Effros, Ruan: Multivariable multipliers for groups and their operator algebras, Proceedings of Symposia in Pure Mathematics, Volume 51 (1990), part 1, 197-218.

Date | Material covered | Reference | Video |

January 25 | Introduction/Invariant measure and Vitali sets | Lecture 1 | |

January 27 | Banach Tarski Paradox/Amenable groups, paradoxical decomposition | 2.6 in [BO] | Lecture 2 |

January 29 | Amenable groups II/Folner sequences, almost invariant vectors. | 2.6 in [BO] | Lecture 3 |

February 1 | Operator algebras associated to amenable groups, Relative property (T) | 12.1 in [BO] | Lecture 4 |

February 3 | Relative property (T) for (Z^2, SL(2, Z) \ltimes Z^2) | 12.1 in [BO] | Lecture 5 |

February 5 | Positive definite functions/cocycles and Schoenberg's theorem | Appendix D in [BO] | Lecture 6 |

February 8 | Characterizations of relative property (T) | 12.1 in [BO] | Lecture 7 |

February 10 | Property (T) and reduced cohomology | 12.1 in [BO] | Lecture 8 |

February 12 | Property (T) for SL(3, Z) | 12.1 in [BO] | Lecture 9 |

February 15 | Haagerup's property for free groups | 12.2 in [BO] | Lecture 10 |

February 17 | Haagerup's property and wreath products | 12.2 in [BO] | Lecture 11 |

February 19 | Finite von Neumann algebras | Review of selected sections in [AP] | Lecture 12 |

February 22 | Completely positive maps/Correspondences | 13.1 in [AP] | Lecture 13 |

February 24 | Applications to II_1 factors | 14.4 in [AP] | Lecture 14 |

February 26 | Property (Gamma) and inner-amenability | 15.4 in [AP] | |

February 26 | Hyperbolic graphs/groups | 5.3 in [BO] | |

February 26 | Hyperbolic graphs/groups II | 5.3 in [BO] | |

February 26 | Amenable actions | 4.3 in [BO] | |

February 26 | Exact groups | 5.1 in [BO] | |

March 1 | Biexact groups | 15.2 in [BO] | |

March 3 | Wreath products and Haagerup's property | 12.2 in [BO] | |

March 5 | Wreath products and biexactness | 15.3 in [BO] | |

March 8 | Acylindrically hyperbolic groups | [Os] | |

March 10 | Acylindrical actions on Quasi-trees | [Ba] | |

March 12 | Quasi-cocycles from quasi-trees | [BFF] | |

March 15 | Herz-Schur multipliers | Appendix D in [BO] | |

March 17 | Herz-Schur multipliers II | Appendix D in [BO] | |

March 19 | Weak amenability | 12.3 in [BO] | |

March 22 | Weak amenability for free groups | 12.3 in [BO] | |

March 24 | Weak amenability for hyperbolic groups | [Oz] | |

March 26 | Non-weak amenability for wreath products | 12.3 in [BO] | |

March 29 | Operator systems and completely positive maps | Chapter 3 in [P] | |

March 31 | Arveson's Extension Theorem | Chapter 7 in [P] | |

April 5 | Operator spaces and completely bounded maps | Chapter 8 in [P] | |

April 7 | Wittstock's Extension Theorem | Chapter 8 in [P] | |

April 9 | The Haagerup tensor product | Chapter 17 in [P] | |

April 12 | Completely bounded multilinear maps | Chapter 17 in [P] | |

April 14 | Multivariable Herz-Schur multipliers | [ER] | |

April 16 | Multivariable Herz-Schur multipliers II | [ER] | |

April 19 | Applications to von Neumann algebras I | ||

April 21 | Applications to von Neumann algebras II | ||

April 23 | |||

April 26 | |||

April 28 | |||

April 28 |