Representation Theory & Noncommutative Geometry

These short courses of short lectures will lie somewhere between the This is what I do talks
and the language schools.
They will aim to be informal and understandable introductions to subjects of likely interest to many members of the RTNCG community.
There will be few prerequisites, and questions and digressions will be welcomed. Meetings will generally be held during Sococo tea times.

January-February 2023

Let $G/H$ be a reductive symmetric space. These short (30-minute) lectures will be about the problem of constructing a C*-algebra of operators on $L^2(G/H)$ that reflects the decomposition of the regular representation of $G$ on $G/H$ into irreducible representations. Our attempts to solve the problem have been partially successful, as we shall describe, but they have also generated a rapidly expanding list of questions in representation theory that appear to us to be both interesting and difficult. There are also many questions in C*-algebra theory, and in particular in K-theory, that remain unanswered. Joint work with Alexandre Afgoustidis and Peter Hochs.

**Next talk:** February 6 at
11:00 EDT

1/30 |
Two C*-algebras for the regular representation, by N. Higson | Recording | Notes | |

2/06 |
C*-module approach to the tempered dual, by S. Nishikawa | |||

2/13 |
Symmetric spaces and deformation spaces, by N. Higson | |||

2/20 |
Some rank-one examples, by S. Nishikawa |

November-December 2022

The representation theory of locally compact groups and in particular reductive Lie groups developed in great ways in this period. We shall present one view (very personal - this period no doubt invites many such), where some ideas of a lasting value will be presented - not always in chronological order - also with connections to interpretations in mathematical physics. One omission will be the œuvre of Harish-Chandra, maybe for experts to present later. Same remark for the work of David Vogan, the theory of lowest $K$-types and cohomological induction.

Summaries of the lectures.

11/14 |
George Mackey and Irving Segal | Recording | Slides | |

11/21 |
Anthony W. Knapp and Elias M. Stein | Recording | Slides | |

12/05 |
Bertram Kostant | Recording | Slides | |

12/12 |
Edward Nelson and Leonard Gross | Recording | Slides |

September-October 2022

When operator algebras and the representation theory of locally compact groups both took off around 1950, they were very much two faces of the same subject — they have common origins in the rise of spectral theory in the wake of Hilbert, the mathematical analysis of quantum physics in the wake of Von Neumann, and other topics. I will discuss some of the main episodes of their common early history, beginning with the Peter-Weyl theorem around 1926, and stopping before the appearance of the Plancherel formula of Segal and Mautner. No special historical knowledge will be required.

For additional information and further reading, see this companion page.

Compact Lie groups in the 1920s | Recording | Slides | |

Abelian locally compact groups in the 1930s | Recording | Slides | |

Beginnings of operator algebras | Recording | Slides | |

General theory of locally compact groups and other topics | Recording | Slides |

April-May 2022

I plan to introduce some topics on proper actions with emphasis on their relation to representation theory. No special background knowledge will be required. The lectures will be short without details, but with elementary examples so that they fit into teatime. The lectures are loosely related but are mostly independent of one another. The first two topics are of more geometric nature, and the last two are of more analytic nature.

Discontinuous dual and properness criterion | Recording | Slides | |

The Mackey analogy and proper actions | Recording | Slides | |

Tempered subgroups à la Margulis | Recording | Slides | |

Tempered homogeneous spaces | Recording | Slides |

February-March 2022

My goal will be to introduce some topics in index theory and K-theory that might be of interest in representation theory.
No background knowledge will be required from index theory or K-theory.

Some overall principles: The lectures will be short. They will be independent of one another, mostly. They will be aimed at representation theorists.
They will avoid details. Finally, I will lie wherever necessary in order to tell the truth.