Geometric Representation Theory (Lecture 1)
Posted by John Baez
This fall, the socalled Quantum Gravity Seminar at U. C. Riverside will actually tackle geometric representation theory — the marvelous borderland where geometry, groupoid theory and logic merge into a single subject. And there are two other new things about this seminar.
First, it will be jointly run by John Baez and James Dolan. In addition to explaining wellknown stuff, we’ll report on research we’ve done with Todd Trimble over the last few years. Second, we plan to offer videos as well as written notes of the seminar. We’re still working the bugs out of the technology, so please bear with us.
As usual, the seminar will meet on Tuesdays and Thursdays, and you can ask questions and discuss things here at the $n$Category Café.
This week, I kicked off the proceedings with a gentle introduction to a few of the main themes.

Lecture 1 (Sept. 27) 
John Baez on some of the basic ideas of geometric representation theory. Classical
versus quantum; the category of sets and functions versus the
category of vector spaces and linear operators. Group representations
from group actions. Representations of the symmetric group $n!$
from types of structure on $n$element sets. Representations of
the general linear group $GL(n,F_q)$ from types of structure
on the $n$dimensional vector spaces over the field with $q$ elements, $F_q$. Uncombed Young diagrams $D$, and ‘$D$flags’ as structures either on $n$element sets or
$n$dimensional vector spaces. Irreducible representations of $n!$ versus representations coming from the actions of $n!$ on sets of $D$flags. Counting $D$flags:
$q$factorials and their limit as $q \to 1$. The ‘field with one element’.
Projective geometry.

Streaming
video in QuickTime format; the URL is
http://mainstream.ucr.edu/baez_9_27_stream.mov  Streaming video (alternate format)
 Downloadable video (556 megabytes)
 Lecture notes by Alex Hoffnung

Streaming
video in QuickTime format; the URL is
Videos
We’re offering the videos in streaming and/or downloadable form, both as .mov files. Downloading them takes a long time, but you may need to do this, since the streaming videos seem to work well only if you have a good internet connection.
.mov files can best be played using a free program called QuickTime. If you have QuickTime and your web browser has .mov files associated to this program, you should be able to click on the first “streaming video” link above and watch the video. An alternate method is to launch the QuickTime player on your computer, click on “File” and then “Open URL”, and type in the URL provided. This has the advantage that you can easily make the picture bigger.
If you can handle URL’s that begin with rtsp, you can instead go the corresponding URL of that form, e.g. rtsp://mainstream.ucr.edu/baez_9_27_stream.mov. This may also have advantages, but at present my computer gags on such URL’s.
If you encounter problems or — even better — know cool tricks to solve such problems, please let us know about them here!
Errata
If you catch mistakes, let me know and I’ll add them to the list of errata.
Re: Geometric Representation Theory (Lecture 1)
This looks great, but I wonder if there’s any chance of making the video files available for download rather than just streaming? My effective bandwidth seems to be a factor of 4 too small to receive the stream – and trying to watch a lecture that plays for 5 seconds then halts for 15 is pretty painful – but if I could download the files to play smoothly I wouldn’t care how long that took.