## January 15, 2008

### Geometric Representation Theory (Lecture 21)

#### Posted by John Baez

We’re back! In the fall quarter of the Geometric Representation Theory Seminar, James Dolan and I developed the basic idea of groupoidification. In the winter quarter we’ll apply it to examples, starting with three closely related ones:

• the $q$-deformed harmonic oscillator,
• the Hall algebra of a quiver,
• the Hecke algebra of a Dynkin diagram.

As before, we’ll report on research we’ve done with Todd Trimble. Also as before, you’ll be able to see videos and handwritten notes of the seminar, and discuss them here at the $n$-Category Café.

• Lecture 21 (Jan. 8) - John Baez on groupoidifying and $q$-deforming the quantum harmonic oscillator: overall battle plan. The quantum harmonic oscillator is all about the polynomial algebra $\mathbb{C}[z_1, \dots, z_n]$. If we groupoidify this polynomial algebra, we get the groupoid of $n$-tuples of finite sets, which is also the groupoid of finite sets equipped with $n$-stage flag. If we $q$-deform the polynomial algebra, we get a certain noncommutative algebra $\mathbb{C}_q[z_1, \dots, z_n]$. If both groupoidify and $q$-deform it, what do we get? A guess: the groupoid of finite-dimensional vector spaces with $n$-stage flag over the finite field with $q$ elements, $F_q$.

Review of the harmonic oscillator and how to quantize it. The harmonic oscillator hamiltonian. Annihilation and creation operators.

• Answers to homework by John Huerta: the ground state of the harmonic oscillator Hamiltonian; the commutation relations between annihilation operators, creation operators, and the harmonic oscillator hamiltonian.
• Answers to homework by Christopher Walker.

Posted at January 15, 2008 1:01 AM UTC

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### Re: Geometric Representation Theory (Lecture 21)

When I download the linked .mov file, my browser just retrieves a 57-byte file which points to the streaming video. Playing that in Quicktime then just tries to stream the video from the server, so I face the original bandwidth problems.

But I have to confess I’m only half-way through the last quarter’s lectures anyway!

Posted by: Greg Egan on January 15, 2008 10:47 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 21)

Hmm… I thought it was slowly downloading, but all I really got was that “57-byte file which points to the streaming video”. So, I’ll have to complain a bit.

But I have to confess I’m only half-way through the last quarter’s lectures anyway!

Yeah, it doesn’t seem like many people are actually watching the videos — it’s a big time commitment. So, I’ll probably quit making them when we’re done with this ‘geometric representation theory’ stuff, maybe at the end of this year or next.

It’ll be good to have something to point to, though, when people ask — as I’m sure they will someday — “who was James Dolan?”

Posted by: John Baez on January 16, 2008 1:37 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 21)

I’ll probably quit making them when we’re done with this ‘geometric representation theory’ stuff, maybe at the end of this year or next.

No!! That would be a terrible backwards step! I haven’t watched them… but I’m going to! The value of the videos isn’t related to the number of people who watch them immediately, but to the number who watch them ever.

Posted by: Jamie Vicary on January 16, 2008 1:51 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 21)

…but to the number who watch them ever.

Definitely!

Posted by: Bruce Bartlett on January 16, 2008 4:42 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 21)

Yes, don’t give up! I’ve watched three of them.

What you might do is encourage the note-takers to write more extensive notes, and perhaps even to jot down occasionally what time into the seminar something is said, so that people could tune into portions they find particularly interesting.

Posted by: David Corfield on January 16, 2008 4:10 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 21)

Yes, I’ve only downloaded the videos once (and not all of them), but I’ve watched bits of some of them several times. But more detailed notes would be good. That would reduce the need to make my own.

Posted by: Tim Silverman on January 16, 2008 8:38 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 21)

I agree that more detailed notes would be nice. But, it’s unlikely to happen.

When Derek Wise was around I was lucky, since his handwriting was great and he stuck very closely to what I wrote on the board… except when it needed a bit of clarification, which he then added. This encouraged me to write a lot of long, detailed sentences on the blackboard, knowing that a beautiful copy would find its way to the world at large.

Now Derek is gone — and what’s more, I’m coteaching this seminar with Jim Dolan. Jim writes very little on the blackboard, but he talks a lot. So, notes that only record what he writes are almost impossible to follow. This is what pushed me to start videotaping the seminars.

Given that the seminar was being videotaped, and half of it was being taught by someone who doesn’t write long, detailed sentences on the blackboard, I decided to adopt a more talky, less writerly lecture style myself.

So, it’s a bit difficult for anyone to take notes that fully capture what Jim and I are saying. Alex Hoffnung takes very legible, not very detailed notes that stick closely to what’s actually written on the board. Apoorva Khare takes less legible, more detailed notes that include more things that are said but not written. But, to really follow the seminar, it’s probably good to watch the videos.

Later, of course, most of this stuff will turn into a bunch of papers that’ll show up on the arXiv. So, people who don’t like watching videos can always wait for those.

Posted by: John Baez on January 29, 2008 4:38 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 21)

Downloadable videos are now available for this and other sessions of the winter seminar. See if they work for you!

Posted by: John Baez on January 29, 2008 4:40 AM | Permalink | Reply to this

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