## July 18, 2005

### Streetfest Workshop I

#### Posted by Guest

Greetings from Canberra!

It is very pleasant here. There was a good frost this morning and some black swans with their chicks down by the lake. Most of the Streetfest participants moved down here yesterday in buses or cars or planes, mostly uneventfully, although Tim Porter found himself driving on a dirt road through the mountains after realising that the main highway south actually went to Melbourne. The workshop started this morning. We’re so busy that there’s almost no time to tell you what’s going on.

Kapranov was up first: the prequel to last week’s talk on NC Fourier transforms. At the end there was a little discussion with some people wondering exactly how this connects to Connes’ NCG.

Panov spoke about model cats, homotopy colimits and toric topology. Toric topology is the study of torus actions on manifolds or complexes with a rich combinatorial structure in the orbit quotient. First he defined ‘face rings’ which are Stanley-Reisner algebras of simplicial complexes. Ross Street later highly recommended Stanley’s revolutionising of combinatorics … something I must look into later. The Poincare series for $R\left[K\right]$ was defined, and Panov listed some problems that could be attacked with this machinery: the Charney-Davis conjecture, the question of when ${\mathrm{Ext}}_{k\left[K\right]}\left(k,k\right)$ has rational Poincare series, and something called the g-conjecture - whatever that is.

Then he defined the David-Januszkiewicz space $\mathrm{DJ}\left(K\right)$, which seems to be important because it turns up in a theorem (Panov, Ray, Vogt) giving a homotopy commutative diagram involving the loop functor $\Omega :\mathrm{Top}\to \mathrm{TMon}$ into the monoid category.

Marni

Posted at July 18, 2005 4:08 AM UTC

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### Re: Streetfest Workshop I

Hi Marni,

probably you do not have the time to check for, read and reply to my comments, but anyway. You wrote

At the end there was a little discussion with some people wondering exactly how this connects to Connes’ NCG.

Was there any consensus? What did Kapranov himself say?

Posted by: Urs on July 18, 2005 11:23 AM | Permalink | Reply to this

### Re: Streetfest Workshop I

Hi, Urs!

No, there wasn’t exactly any “consensus” regarding a relation between Kapranov’s work and noncommutative differential geometry - mainly because the only people who seemed to understand this suggestion by Getzler were Kapranov and presumably Getzler himself!

In particular, there was not any obvious relation between this comment and the sort of relation you like to envisage between noncommutative geometry and higher gauge theory. Getzler thought some of the homological algebra in Kapranov’s talk reminded him of Hochschild cohomology, and wanted Kapranov to push it to include more ideas from cyclic cohomology - but the connection was over my head.

The beautiful SIMPLE idea in Kapranov’s talk was a noncommutative analog of the Fourier transform sending Laurent series in n noncommuting variables to measures on the space of paths in n-dimensional space. And, beautifully, the noncommutative Taylor series for

$\mathrm{exp}\left(-{z}_{1}^{2}+\dots +{z}_{n}^{2}\right)$

turns out to give Wiener measure on paths! In other words, a “Gaussian function of n noncommuting variables” gives a new way of thinking about path integrals! It’s very cool and maybe I’ll explain it in This Week’s Finds someday.

Best, John

Posted by: John Baez on July 19, 2005 4:28 AM | Permalink | Reply to this

### Re: Streetfest Workshop I

Hi, just a quick comment:

measures on the space of paths in n-dimensional space

Thanks for that piece of information. Do you see any relation to our ideas on surface holonomy?

Can I find Kapranov’s ideas written down in detail anywhere?

Posted by: Urs (from Aberystwyth/Wales) on July 22, 2005 5:29 PM | Permalink | Reply to this

### Vacation

Maybe I should mention that I’ll be on vacation from tomorrow on up to August 3. So if you don’t hear anything from me the next days, that’s why.

Posted by: Urs on July 18, 2005 11:54 AM | Permalink | Reply to this

### Re: Streetfest Workshop I

Look I was searching the web and found my name in this theorem. I would like to know more if at all posible.

THANX!!!

Posted by: David Januszkiewicz on May 31, 2006 1:17 AM | Permalink | Reply to this

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