Tuesday, July 26, 2005

Days Three and Four...

Well there's only so long I can manage without a night of nine hours of sleep, consequently my mind is now mush, and there are only a few constructive comments I can make about the last few talks.

Let's be chronoloigical and go over yesterday's talks first. Werner Nahm greeted us in the morning with a talk entitled "Mirror symmetry for cohomology with values in vector bundles?" in which he described a symmetry of vanishing cohomology classes on the Hodge diamond similar to that of mirror symmetry (a "symmetry" in the Hodge diamond that predicts the existence of a Calabi-Yau manifold for cohomology groups H^{p,q} and H^{q,p} if either one is known to exist...in most cases, for some detail see Diego Matessi's notes in postscript). His aim was to try and reduce the non-vanishing cohomology groups down to a (reflectively) symmetric set of points on the Hodge diamond. The cohomology class H^{p,q}=H^{p,q}(X,V) where X is a compact Kahler manifold and V is a holonomic vector bundle. In the construction presented by Nahm V was taken to be an ample vector bundle (which was defined by its projection being an ample line bundle). For more detail see "Vanishing theorems for products of exterior and symmetric powers" by Laytimi and Nahm.

Since I am so very tired let me mention one of the lighter moments of the talk. At one point Nahm took up the comedic baton from Terry Gannon (more later, in response to comments from Clifford Johnson) by saying that really some of the theorems he was going to discuss were best thought about after two guinnesses: they were two guiness problems. This was a passing comment and no more was thought about it. However, during the interval his former student Katrin Wendland nipped out and purchased a fourpack of guinnesses and Werner was encouraged by the audience to drink them before giving the second-half of his talk - he seemed quite keen to do this, but held back until after the end of the talk when he could be seen supping from one of the cans, just before lunchtime. As a passing comment I should say that there are many of Werner's former students (as well as their students) here; there are at least three generations of the Nahm PhD-advisor/family-tree and certainly at least six members of the clan are here. Which makes this a very nice family reunion, for them.

The first afternoon talk yesterday was given Alessio Corti and was entitled "Examples of orbifold quantum cohomology" - I'm afraid due to my own lack of knowledge I wasn't able to get a lot out of this talk, however Alessio did tell us about stacks, and "stacky fans" so this vocabulary is a start at least (I am being very optimistic here - I have no idea how they might be useful to me, and unfortunately as a consequence I cannot get very excited about them). However I do expect some insights into Alessio's talk can be gained by reading (and understanding) his paper with Abramovich and Vistoli entitled "Twisted bundles and admissible covers".

The second afternoon talk was by Miles Reid and entitled "Orbifold RR and plurigenera", where RR stands for the Riemann-Roch theorem.

To end yesterday we were "treated" to an extra, unpublicised talk on a recent paper by Calin-Iuliu Lazaroiu entitled "Topological D-branes and noncommutative geometry". Unfortunately, to cap a demoralising afternoon, this talk was also outside of my comfort zone. My one piece of terminology I picked up was the definition of a necklace in a quiver diagram, which is, as you may guess, a closed loop on a quiver diagram. I suspect if one was inclined to make a serious investigation of noncommutative geometry one could do worse than looking through the work of Lieven Le Bruyn, who was cited a couple of times during the talk, or even consider buying his self-published textbook via NeverEndingBooks. At the moment Le Bruyn himself recommends reading the Lectures on Noncommutative Geometry by Victor Ginzburg.

To complete my catch-up on what we've been listening to at Durham, I must mention this morning's very clear exposition on the topic "D-branes in Poisson sigma models" by Giovanni Felder. The talk was based on work completed with Alberto Cattaneo in the paper "Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model".

Finally, let me add my support to the notion expressed in previous comments that Tony Gannon is a comedy genius. It was pointed out that I failed to indicate this in my write-up of his talk earlier in the workshop, so permit me to correct this by relaying just one of the examples of his talent here. So, during his talk Gannon was explaining to us his ideas about category theorists, he was telling us (during an aside in the middle of his talk) that category theorists want to change the basis in which the logical structure of mathematics is framed. He said that they preferred not set theory but category theory, and he compared the two areas by saying set theorists like to describe maths using "nouns" while categorists prefer to use "verbs". He then made a leap, and told us his theory that category theorists are really like beavers. He said that the thing that makes beavers build dams is the sound of trickling water - they don't like it and they build a dam to make it stop. He said that if you took a tape recording of trickling water down to a riverbank where there were beavers and left the tape playing, that you could come back the next day andfind the tape recorder covered in pieces of wood. Somehow this idea reminded him of categorists, but having said all that he went on to praise the work and successes of category theory in a most appreciative way. Terry Gannon also owns a red t-shirt featuring a bear wearing green sunglasses, and he gets my thumbs-up for his very entertaining talking style.


cvj said...

Nice post Paul. I hope that you did not mind me reminding you of Terry's comedic turn. The entire talks were just so good on all three: maths, physics and comedy that it would have been a shame if they went unremarked.



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