We had a field day today (Tuesday October 15). Walter van Suijlekom got us rapidly into business with a full-fledged discussion of the “Inner perturbations (it used to be *fluctuations*) of non-commutative geometry”. He faced squarely the subtleties in constructing gauge theories with the non-commutative tools. We learned (or re-learned) a couple of things, in particular about the semigroup structure of the gauge potentials, and the order-1 condition, which is most honored in the breaching. The discussion was lively, and the comments by Alain Connes on the above, very illuminating. The comment on the dressed propagator, a bit mystifying as usual. After that Igor Cherednikov gave a profound talk on the structure of nucleons from the viewpoint of Wilson loop theory. It was quite hard-going, but one could see at work a strong machinery there, toiling against formidable obstacles. It is fair to say that NCG has had nothing to say about QCD in the non-perturbative realm, so far. And then the moment for the study groups to take off had come. Two of the four proposals were deserted or nearly-deserted, so most everybody coalesced around the group headed by Fedele Lizzi (higgs vacuum stability) or the group headed by John Barrett (Lorentzian NCG). From the first, well attended by phenomenologists, it is apparent that the question of the meta-stability of the higgs potential (which is almost a relief to particle physicists, since after all the mass region for meta-stability is narrow) has preoccupied a lot the model-builders of NCG. As Fedele said, discussion of that entailed going into all the nitty-gritty of the spectral action of Connes and Chamseddine. From the second, I retain that the subject still keeps jealous watch on its approaches. The plenary discussion was enlivened by consideration of relativistic equations with characteristic initial data, surely more in the spirit of spectral geometry. And so the play continues.

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