By Daniel Iagolnitzer

Over one thousand mathematicians participated within the Paris foreign convention on Mathematical Physics and its satellite tv for pc convention on topology, strings and integrable versions. This quantity includes the various highlights, together with issues corresponding to conformable box idea and common relativity.

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In particular the odd elements fx G M, p = p* with ||ja|| < 1 correspond exactly to a single Beltrami differential v(zyz)dz/dz with ||v ||o o < 1, and v measurable, by the equality: 0 v* /z = v 0 Now by proposition 4 c) all the relevant perturbations of a Fredholm module by the action of /xev(M) are obtained using the elements g(fi)yg, odd, of proposition 4 c). ) One checks by a direct calculation that for any g(g) G gev(M) the perturbed Fredholm module (Hyg(g)(F)) over A = C(P\(C)) is canonically isomorphic to the Fredholm module over A associated to the perturbed conformal structure on Pi(C) associated to the measurable Beltrami differential v(zyz)dz/dz.

The length, thickness and precise shape are of no interest. What is significant is the extent tp which the string is really knotted and whether it can be to some extent untangled. For example given two superficially different knots, can I twist one around so that it eventually looks like the other? In particular can I make it look like a circle (which is called the “unknot”)? In practical terms if we are presented with a tangle of string (without loose ends) it may not be clear how many pieces there are.

The cyclic cohomology construction then gives back the usual differential forms, the de Rham complex of currents etc— But it also applies to the highly non com mutative examples of group rings, in which case the group cocycles give rise to cyclic cocycles, with direct application to the Novikov conjecture for the homotopy invariance of the higher signatures of non simply connected manifolds with given fundamental group (cf. [Co]). If one wants to go beyond differential topology and reach the geometric struc ture itself, including the metric and the real analytic aspects, it turns out that the most fruitful point of view is that of spectral geometry.