By Daniel Iagolnitzer
Over a thousand mathematicians participated within the Paris foreign convention on Mathematical Physics and its satellite tv for pc convention on topology, strings and integrable types. This quantity includes a few of the highlights, together with subject matters corresponding to conformable box conception and normal relativity.
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With E replaced by a 4-dimensional conformal manifold. The action we shall get will be conformally invariant by construction and intimately related to the Einstein action of gravity. In general, given an even dimensional conformal manifold T, dim E = n = 2m, we let H = L2 ^E , Ajg T*^ be the Hilbert space of square integrable forms of middle dimension, in which functions on E act as multiplication operators. We let F — 2P - 1 be the operator in H obtained from the orthogonal projection P on the image of d.
The first example of such a triple is provided by the Dirac operator on a compact Riemannian (Spin) manifold. In that case H is the Hilbert space of L2 spinors on the manifold M, A is the algebra of (smooth) functions acting in H by multiplication operators and D is the (selfadjoint) Dirac operator. One can easily check in this case that no information has been lost in trading the geometric space M for the spectral triple (A,H,D ) indeed (cf. [Co]) one recovers: 1 ) The space M as the spectrum Spec(*4), of the norm closure of the algebra A of operators in H.
This, however, describes the situation of some 50 years ago. In recent times, in the past 20 years or so, there have been new and unexpected developments linking Physics and Geometry in a quite different way. These developments are extensive and cover a wide range, some of which will be described in detail at this Congress. In picking on the subject of Knots I have isolated one aspect of this new interaction. In some ways it is typical and it is also the easiest to describe because knots are familiar objects whose intricacy is intuitively easy to understand.
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