By Daniel Iagolnitzer

Over one thousand mathematicians participated within the Paris overseas convention on Mathematical Physics and its satellite tv for pc convention on topology, strings and integrable types. This quantity comprises a number of the highlights, together with themes similar to conformable box concept and normal relativity.

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Then B is the graph of a partial isometry S : H q —> H q (resp. S * : Ho’ —► and 1 —(SS* + S*S) is the orthogonal projection on the finite dimensional space of harmonic forms. Proof. e. harmonic forms, and the image of d + d*. Thus any u e H q , orthogonal to harmonic forms, can be written as u = da, a G L 2 (V, Aj^"1! 1*), using the formula d* = - * d* for the adjoint of d. Moreover the equality cj = -Mp da determines da uniquely since Il H 2 d a ll2 = Il á a ll2 G Aq "1 T*). f,F , 7 ) over C 00 (V').

34 (1869), W itten, E. Some geometrical applications of quantum field theory, Proc. Int. Congress of Mathematical Physics, Swansea, 1988. Quantized Calculus and Applications Alain Connes Institut des Hautes Etudes Scientifiques 35 Route de Chartres, F91440 Bures sur Yvette, France Our aim in this paper is to give a general introduction to noncommutative geometry and describe in some detail an example of the quantized calculus. Many of the tools of the differential calculus acquire their full power when formulated at the level of variational calculus where the original spaqe X one is dealing with, is replaced by a functional space F (X ) of functions or fields on X .

8 ) but, by [W02] it is independent of the choice of local coordinates and defines a trace, TV,, on the algebra of scalar pseudodifferential operators. When we consider a vector bundle E over a manifold E y and a pseudodifferential operator P acting on sections of E , we compute T r,(P ) as follows. Choose local coordinates xJ and local basis of sections a* for the bundle E. Then P appears as a matrix P/ of scalar pseudodifferential operators: P ( f k ak) = ( P * f k) a e. The expression TV,(P) = Tr^P^) is then independent of the choice of the local basis (ak) of E and defines a trace.