By Daniel Iagolnitzer

ISBN-10: 1571460306

ISBN-13: 9781571460301

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 comprises many of the highlights, together with subject matters resembling conformable box thought and basic relativity.

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**Extra info for 11th International Congress of Mathematical Physics**

**Example text**

E. X compact) the operator D will have discrete spectrum, with (real) eigenvalues An, |An| —>oo, when n —>oo. 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.

Thus for the trefoil knot the Jones polynomial turns out to be: J{t) —— - f - + t . Taking mirror images replaces t by t~l and produces a quite different polynomial, demonstrating that the trefoil knot and its mirror image are indeed different knots. Like the Alexander polynomial there is a simple recurrence formula for the Jones polynomial of a link. With a similar notation to that in the previous section: t 1 J+ —tJ - = (t 1/ 2 —t 1^2)Jo • The Jones polynomial has been extensively studied and we now know: - ( 1 ) it is the first of a whole family of new polynomial invariants (related to Lie groups), - (2) it can be used to solve the Tait conjectures, - (3) it is not related to classical topological machinery (such as winding num bers), - (4) it is related to algebraic aspects of various parts of physics.

There is a great deal more behind the scenes, though much of it is too technical to explain without extensive formulae. In what follows I shall avoid all such technicalities. 2 The Classical Story In classical physics the fundamental connection with Geometry lies in the in terpretation of force as curvature. This takes many forms beginning with the Newton view that, in the absence of external forces, a particle will move in a straight line with uniform velocity. Any deviation from uniform motion is there fore due to a force and the amount of deviation (or curvature) measures the strength of the force.

### 11th International Congress of Mathematical Physics by Daniel Iagolnitzer

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