New Trends in Geometry: Their Role in the Natural and Life by Claudio Bartocci, Luciano Boi, Corrado Sinigaglia

By Claudio Bartocci, Luciano Boi, Corrado Sinigaglia

This quantity makes a speciality of the interactions among arithmetic, physics, biology and neuroscience via exploring new geometrical and topological modelling in those fields. one of the highlights are the vital roles performed by means of multilevel and scale-change techniques in those disciplines. the mixing of arithmetic with physics, in addition to molecular and mobilephone biology and the neurosciences, will represent the hot frontier of twenty first century technology, the place breakthroughs usually tend to span throughout conventional disciplines.

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Points in each slice v can be thus parametrised by Minkowskian coordinates x0 , x1 , x2 , x3 (rescaled by ev on v ). This explains why the anti-de Sitter coordinates (v, x0 , x1 , x2 , x3 ) are also called Poincaré coordinates. The coordinate system covers only one-half of the anti-de Sitter manifold; the anti-de Sitter metric takes the following form: ds2 = (dX0 )2 − (dX1 )2 − (dX2 )2 − (dX3 )2 − (dX4 )2 + (dX5 )2 =e 2v (dx02 − dx12 − dx22 − dx32 ) − dv2 . AdS5 (10) The slices v are often called branes.

We can think of as evolving in a given spacetime M. From a mathematical point of view this is strictly related to the theory of minimal (or maximal) surfaces in M, (harmonic map theory). Also for strings it is worth recalling some basic features of the characters of their classical motion in a given spacetime geometry. In the simplest situation, the motion of a string is described by a parameterized surface which maps an oriented cylinder in a four-dimensional spacetime. , the basic shape of the motion of a closed string is itself an abstract surface.

The de Sitter spacetime has a boundary at timelike infinity (while timelike infinity of the Minkowski manifold is a point). The cone C also provides a description of this boundary, which may be used instead of a Penrose diagram. The de Sitter kinematical group coincides with the Lorentz group of the ambient spacetime SO(1, 4). As for the sphere, there are no commutative translations on the de Sitter manifold. This fact is a source of considerable technical difficulties in the study of de Sitter quantum field theory.

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