Algebraic Geometry Santa Cruz 1995, Part 2 by Kollar J., Lazarsfeld R., Morrison D. (eds.)

By Kollar J., Lazarsfeld R., Morrison D. (eds.)

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Singularities are also known to form, in some special cases, for solutions to the Einstein field equations in General Relativity. Moreover one expects this to happen, in general, in the presence of strong gravitational fields. It is also widely expected that the general solutions of the incompressible Euler equations in three space dimensions, modeling the behavior of inviscid fluids, break-down in finite time. Some speculate that the break-down may have something to do with the onset of turbulence for fluids with very high Reynolds numbers l .

The Minkowski space-time itself. which will be denoted by Mn+l, consists of a copy of Rn+l together with a metric m and a distinguished class of global coordinates XCi, Q = 0,1, ... , n, relative to which the line element of the metric takes the form, We split XCi into the time variable XO = t and the space variables x = (xl, ... ,xn). A relativistic field theory on M is specified by a Lagrangean L which depends on the metric g and also on a collection of fields H = (H 1 ... , Hm) corresponding to some matter- fields present in the space-time.

In this case the general consensus is that all smooth solutions remain smooth for all times. The problem is still open. E's today. It is intimately tied to the basic mathematical question of understanding what we actually mean by solutions and, from a physical point of view, to the issue of understanding the very limits of validity of the corresponding physical theories. Thus in the case of the Burger equation, for example, the problem of singularities can be tackled by extending our concept of solutions to accommodate "shock waves" .

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