Mathematical Physics Seminar
Thursday 19.XI.2009
Time: 15:15-16:15
Room: 431a
Instytut Fizyki, ul. Reymonta 4, 30-059 Kraków
Pawel Nurowski
(Uniwersytet Warszawski, Poland)
Conformal geometry of differential equations
Abstract:
Given two differential equations, it is often useful to know invariants which guarantee that
there exists a transformation of variables (independent, dependent or both) that transforms
one of the equations into the other. Recently, it has been observed that various classes of
ODEs and PDEs, when considered modulo some specific kinds of transformations of the
variables, fall into nonequivalent classes of equations, whose local invariants are conformal
invariants of appropriately defined pseudo-riemannian metrics on manifolds. In this talk we
provide some examples of this phenomenon. The most striking of them associates a conformal
5-dimensional geometry of signature (2,3), with the equation
z'=F(x, y, y', y'', z). This conformal
geometry has Cartan normal conformal connection reduced from the so(3,4) Lie algebra to the
exceptional g2 Lie algebra. This implies in particular that Cartan's invariants of 2-dimensional
nonintegrable distributions in dimension five are just conformal invariants of this (2,3)-signature
conformal geometry.
(Please notice the exceptional time and venue this week.)
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