##### Department of Mathematics,

University of California San Diego

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### Food for Thought Seminar

## Raul Gomez

## The Symmetry Group of Differential Equations

##### Abstract:

While he was studying partial differential equations, Sophus Lie came up with the idea of trying to solve them by using their symmetry group. His idea was to apply Galois Theory to differential equations instead of polynomials. Lie's key observation was that these symmetry groups are locally determined by their Lie algebras. Normally Lie groups of differential equations are only locally defined , i.e. they are only defined in a neighborhood of the identity element. However if we enlarge the manifold where the group is acting we can find a globally defined group action whose restriction to the original manifold is the original action. In this talk we will calculate the symmetry group of the line equation $y''=0$ and see that, despite the simplicity of this equation, the symmetry group is not globally defined! However, the action can be enlarged to a well defined action on $RP^2$. We will do the same with Maxwell's equations obtaining, in this way, a conformal model of the universe where the symmetry group of Maxwell's equations is well defined.

### November 30, 2006

### 12:00 PM

### AP&M 7321

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