Bernard Julia (ENS Paris):

E families of exceptional groups: from Painlevé analysis to invariant theory and supergravity models

The Painlevé classification of ordinary differential equations whose only movable singularities are poles is still incomplete. A new perspective is obtained by considering “difference Painlevé” equations. For order two equations, whereas in the continuous case the Cvitanovic-Deligne-Gross exceptional symmetries that are simply laced seemed to be prominent, the multiplicative discrete list admits Okamoto-Sakai spaces of initial conditions corresponding precisely to the exceptional E-family of Manin-Cremmer-Julia (root lattices in Del Pezzo middle cohomologies resp. U-dualities of supergravities). The q-Painlevé Baecklund symmetries are the (abelian) translation subgroups of the affine linear groups corresponding to spacetime linear unimodular changes of (transverse) coordinates of maximal “toric” supergravities. The dictionaries are being developed, the main objects and a natural program will be presented. Specific aspects include the relation between “nonlinear confluence” and “inverse dimensional reduction” and the algebraic consequences of discrete multiplicativity.
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