Bernard Julia (ENS Paris):
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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.