L3

  I am going to discuss a special class of logarithmic solutions to WDVV equations. This type of solutions appeared in Seiberg-Witten theory is defined by a finite set of covectors, the V-systems. The V-systems introduced by Veselov have remarkable properties. They contain Coxeter root systems, and they are closed under taking subsystems and restrictions. The corresponding solutions are almost dual in Dubrovin's sense to the Frobenius manifolds structures on the orbit spaces of Coxeter groups and their restrictions to discriminants. Another source of V-systems is generalized root systems. The talk will be based on joint work with Veselov.    

    I sketch, using Kichenassamy - Rendall ideas, a simplified and generalized proof of the Fuchs theorem for differential equations with a singularity. I use the theorem to construct solutions of polarized and half polarized U(1) symmetric vacuum spacetimes with "Asymptotically Velocity Terms Dominated" (AVTD) behaviour near the singularity. I show that the definition I give of half polarization for U(1) symmetric vacuum space-times is a necessary and sufficient condition for non polarized such spacetimes to have this AVTD behaviour.