Toplogies of the zero sets of random real projective hyper-surfaces and of monochromatic waves.

Nazarov and Sodin have shown that the zero set of a random real homogeneous polynomial in n+1 variables and of large degree has many components and the same is true for the random harmonic such polynomial ("mono-chromatic waves") .We show that for each of these models the topologies of the components of the zero set are equidistributed w.r.t universal probability measures on the space of topologies, and that the same is true for the nesting configurations of the components w.r.t universal probability measures on the space of rooted trees.The support of these probability measures is determined . Numerical simulations by Barnett and Jin indicate a dramatic transition from n=2 to n=3 as far as there being a dominant percolating component and we speculate as to its impact on our universal measures.

Joint works with I.Wigman and with Y.Canzani.