Title: Integral geometry of the complex projective space
Speaker: Andreas Bernig (Professor, Goethe Universität Frankfurt)
Abstract: Two complex submanifolds of the complex projective space of complementary dimension and in general position will intersect in a constant number of points which is given by Bezout’s theorem. If we take two real submanifolds of complementary dimension, the number of intersection points will no longer be constant and one may ask about the average number of intersection points. More generally, given two geometric objects A,B in complex projective space (compact submanifolds with boundaries or corner; sets of positive reach or subanalytic sets) and some isometry invariant functional (for instance Euler characteristic or volume), one may ask about the expected value of this functional applied to A intersected with gB, where g is an element of the isometry group. The solution to this old problem was recently obtained, in collaboration with Joseph Fu (Univ. of Georgia) and Gil Solanes (UA Barcelona) in the form of a kinematic formula in complex projective space.