Kinematic formulas in Hermitian space forms and applications

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Hermitian Integral Geometry is the part of integral geometry that concerns Hermitian vector spaces, complex projective and complex hyperbolic spaces, with the actions of their respective isometry groups. A central question is to determine the average number of intersection points of two random geometric subsets in these spaces. In the complex space forms (hermitian vector spaces, complex projective and complex hyperbolic spaces), the corresponding kinematic formulas have been found recently by Bernig-Fu and Bernig-Fu-Solanes, using the algebraic structure on the space of valuations introduced by S. Alesker as well as new algebraic structures on the space of curvature measures. As a result of the computations, the kinematic formulas turn out to be formally identical in these spaces. The first aim of the project is a better understanding of hermitian integral geometry. We will first study the integral geometry of the two-dimensional quaternionic space forms to see whether the kinematic formulas are formally identical, analogous to the complex case. A deeper and more elegant way of stating local kinematic formulas on complex space forms would be by describing the algebra structure (generators and relations) on the dual space to the space of unitarily invariant curvature measures, which is our second aim. The third aim is to describe the multipliers of the Alesker-Fourier transform on smooth translation invariant valuations.