Kinematic formulas in Hermitian space forms and applications

厄密积分几何是关于厄密矢量空间、复射影空间和复双曲空间及其各自等距群的作用的积分几何的一部分。一个核心问题是确定这些空间中两个随机几何子集的交点的平均值。在复空间形式（厄米向量空间、复射影空间和复双曲空间）中，最近berni - fu和berni - fu - solanes利用S. Alesker引入的赋值空间上的代数结构和曲率测度空间上的新代数结构，找到了相应的运动学公式。计算结果表明，在这些空间中，运动学公式在形式上是相同的。该项目的第一个目标是更好地理解厄米积分几何。我们将首先研究二维四元数空间形式的积分几何，看看运动公式是否在形式上相同，类似于复数情况。在复杂空间形式上表述局部运动学公式的一种更深刻、更优雅的方式是将对偶空间上的代数结构（生成函数和关系）描述为具有一元不变曲率测度的空间，这是我们的第二个目标。第三个目的是描述平滑平移不变值上的Alesker-Fourier变换的乘数。

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.