Discrete Morse theory for random complexes - IST Austria 2017

Thesis

Abstract Acknowledgments List of publications 1 Introduction 2 Results 3 Blaschke- Petkantschin formulas 4 Constants 5 Poisson-Delaunay, Poisson-Čech and weighted Poisson-Delaunay complexes 6 Poisson-Delaunay complexes of higher order 7 Random inscribed polytops 8 Future directions Bibliography Index

The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud.

In particular, we consider a Poisson point process in R^n

and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the Čech complex.

Further, we examine theDelaunay complex of a Poisson

point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes.

Each of the complexes in question can be endowed with a radius function,

which maps its cells to the radii of appropriately chosen circumspheres,

called the radius of the cell.

Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells.

For all considered complexes, we are able to generalize and obtain up to constants

the distribution of radii of typical intervals of all types.

In low dimensions the constants

can be computed explicitly, thus providing the explicit

expressions for the expected numbers of cells.

In particular, it allows to find the expected density of simplices

of every dimension for a Poisson point process in R^4,

whereas the result for R^3 was known already in 1970's.