M.Sc Student | Igor Khmelnitsky |
---|---|

Subject | D-collapsibility and its applications |

Department | Department of Mathematics |

Supervisor | Full Professor Meshulam Roy |

Full Thesis text |

A classical result of Helly asserts
that given a finite family {K_{1}, K_{2},?, K_{n}} of
convex sets in R^{d }such that an intersection of any sub family of
size<d is not empty, then the intersection of the entire family is not
empty. Helly’s theorem and its many extensions and generalizations form a
central area of study in discrete geometry and its applications. A key
ingredient that plays an important role in a variety of Helly type theorems is
the notion of d-collapsibility.

In this work we will define d-collapsibility as a d-dimensional extension of the notion of chordality in graphs. Following the review of some basic definitions and examples, we will present a number of results we obtained concerning the combinatorics of d-collapsible complexes. These include estimates on the collapsibility of intersection, union and joins of complexes. We conclude this work by relating d-collapsibility to an old result by Vorobév concerning consistent measures and their extensions.