Special Seminar in Computing and Mathematical Sciences

Connectivity and percolation (the formation of "giant" components) are two well-studied phenomena in random graphs. In recent years, there has been an ongoing effort to generalize these phenomena to higher dimensions using random simplicial complexes. Simplicial complexes are a natural generalization of graphs that consist of vertices, edges, triangles, tetrahedra, and higher dimensional simplexes. The generalized notions of connectivity and percolation are based on the language of homology - an algebraic-topological structure representing cycles of varying dimensions. 

In this talk we will mainly focus on random geometric complexes. Such complexes are generated from vertices given by a random point process, with simplexes added according to their geometric configuration. We will discuss recent results analyzing phase transitions (i.e. rapid changes) related to these topological phenomena. We will also discuss the relevance of these results in the field of Topological Data Analysis (TDA).